Last Update

OPML feed of all feeds.

Subscribe to the Atom feed, RSS feed to stay up to date.

Thank you to arXiv for use of its open access interoperability.

Note: the date of arXiv entries announced right after publication holidays might incorrectly show up as the date of the publication holiday itself. This is due to our ad hoc method of inferring announcement dates, which are not returned by the arXiv API.

Powered by Pluto.

Source on GitHub.

Maintained by Nima Anari, Arnab Bhattacharyya, Gautam Kamath.

Theory of Computing Report

Thursday, November 05

TR26-305 | Nonnegative Rank Bounds for the Algebraic Monotone Calculus | Fedor Part

from ECCC Papers
In Hrubeš' Algebraic Monotone Calculus AMC a DNF tautology $A$ is proved by a monotone arithmetic circuit computing $\chi(A)-\epsilon\cdot\chi(1)$ for some $\epsilon>0$, where $\chi(A)$ is the set-multilinear polynomial whose coefficient at $\mathbf x^{\sigma}$ counts the terms of $A$ satisfied by $\sigma$, and $\chi(1)=\prod_{i\le n}(x_i^0+x_i^1)$. Proof size is circuit size, and $\epsilon$ is existentially quantified, so a lower bound must hold at arbitrarily small $\epsilon$. We prove such bounds for bounded-depth fragments by two nonnegative rank measures of $\chi(A)-\epsilon\cdot\chi(1)$: its monotone tensor rank, and the nonnegative rank of a flattening of its coefficient tensor. At depth three, a proof decomposes the coefficient tensor into nonnegative rank-one tensors. Restricting the summands to nonnegative multiples of subcube indicator tensors yields conical juntas and a system polynomially equivalent to Sherali–Adams when size counts conjunctions over the twin variables $x_i,\bar x_i$. For arbitrary nonnegative rank-one summands, we adapt Rothvoß' hyperplane separation method to monotone tensor rank and prove that Tseitin formulas on constant-degree expanders of sufficiently large girth require proofs of size $2^{\Omega(\epsilon n)}$ for $\epsilon\in(0,1]$ with $\epsilon=\omega(\log n/n)$. This cannot reach every $\epsilon>0$: at shift $2^{-\Theta(n\log n)}$ Tseitin acquires polynomial-size depth-three proofs, which separates the general depth-three fragment from the subcube one exponentially. We compose Tseitin formulas with a constant-size index gadget to obtain lower bounds for every $\epsilon\ge2^{-\Omega(n)}$, converting a Sherali–Adams *degree* bound into a depth-three *size* bound. Depth three is essential to all of this: we also show that Tseitin has monotone proofs of size $O(n^3)$ and depth $O(\log n)$ at *every* $\epsilon\in(0,1]$. Beyond depth three, we bound the nonnegative rank of a flattening by $L^{2^t}$ for a monotone formula with $L$ leaves and *mixed-product depth* $t$. Here $t$ is the maximum number, along a root-to-leaf path, of product gates whose two inputs each involve both sides of the bipartition; the Tseitin proofs just mentioned have $t=0$ for every bipartition, so this bound is empty for Tseitin. We give two constructions turning hard matrices into hard DNFs. Approximate extension complexity yields bounds for perfect matching and odd factor instances. Linear programming integrality gaps for planted constraint satisfaction yield an explicit family of planted Tseitin DNFs hard at inverse-polynomial shift.
In Hrubeš' Algebraic Monotone Calculus AMC a DNF tautology $A$ is proved by a monotone arithmetic circuit computing $\chi(A)-\epsilon\cdot\chi(1)$ for some $\epsilon>0$, where $\chi(A)$ is the set-multilinear polynomial whose coefficient at $\mathbf x^{\sigma}$ counts the terms of $A$ satisfied by $\sigma$, and $\chi(1)=\prod_{i\le n}(x_i^0+x_i^1)$. Proof size is circuit size, and $\epsilon$ is existentially quantified, so a lower bound must hold at arbitrarily small $\epsilon$. We prove such bounds for bounded-depth fragments by two nonnegative rank measures of $\chi(A)-\epsilon\cdot\chi(1)$: its monotone tensor rank, and the nonnegative rank of a flattening of its coefficient tensor. At depth three, a proof decomposes the coefficient tensor into nonnegative rank-one tensors. Restricting the summands to nonnegative multiples of subcube indicator tensors yields conical juntas and a system polynomially equivalent to Sherali–Adams when size counts conjunctions over the twin variables $x_i,\bar x_i$. For arbitrary nonnegative rank-one summands, we adapt Rothvoß' hyperplane separation method to monotone tensor rank and prove that Tseitin formulas on constant-degree expanders of sufficiently large girth require proofs of size $2^{\Omega(\epsilon n)}$ for $\epsilon\in(0,1]$ with $\epsilon=\omega(\log n/n)$. This cannot reach every $\epsilon>0$: at shift $2^{-\Theta(n\log n)}$ Tseitin acquires polynomial-size depth-three proofs, which separates the general depth-three fragment from the subcube one exponentially. We compose Tseitin formulas with a constant-size index gadget to obtain lower bounds for every $\epsilon\ge2^{-\Omega(n)}$, converting a Sherali–Adams *degree* bound into a depth-three *size* bound. Depth three is essential to all of this: we also show that Tseitin has monotone proofs of size $O(n^3)$ and depth $O(\log n)$ at *every* $\epsilon\in(0,1]$. Beyond depth three, we bound the nonnegative rank of a flattening by $L^{2^t}$ for a monotone formula with $L$ leaves and *mixed-product depth* $t$. Here $t$ is the maximum number, along a root-to-leaf path, of product gates whose two inputs each involve both sides of the bipartition; the Tseitin proofs just mentioned have $t=0$ for every bipartition, so this bound is empty for Tseitin. We give two constructions turning hard matrices into hard DNFs. Approximate extension complexity yields bounds for perfect matching and odd factor instances. Linear programming integrality gaps for planted constraint satisfaction yield an explicit family of planted Tseitin DNFs hard at inverse-polynomial shift.

TR26-304 | Combinatorial limitations of list-decoding Reed-Solomon codes | Omar Alrabiah, Rohan Goyal, Venkatesan Guruswami

from ECCC Papers
For infinitely many rates $R \in (0,1)$, we show that full-length Reed-Solomon codes over fields of low characteristic can contain super-polynomially many codewords in a Hamming ball of radius just exceeding the Johnson radius. Over prime fields, we exhibit full-length Reed-Solomon codes of constant rate $R$ with polynomially many codewords in a Hamming ball of relative radius bounded strictly below $1-R$. This refutes a recent conjecture that, over prime fields, list sizes below capacity are always bounded by a constant. Together, these results demonstrate that certain restrictions in recent works on explicit Reed-Solomon codes achieving list-decoding capacity, related to the choice of field or evaluation points or to the list-size guarantee, are inherent.
For infinitely many rates $R \in (0,1)$, we show that full-length Reed-Solomon codes over fields of low characteristic can contain super-polynomially many codewords in a Hamming ball of radius just exceeding the Johnson radius. Over prime fields, we exhibit full-length Reed-Solomon codes of constant rate $R$ with polynomially many codewords in a Hamming ball of relative radius bounded strictly below $1-R$. This refutes a recent conjecture that, over prime fields, list sizes below capacity are always bounded by a constant. Together, these results demonstrate that certain restrictions in recent works on explicit Reed-Solomon codes achieving list-decoding capacity, related to the choice of field or evaluation points or to the list-size guarantee, are inherent.

TR26-303 | Quadratic Extractors | Gautam Chandrasekaran, Jesse Goodman, David Zuckerman

from ECCC Papers
A quadratic source is a random variable $X\sim\mathbb{F}_2^n$ generated by applying $n$ quadratics to a hidden uniform string. Previously, the best-known explicit extractor for this model required min-entropy roughly $k=n-\sqrt{\log n}$. In this paper, we give an explicit construction for min-entropy $k=\mathrm{polylog}(n)$. Even for the special case of $2$-local sources, where each bit of $X$ depends on two hidden uniform bits, achieving an entropy requirement below $k=\sqrt{n}$ was an important open problem. This is because such sources provably cannot be reduced (in a black-box manner) to our most powerful general-purpose seedless extractor, known as a sumset extractor. Our new extractor bypasses this barrier, providing the first explicit construction to go beyond sumset extraction.
A quadratic source is a random variable $X\sim\mathbb{F}_2^n$ generated by applying $n$ quadratics to a hidden uniform string. Previously, the best-known explicit extractor for this model required min-entropy roughly $k=n-\sqrt{\log n}$. In this paper, we give an explicit construction for min-entropy $k=\mathrm{polylog}(n)$. Even for the special case of $2$-local sources, where each bit of $X$ depends on two hidden uniform bits, achieving an entropy requirement below $k=\sqrt{n}$ was an important open problem. This is because such sources provably cannot be reduced (in a black-box manner) to our most powerful general-purpose seedless extractor, known as a sumset extractor. Our new extractor bypasses this barrier, providing the first explicit construction to go beyond sumset extraction.

The determinantal complexity of the 4*4 permanent is 15

from arXiv: Computational Complexity

Authors: Karthik Sheshadri

The best known lower bound on the determinantal complexity of the 4x4 permanent is 10. In this short note, I prove that this value is exactly 15: the 4x4 permanent is the determinant of a 15x15 matrix of affine linear forms, and cannot be that of any smaller matrix. To do so, I must establish both a lower and upper bound. To establish the upper bound, I use Grenet's construction of size 2^n-1. Hence that construction is optimal at n=4 (and n=3). This now constitutes the first knwon exact value of the determinantal complexity of the permanent at n=4. Only the value at n=3 was exactly known, prior to this note. The general quadratic bound only gives 8 for n=4. To establish the lower bound, I work from a normal form at a zero of the permanent. In the first row (and column) of the normal form, the linear forms vanish on a linear space inside the singular locus of the permanent. A hessian inequality bounds the size of the matrix in terms of that space. Then, a matrix of sie 14 foces the space into any of nine corordinate types. The forcing is either up to symmetry, or else into a family that degenerates to one of them. I then rule out every case. All involved computations are exact. In respect of the claim, I have made all certificates public.

Authors: Karthik Sheshadri

The best known lower bound on the determinantal complexity of the 4x4 permanent is 10. In this short note, I prove that this value is exactly 15: the 4x4 permanent is the determinant of a 15x15 matrix of affine linear forms, and cannot be that of any smaller matrix. To do so, I must establish both a lower and upper bound. To establish the upper bound, I use Grenet's construction of size 2^n-1. Hence that construction is optimal at n=4 (and n=3). This now constitutes the first knwon exact value of the determinantal complexity of the permanent at n=4. Only the value at n=3 was exactly known, prior to this note. The general quadratic bound only gives 8 for n=4. To establish the lower bound, I work from a normal form at a zero of the permanent. In the first row (and column) of the normal form, the linear forms vanish on a linear space inside the singular locus of the permanent. A hessian inequality bounds the size of the matrix in terms of that space. Then, a matrix of sie 14 foces the space into any of nine corordinate types. The forcing is either up to symmetry, or else into a family that degenerates to one of them. I then rule out every case. All involved computations are exact. In respect of the claim, I have made all certificates public.

A Width Hierarchy of Monotone Algebraic Branching Programs

from arXiv: Computational Complexity

Authors: Julian Dörfler, Alexander Rogovskyy

It is well known that $\mathsf{VF} = \mathsf{VBP}_{3}$ (Ben-Or-Cleve'92), i.e. the width of any constant-width algebraic branching program can be reduced to width $3$ with only a polynomial overhead to size. We prove that this is not the case for the monotone equivalents. In particular, we prove that there is an infinite hierarchy $\mathsf{mVBP}_{2} \subsetneq \mathsf{mVBP}_{3} \subsetneq \ldots$ by showing that $\mathrm{imm}_{k,n}$ requires width $w$ mABPs to have depth at least $\left(\frac{k}{w-1}\right)^{\frac{1}{2}\left\lfloor n^{\frac{1}{(w-2)}} \right\rfloor - 1}$ whenever $3 \leq w < k$. We prove this lower bound using support-based techniques. Since this lower bound is exact instead of asymptotic, it allows some width separations even for superconstant width. Moreover, it allows us to show that the expressive power of poly-sized monotone formulas of depth $2d$ is not reached by poly-sized mABPs of width $d-1$ but a width of $2d+1$ is sufficient. We contrast this to unbounded-depth monotone formulas, which are proven to be stronger than constant-width mABPs, but weaker than $\log$-width mABPs and incomparable for most widths in between. Finally, we generalize Snir's framework (Snir'80) for showing lower bounds on the number of leaves of monotone formulas to monotone formulas of multiplication depth $d^{\times}$, giving a lower bound of $n \cdot k^{d^{\times}(n^{\frac{1}{d^{\times}}} - 1)}$ for the iterated matrix multiplication polynomial and $\left(\left(e+o(1)\right) \ln^{\circ(d^{\times}-1)}(n)\right)^n$ for the permanent.

Authors: Julian Dörfler, Alexander Rogovskyy

It is well known that $\mathsf{VF} = \mathsf{VBP}_{3}$ (Ben-Or-Cleve'92), i.e. the width of any constant-width algebraic branching program can be reduced to width $3$ with only a polynomial overhead to size. We prove that this is not the case for the monotone equivalents. In particular, we prove that there is an infinite hierarchy $\mathsf{mVBP}_{2} \subsetneq \mathsf{mVBP}_{3} \subsetneq \ldots$ by showing that $\mathrm{imm}_{k,n}$ requires width $w$ mABPs to have depth at least $\left(\frac{k}{w-1}\right)^{\frac{1}{2}\left\lfloor n^{\frac{1}{(w-2)}} \right\rfloor - 1}$ whenever $3 \leq w < k$. We prove this lower bound using support-based techniques. Since this lower bound is exact instead of asymptotic, it allows some width separations even for superconstant width. Moreover, it allows us to show that the expressive power of poly-sized monotone formulas of depth $2d$ is not reached by poly-sized mABPs of width $d-1$ but a width of $2d+1$ is sufficient. We contrast this to unbounded-depth monotone formulas, which are proven to be stronger than constant-width mABPs, but weaker than $\log$-width mABPs and incomparable for most widths in between. Finally, we generalize Snir's framework (Snir'80) for showing lower bounds on the number of leaves of monotone formulas to monotone formulas of multiplication depth $d^{\times}$, giving a lower bound of $n \cdot k^{d^{\times}(n^{\frac{1}{d^{\times}}} - 1)}$ for the iterated matrix multiplication polynomial and $\left(\left(e+o(1)\right) \ln^{\circ(d^{\times}-1)}(n)\right)^n$ for the permanent.

Competitive Equilibrium for Linear Chores is CLS-complete

from arXiv: Computational Complexity

Authors: Qian Liu, Shinuo Ma, Yixin Tao, Minji Yang, Weiqiang Zheng

The competitive equilibrium (CE) is a fundamental concept for markets, with applications to efficient and fair allocation of resources. We consider the fundamental question of computing a CE in the most basic Fisher market model where agents have linear preferences. While it is known that we can compute a CE in strongly polynomial time with linear goods, there is no known polynomial-time algorithm for CE with linear chores. Computing a CE for linear chores is known to be in CLS with no hardness results. In this paper, we prove that computing an exact CE in Fisher markets with divisible chores and linear disutilities is CLS-hard and thus CLS-complete. To the best of our knowledge, this is the first CLS-completeness result for a market equilibrium problem. On the technical side, we use the nonconvex dual Eisenberg-Gale (EG) program whose Karush-Kuhn-Tucker (KKT) points are CE [Chaudhury, Kroer, Mehta, and Nan, 2024], and the CLS-complete problem of computing a KKT point of a quadratic program [Fearnley, Goldberg, Hollender, and Savani, 2024]. We show that from any exact equilibrium of the constructed market, we can recover in polynomial time an $\varepsilon$-KKT point of the input quadratic program over the unit box.

Authors: Qian Liu, Shinuo Ma, Yixin Tao, Minji Yang, Weiqiang Zheng

The competitive equilibrium (CE) is a fundamental concept for markets, with applications to efficient and fair allocation of resources. We consider the fundamental question of computing a CE in the most basic Fisher market model where agents have linear preferences. While it is known that we can compute a CE in strongly polynomial time with linear goods, there is no known polynomial-time algorithm for CE with linear chores. Computing a CE for linear chores is known to be in CLS with no hardness results. In this paper, we prove that computing an exact CE in Fisher markets with divisible chores and linear disutilities is CLS-hard and thus CLS-complete. To the best of our knowledge, this is the first CLS-completeness result for a market equilibrium problem. On the technical side, we use the nonconvex dual Eisenberg-Gale (EG) program whose Karush-Kuhn-Tucker (KKT) points are CE [Chaudhury, Kroer, Mehta, and Nan, 2024], and the CLS-complete problem of computing a KKT point of a quadratic program [Fearnley, Goldberg, Hollender, and Savani, 2024]. We show that from any exact equilibrium of the constructed market, we can recover in polynomial time an $\varepsilon$-KKT point of the input quadratic program over the unit box.

Mortality Is Undecidable for $2 \times 2$ Integer Matrices

from arXiv: Computational Complexity

Authors: Thanin Quartz, Ivan O. Shevchenko, Hans De Sterck, Jun Liu

The matrix mortality problem asks whether a given finite family of matrices can be multiplied, in any order and with possible repetition, to get the zero matrix. It is one of the most basic decision problems about matrix products, and it arises naturally in the study of switched linear systems and of finite automata. Mortality is trivially decidable in dimension one and has been known to be undecidable for integer matrices in dimensions three and higher since 1970; only the problem in dimension $2$ is left open. We resolve this problem by proving that mortality is undecidable for families of 5,799 distinct $2\times2$ integer matrices. All but one of the matrices are fixed and nonsingular, with determinants $\pm p$ for a single fixed prime $p$. The input affects only the remaining rank-one matrix, which encodes this input into its kernel. Our proof encodes the computation of a fixed universal machine into products of $2\times2$ matrices. We represent each configuration of the computation by a single rational number, which we identify with a line through the origin in $\mathbb{Q}^2$, and each nonsingular matrix carries out a small piece of one computation step. The main difficulty is that the matrices may be multiplied in any order, so we design them to check their own work: a "wrong'' step puts a factor of a prime number $p$ into the denominator of the rational number corresponding to the current computation step, where no later matrix can remove it. As a result, products of these matrices can go from one encoded configuration to another only by following a valid computation. The rank-one matrix then turns the question of whether the machine halts into the question of whether some product equals zero. A companion formalization in the Lean proof assistant verifies the complete reduction.

Authors: Thanin Quartz, Ivan O. Shevchenko, Hans De Sterck, Jun Liu

The matrix mortality problem asks whether a given finite family of matrices can be multiplied, in any order and with possible repetition, to get the zero matrix. It is one of the most basic decision problems about matrix products, and it arises naturally in the study of switched linear systems and of finite automata. Mortality is trivially decidable in dimension one and has been known to be undecidable for integer matrices in dimensions three and higher since 1970; only the problem in dimension $2$ is left open. We resolve this problem by proving that mortality is undecidable for families of 5,799 distinct $2\times2$ integer matrices. All but one of the matrices are fixed and nonsingular, with determinants $\pm p$ for a single fixed prime $p$. The input affects only the remaining rank-one matrix, which encodes this input into its kernel. Our proof encodes the computation of a fixed universal machine into products of $2\times2$ matrices. We represent each configuration of the computation by a single rational number, which we identify with a line through the origin in $\mathbb{Q}^2$, and each nonsingular matrix carries out a small piece of one computation step. The main difficulty is that the matrices may be multiplied in any order, so we design them to check their own work: a "wrong'' step puts a factor of a prime number $p$ into the denominator of the rational number corresponding to the current computation step, where no later matrix can remove it. As a result, products of these matrices can go from one encoded configuration to another only by following a valid computation. The rank-one matrix then turns the question of whether the machine halts into the question of whether some product equals zero. A companion formalization in the Lean proof assistant verifies the complete reduction.

Fourier Entropy Bounds via Approximations

from arXiv: Computational Complexity

Authors: Manaswi Paraashar, Nitin Saurabh

We establish upper bounds on the Shannon entropy $\mathbb H(\widehat f^{\,2})$ of the Fourier distribution of Boolean functions. We prove that $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\log\bigl(1+\|\widehat f\|_{1,1/3}\bigr)\right), $$ where $\|\widehat f\|_{1,1/3}$ denotes the minimum spectral norm of a pointwise $1/3$-approximant to $f$. This implies $\mathbb H(\widehat f^{\,2})=O(\mathsf R^\oplus(f))$, where $\mathsf R^\oplus(f)$ is bounded-error randomized parity-query complexity. We also bound Fourier entropy by the smaller of the two one-sided average unambiguous parity-certificate complexities. More generally, for any exact representation of indicator of $f$ as $\sum_C w_C\mathbf1_C$, with nonnegative weights and affine subspaces $C$, we prove $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\sum_C w_C\,2^{-k_C}k_C\right), \qquad k_C=\operatorname{codim}(C). $$ The quantity on the right is the average conical junta degree of the representation. These results strengthen earlier bounds of (Arunachalam et al. 2021) by extending approximation-based entropy estimates to Shannon entropy and allowing one-sided and overlapping exact representations. The proofs use an entropy bound from a sequence of increasingly accurate approximants of $f$ with \emph{small} spectral norm. Our techniques also find an application in quantum communication complexity, where it improves a lower bound for the bounded-error quantum communication complexity of XOR functions.

Authors: Manaswi Paraashar, Nitin Saurabh

We establish upper bounds on the Shannon entropy $\mathbb H(\widehat f^{\,2})$ of the Fourier distribution of Boolean functions. We prove that $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\log\bigl(1+\|\widehat f\|_{1,1/3}\bigr)\right), $$ where $\|\widehat f\|_{1,1/3}$ denotes the minimum spectral norm of a pointwise $1/3$-approximant to $f$. This implies $\mathbb H(\widehat f^{\,2})=O(\mathsf R^\oplus(f))$, where $\mathsf R^\oplus(f)$ is bounded-error randomized parity-query complexity. We also bound Fourier entropy by the smaller of the two one-sided average unambiguous parity-certificate complexities. More generally, for any exact representation of indicator of $f$ as $\sum_C w_C\mathbf1_C$, with nonnegative weights and affine subspaces $C$, we prove $$ \mathbb H(\widehat f^{\,2}) =O\!\left(\sum_C w_C\,2^{-k_C}k_C\right), \qquad k_C=\operatorname{codim}(C). $$ The quantity on the right is the average conical junta degree of the representation. These results strengthen earlier bounds of (Arunachalam et al. 2021) by extending approximation-based entropy estimates to Shannon entropy and allowing one-sided and overlapping exact representations. The proofs use an entropy bound from a sequence of increasingly accurate approximants of $f$ with \emph{small} spectral norm. Our techniques also find an application in quantum communication complexity, where it improves a lower bound for the bounded-error quantum communication complexity of XOR functions.

Improved Exact and Approximate Quantum Fourier Transforms from Multiplication

from arXiv: Computational Complexity

Authors: Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

In the world of quantum computing, quantum Fourier transform (QFT) has been one of the most important subroutines for quantum algorithms, as exemplified by Shor's factoring algorithms and phase estimation (which in itself has been used as a primitive in various quantum algorithms). In fact most of the well known quantum algorithms can be divided into two classes, QFT based algorithms and search based algorithms. The usual circuit for QFT (covered in almost every basic quantum algorithms course) needs $O(q^2)$ gates (and $O(q)$ qubits) to perform the exact Fourier transform over $\mathbb{F}_{2^q}$. Cleve and Watrous (FOCS 2000) improved this gate count to $O(q (\log q)^2 (\log \log q))$ using the integer multiplication algorithm in $O(n (\log n) (\log \log n))$ time for two $n$-bit integers (best known at that time). Our first main result shows that this gate count can be improved to $O(q (\log q))$. In fact, if there is a multiplication circuit, to multiply two $n$-bit integers, with $O(n (\log n)^α)$ gates; we get a QFT circuit with $O(q (\log q)^α)$ gates (improving upon $O(q (\log q)^{α+ 1})$) gates from the construction of Cleve and Watrous). Given the recent claim by OpenAI about integer multiplication, this opens up the possibility of having a QFT circuit with $o \left( q (\log q)\right)$ gates. On approximating the QFT over $\ZZ/(2^q \ZZ)$, the result of Ahokas, Cleve and Hales (ERATO 2003) showed that an approximating circuit can be built with $O \left( q (\log \log(q/\varepsilon))^{α+1} \right)$ gates, where $ε$ is the approximating parameter. Our second main result again improves this to $O \left( q (\log \log(q/\varepsilon))^α \right)$ gates.

Authors: Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Nitin Saurabh

In the world of quantum computing, quantum Fourier transform (QFT) has been one of the most important subroutines for quantum algorithms, as exemplified by Shor's factoring algorithms and phase estimation (which in itself has been used as a primitive in various quantum algorithms). In fact most of the well known quantum algorithms can be divided into two classes, QFT based algorithms and search based algorithms. The usual circuit for QFT (covered in almost every basic quantum algorithms course) needs $O(q^2)$ gates (and $O(q)$ qubits) to perform the exact Fourier transform over $\mathbb{F}_{2^q}$. Cleve and Watrous (FOCS 2000) improved this gate count to $O(q (\log q)^2 (\log \log q))$ using the integer multiplication algorithm in $O(n (\log n) (\log \log n))$ time for two $n$-bit integers (best known at that time). Our first main result shows that this gate count can be improved to $O(q (\log q))$. In fact, if there is a multiplication circuit, to multiply two $n$-bit integers, with $O(n (\log n)^α)$ gates; we get a QFT circuit with $O(q (\log q)^α)$ gates (improving upon $O(q (\log q)^{α+ 1})$) gates from the construction of Cleve and Watrous). Given the recent claim by OpenAI about integer multiplication, this opens up the possibility of having a QFT circuit with $o \left( q (\log q)\right)$ gates. On approximating the QFT over $\ZZ/(2^q \ZZ)$, the result of Ahokas, Cleve and Hales (ERATO 2003) showed that an approximating circuit can be built with $O \left( q (\log \log(q/\varepsilon))^{α+1} \right)$ gates, where $ε$ is the approximating parameter. Our second main result again improves this to $O \left( q (\log \log(q/\varepsilon))^α \right)$ gates.

An Engineering Reading of the Bounded Halting Problem and Observable Properties of a Universal Machine

from arXiv: Computational Complexity

Authors: Sergey Salishev

The halting problem asks whether a program eventually halts, with unrestricted resources. Engineering systems instead certify bounded halting: given a program, an input, and a step budget T, decide whether the program halts within T steps. This paper studies two of its uses. First, it defines an exact finite-budget problem, placing the simulated program and the checker on the same unit-cost timescale, so the checker can be embedded in an adversarial program. A diagonal argument with explicit additive overhead shows that no fixed deterministic checker decides every instance within any fixed fraction of T below one, for large T. In the nondeterministic case, T bounds the branch depth allowed for a stable observable decision. Second, bounded-halting instances give adversarial tests for finitary universality, since a universal model must support arbitrary finite computations as T grows. The tests check resource growth with T, the absence of a fixed deliberation bound, and later access to intermediate data. They separate a fixed model with extendable external resources, a model family that grows with T, and a larger system built around a fixed component. For machine-learning systems, this identifies the resource growth a universality claim assumes and whether it fails for deployed models.

Authors: Sergey Salishev

The halting problem asks whether a program eventually halts, with unrestricted resources. Engineering systems instead certify bounded halting: given a program, an input, and a step budget T, decide whether the program halts within T steps. This paper studies two of its uses. First, it defines an exact finite-budget problem, placing the simulated program and the checker on the same unit-cost timescale, so the checker can be embedded in an adversarial program. A diagonal argument with explicit additive overhead shows that no fixed deterministic checker decides every instance within any fixed fraction of T below one, for large T. In the nondeterministic case, T bounds the branch depth allowed for a stable observable decision. Second, bounded-halting instances give adversarial tests for finitary universality, since a universal model must support arbitrary finite computations as T grows. The tests check resource growth with T, the absence of a fixed deliberation bound, and later access to intermediate data. They separate a fixed model with extendable external resources, a model family that grows with T, and a larger system built around a fixed component. For machine-learning systems, this identifies the resource growth a universality claim assumes and whether it fails for deployed models.

Exhaustive Search Is Hard to Beat in Constant Depth for Hyperclique and Orthogonal Vectors

from arXiv: Computational Complexity

Authors: Nikolai Chukhin

A hyperclique in an $r$-uniform hypergraph is a set of vertices in which every $r$-subset is an edge. A simple exhaustive search algorithm detects a hyperclique on $\ell$ vertices in an $r$-uniform hypergraph on $m$ vertices by checking all $\binom{m}{\ell}$ sets of $\ell$ vertices. This algorithm can be expressed as a depth-two circuit with at most $m^\ell$ gates and wires. For hypercliques of *constant* size, Amano (2010) showed that this circuit is almost optimal: for large $r$, constant-depth circuits need $m^{\ell(1-O(\log r/r))}$ wires. We prove that this circuit remains hard to beat even for hypercliques of *polynomial* size. More precisely, circuits of depth $h$ need $m^{Ω(\ell)}$ gates for $\ell\leq m^{1/h-\varepsilon}$, and $m^{\ell(1-O(\log r/r))}$ gates for $\ell=m^{o(1)}$ that tends to infinity. Our hyperclique bound and the method behind it also lead to lower bounds for two central problems of fine-grained complexity. First, we extend the bound $n^{k/4}$ of Rossman (2008) for $k$-Clique from cliques of *constant* size to cliques of *polynomial* size: circuits of depth $h$ detecting a clique on $k$ vertices in an $n$-vertex graph need $n^{Ω(k)}$ gates for $k\leq n^{1/(h-1)-\varepsilon}$. Second, a reduction of Choudhury, Limaye, Sreenivasaiah, and Srinivasan (2026) turns our hyperclique bound into a lower bound of $n^{k-\varepsilon}$ gates for $k$-Orthogonal Vectors ($k$-OV) at every fixed depth for vectors of dimension $\log^{O(1)} n$; this proves the $k$-OV Conjecture for constant-depth circuits in polylogarithmic dimension. Finally, for $2$-OV in logarithmic dimension, we give a simple proof that depth-three circuits need $Ω(n^2)$ wires, which answers a question of Kane and Williams (2019).

Authors: Nikolai Chukhin

A hyperclique in an $r$-uniform hypergraph is a set of vertices in which every $r$-subset is an edge. A simple exhaustive search algorithm detects a hyperclique on $\ell$ vertices in an $r$-uniform hypergraph on $m$ vertices by checking all $\binom{m}{\ell}$ sets of $\ell$ vertices. This algorithm can be expressed as a depth-two circuit with at most $m^\ell$ gates and wires. For hypercliques of *constant* size, Amano (2010) showed that this circuit is almost optimal: for large $r$, constant-depth circuits need $m^{\ell(1-O(\log r/r))}$ wires. We prove that this circuit remains hard to beat even for hypercliques of *polynomial* size. More precisely, circuits of depth $h$ need $m^{Ω(\ell)}$ gates for $\ell\leq m^{1/h-\varepsilon}$, and $m^{\ell(1-O(\log r/r))}$ gates for $\ell=m^{o(1)}$ that tends to infinity. Our hyperclique bound and the method behind it also lead to lower bounds for two central problems of fine-grained complexity. First, we extend the bound $n^{k/4}$ of Rossman (2008) for $k$-Clique from cliques of *constant* size to cliques of *polynomial* size: circuits of depth $h$ detecting a clique on $k$ vertices in an $n$-vertex graph need $n^{Ω(k)}$ gates for $k\leq n^{1/(h-1)-\varepsilon}$. Second, a reduction of Choudhury, Limaye, Sreenivasaiah, and Srinivasan (2026) turns our hyperclique bound into a lower bound of $n^{k-\varepsilon}$ gates for $k$-Orthogonal Vectors ($k$-OV) at every fixed depth for vectors of dimension $\log^{O(1)} n$; this proves the $k$-OV Conjecture for constant-depth circuits in polylogarithmic dimension. Finally, for $2$-OV in logarithmic dimension, we give a simple proof that depth-three circuits need $Ω(n^2)$ wires, which answers a question of Kane and Williams (2019).

The Complexity of Unpredictability

from arXiv: Computational Complexity

Authors: Aryan Agarwala, Dani Dorfman

We initiate the study of middle-bit predictors: functions that take as input a string with one missing bit and try to predict that bit. These are closely related to next-bit predictors, a classical object in pseudorandomness. We study the deterministic complexity of computing an unpredictable distribution (UNPRED). Given a list of middle-bit predictors in circuit form and a parameter $ε>0$, output a multiset of strings such that each predictor succeeds with probability less than $1/2+ε$ on a uniformly sampled string from the multiset. We write $\mathsf{UNPRED}^{\mathsf{TC}^1}$ for the restriction to $\mathsf{TC}^1$ predictors. We show that UNPRED offers a unified route towards resolving several major open problems in catalytic computation, the biggest of which is whether deterministic polynomial time (P and FP) can simulate catalytic logspace (CL and SearchCL). It is believed that $\mathsf{SearchCL}\subseteq\mathsf{FP}$. We prove $$\mathsf{UNPRED}\in\mathsf{FP}\implies\mathsf{SearchCL}\subseteq\mathsf{FP}\implies\mathsf{UNPRED}^{\mathsf{TC}^1}\in\mathsf{FP}.$$ Thus, the questions of whether UNPRED is in FP and whether SearchCL is contained in FP are almost equivalent, up to the restriction to $\mathsf{TC}^1$ circuits. We also show that any deterministic polynomial-time algorithm for $\mathsf{UNPRED}^{\mathsf{TC}^1}$ must exploit properties of $\mathsf{TC}^1$ circuits, either in the algorithm itself or in its analysis. To our knowledge, this would also be the first algorithm to use the structure of $\mathsf{TC}^1$ circuits in this way to solve a natural problem, thus advancing our understanding of Boolean circuits. Together, our results provide strong evidence that middle-bit predictors could be the key to unlocking several major open problems in catalytic computation, pseudorandomness, and Boolean circuit complexity.

Authors: Aryan Agarwala, Dani Dorfman

We initiate the study of middle-bit predictors: functions that take as input a string with one missing bit and try to predict that bit. These are closely related to next-bit predictors, a classical object in pseudorandomness. We study the deterministic complexity of computing an unpredictable distribution (UNPRED). Given a list of middle-bit predictors in circuit form and a parameter $ε>0$, output a multiset of strings such that each predictor succeeds with probability less than $1/2+ε$ on a uniformly sampled string from the multiset. We write $\mathsf{UNPRED}^{\mathsf{TC}^1}$ for the restriction to $\mathsf{TC}^1$ predictors. We show that UNPRED offers a unified route towards resolving several major open problems in catalytic computation, the biggest of which is whether deterministic polynomial time (P and FP) can simulate catalytic logspace (CL and SearchCL). It is believed that $\mathsf{SearchCL}\subseteq\mathsf{FP}$. We prove $$\mathsf{UNPRED}\in\mathsf{FP}\implies\mathsf{SearchCL}\subseteq\mathsf{FP}\implies\mathsf{UNPRED}^{\mathsf{TC}^1}\in\mathsf{FP}.$$ Thus, the questions of whether UNPRED is in FP and whether SearchCL is contained in FP are almost equivalent, up to the restriction to $\mathsf{TC}^1$ circuits. We also show that any deterministic polynomial-time algorithm for $\mathsf{UNPRED}^{\mathsf{TC}^1}$ must exploit properties of $\mathsf{TC}^1$ circuits, either in the algorithm itself or in its analysis. To our knowledge, this would also be the first algorithm to use the structure of $\mathsf{TC}^1$ circuits in this way to solve a natural problem, thus advancing our understanding of Boolean circuits. Together, our results provide strong evidence that middle-bit predictors could be the key to unlocking several major open problems in catalytic computation, pseudorandomness, and Boolean circuit complexity.

Community Search under Member-Specific Tolerances

from arXiv: Computational Complexity

Authors: Yanghao Liu, Fangda Guo, Yuxuan Zhang, Xueqi Cheng

We study community search when each vertex has its own tolerance for differences in position. For positions such as opinions or homes, a latitude-consistent community (LCC) contains the query vertex, is connected, has minimum degree at least k, and bounds its diameter by the smallest member tolerance. Motivated by social judgment theory, the bound exactly characterizes persistent acceptance of every interaction in a closed group under bounded-confidence dynamics with arbitrary positive weights. It also prevents rejection under social-judgment dynamics, whereas natural relaxations preserve neither guarantee. Finding a maximum LCC is NP-hard even in the plane, and with unbounded dimension, deciding whether an LCC exists is NP-complete for every fixed k >= 2. LatSweep turns the coupled bound into a sweep over the smallest tolerance, yielding two feasible heuristics, a (1,2)-bicriteria approximation and an exact branch-and-bound algorithm. Across eight graphs with up to 1.8 billion edges, the exact algorithm proves optimality on all but 9 of 3,700 efficiency queries and usually certifies absence within microseconds when no LCC exists. Its branch-and-bound solver proves more queries than a reimplemented maximum-(k,r)-core algorithm, a mixed-integer program and CP-SAT. In simulations, uneven influence disrupts initially accepted interactions in pairwise groups. Outside ties invalidate the guarantee for many real groups. On two location-based networks, tolerance-agnostic groups often place another member's home beyond a member's estimated travel radius; comparisons with spatial communities of the same size detect no co-visit loss from enforcing every member's tolerance.

Authors: Yanghao Liu, Fangda Guo, Yuxuan Zhang, Xueqi Cheng

We study community search when each vertex has its own tolerance for differences in position. For positions such as opinions or homes, a latitude-consistent community (LCC) contains the query vertex, is connected, has minimum degree at least k, and bounds its diameter by the smallest member tolerance. Motivated by social judgment theory, the bound exactly characterizes persistent acceptance of every interaction in a closed group under bounded-confidence dynamics with arbitrary positive weights. It also prevents rejection under social-judgment dynamics, whereas natural relaxations preserve neither guarantee. Finding a maximum LCC is NP-hard even in the plane, and with unbounded dimension, deciding whether an LCC exists is NP-complete for every fixed k >= 2. LatSweep turns the coupled bound into a sweep over the smallest tolerance, yielding two feasible heuristics, a (1,2)-bicriteria approximation and an exact branch-and-bound algorithm. Across eight graphs with up to 1.8 billion edges, the exact algorithm proves optimality on all but 9 of 3,700 efficiency queries and usually certifies absence within microseconds when no LCC exists. Its branch-and-bound solver proves more queries than a reimplemented maximum-(k,r)-core algorithm, a mixed-integer program and CP-SAT. In simulations, uneven influence disrupts initially accepted interactions in pairwise groups. Outside ties invalidate the guarantee for many real groups. On two location-based networks, tolerance-agnostic groups often place another member's home beyond a member's estimated travel radius; comparisons with spatial communities of the same size detect no co-visit loss from enforcing every member's tolerance.

The CNF Encoding Complexity of Boolean Functions

from arXiv: Computational Complexity

Authors: Andrew Krapivin, Benjamin Przybocki, Bernardo Subercaseaux

We study the minimum number of clauses required to encode a boolean function into a CNF formula with auxiliary variables. Our main result is that if a boolean function can be evaluated in nondeterministic time $t$ (in the RAM model) with $m$ bits of advice such that at most $b$ bits of advice are accessed for each input, then it admits a CNF encoding with $\widetilde{O}(t + \sqrt{mb})$ clauses. This constitutes a nonuniform generalization of Robson's (TCS'91) variant of the Cook--Levin theorem, for which a naïve generalization would yield $\widetilde{O}(t+m)$ clauses; our applications leverage $b = o(m)$ to obtain significantly better bounds. Since preprocessed data structures can be modeled as computation with advice, our result allows us to translate space- and time-efficient data structures to compact CNF encodings. As an application, we derandomize a data structure of Larsen and Williams (SODA'17) to show that every $k$-CNF function on $n$ inputs can be encoded with $n^k/2^{Ω_k(\sqrt{\lg n})}$ clauses. Notably, despite encoding a $k$-CNF function, the encoding itself necessarily contains clauses of unbounded width, since an information-theoretic argument shows that the minimum number of bounded-width clauses to encode such functions is $Ω_k(n^{k}/\lg n)$ in the worst case. In general terms, our results imply that for a wide variety of NP-complete problems, one can reduce to CNF-SAT in a way that results in asymptotically fewer clauses than previously known methods.

Authors: Andrew Krapivin, Benjamin Przybocki, Bernardo Subercaseaux

We study the minimum number of clauses required to encode a boolean function into a CNF formula with auxiliary variables. Our main result is that if a boolean function can be evaluated in nondeterministic time $t$ (in the RAM model) with $m$ bits of advice such that at most $b$ bits of advice are accessed for each input, then it admits a CNF encoding with $\widetilde{O}(t + \sqrt{mb})$ clauses. This constitutes a nonuniform generalization of Robson's (TCS'91) variant of the Cook--Levin theorem, for which a naïve generalization would yield $\widetilde{O}(t+m)$ clauses; our applications leverage $b = o(m)$ to obtain significantly better bounds. Since preprocessed data structures can be modeled as computation with advice, our result allows us to translate space- and time-efficient data structures to compact CNF encodings. As an application, we derandomize a data structure of Larsen and Williams (SODA'17) to show that every $k$-CNF function on $n$ inputs can be encoded with $n^k/2^{Ω_k(\sqrt{\lg n})}$ clauses. Notably, despite encoding a $k$-CNF function, the encoding itself necessarily contains clauses of unbounded width, since an information-theoretic argument shows that the minimum number of bounded-width clauses to encode such functions is $Ω_k(n^{k}/\lg n)$ in the worst case. In general terms, our results imply that for a wide variety of NP-complete problems, one can reduce to CNF-SAT in a way that results in asymptotically fewer clauses than previously known methods.

Fixed-Radius Neighbourhoods in Three-Dimensional Unit-Cube Graphs

from arXiv: Computational Geometry

Authors: Guangjian Zhang

We prove Conjecture 4 of Chan, Chang, Gao, Kisfaludi-Bak, Le, and Zheng (SoCG 2026): for every fixed integer $r\ge0$, the radius-$r$ neighbourhoods of a unit-cube graph in $\mathbb{R}^3$ have bounded VC-dimension. Our bound is $2^{2^{O(r)}}$, independent of the number of vertices. The main ingredient is a VC-dimension bound for reachability through arbitrary fixed-length sequences of three-dimensional generalized dominance relations. Its proof uses groups ordered in two coordinates, with neighbouring groups acting as barriers that preserve this invariant when one layer is removed. Combining these bounds with the existing interval-representation framework, we obtain a Las Vegas algorithm that, for every fixed integer $Δ\ge1$, decides whether a graph given by $n$ cube centres has diameter at most $Δ$ in expected $O_Δ(n^{2-\varepsilon_Δ})$ time, for an explicit constant $\varepsilon_Δ>0$. Finally, for every integer $r\ge7$, we construct unit-cube graphs with rational centres whose radius-$r$ neighbourhoods have VC-dimension $Ω(r^3)$.

Authors: Guangjian Zhang

We prove Conjecture 4 of Chan, Chang, Gao, Kisfaludi-Bak, Le, and Zheng (SoCG 2026): for every fixed integer $r\ge0$, the radius-$r$ neighbourhoods of a unit-cube graph in $\mathbb{R}^3$ have bounded VC-dimension. Our bound is $2^{2^{O(r)}}$, independent of the number of vertices. The main ingredient is a VC-dimension bound for reachability through arbitrary fixed-length sequences of three-dimensional generalized dominance relations. Its proof uses groups ordered in two coordinates, with neighbouring groups acting as barriers that preserve this invariant when one layer is removed. Combining these bounds with the existing interval-representation framework, we obtain a Las Vegas algorithm that, for every fixed integer $Δ\ge1$, decides whether a graph given by $n$ cube centres has diameter at most $Δ$ in expected $O_Δ(n^{2-\varepsilon_Δ})$ time, for an explicit constant $\varepsilon_Δ>0$. Finally, for every integer $r\ge7$, we construct unit-cube graphs with rational centres whose radius-$r$ neighbourhoods have VC-dimension $Ω(r^3)$.

Projected and Reflected Sliced Wasserstein Kernels for Persistence Diagrams

from arXiv: Computational Geometry

Authors: Julián Uribe Castañeda, Mircea Petrache

Persistence diagrams do not naturally live in a Hilbert space. Carrière, Cuturi, and Oudot embedded them in kernel methods through the Sliced Wasserstein kernel, built from the Sliced Wasserstein distance $\mathrm{SW}_1$, which is conditionally negative definite (CND). We study the projected ($\mathrm{SW}_p$) and reflected ($\mathrm{RW}_p$) Sliced Wasserstein distances for every order $p\ge1$, and compare them with the $p$-Wasserstein distance $d_p$ and with a reflected variant $d_p^r$, in which projection onto the diagonal is replaced by reflection across it. We show that $2^{1/p}d_p\le d_p^r\le 2d_p$ with sharp constants, so that $d_1^r=2d_1$ and $d_p^r$ inherits the stability of $d_p$, although it is a metric only for $p=1$. The distances $\mathrm{SW}_1$ and $\mathrm{RW}_1$ are CND metrics, and $\mathrm{RW}_1$ gives rise to a new positive definite kernel, stable with respect to $d_1$. For $p>1$, neither $\mathrm{SW}_p$ nor $\mathrm{RW}_p$ satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that $\mathrm{SW}_2$ and $\mathrm{RW}_2$ are not CND; numerical experiments suggest that the CND property fails for every $p$ above some $p^*\in(1,1.2]$. On diagrams with at most $N$ points, $\mathrm{SW}_p$ and $\mathrm{RW}_p$ are metrically equivalent to $d_p$ and $d_p^r$, with sharp upper constants. On an orbit recognition benchmark, the kernels built from $\mathrm{SW}_1$ and $\mathrm{RW}_1$ reach about 85% accuracy, outperform the persistence scale space kernel, perform on par with the persistence weighted Gaussian kernel, and are about one percentage point below the kernel built from $\mathrm{SW}_2$, which is not positive definite.

Authors: Julián Uribe Castañeda, Mircea Petrache

Persistence diagrams do not naturally live in a Hilbert space. Carrière, Cuturi, and Oudot embedded them in kernel methods through the Sliced Wasserstein kernel, built from the Sliced Wasserstein distance $\mathrm{SW}_1$, which is conditionally negative definite (CND). We study the projected ($\mathrm{SW}_p$) and reflected ($\mathrm{RW}_p$) Sliced Wasserstein distances for every order $p\ge1$, and compare them with the $p$-Wasserstein distance $d_p$ and with a reflected variant $d_p^r$, in which projection onto the diagonal is replaced by reflection across it. We show that $2^{1/p}d_p\le d_p^r\le 2d_p$ with sharp constants, so that $d_1^r=2d_1$ and $d_p^r$ inherits the stability of $d_p$, although it is a metric only for $p=1$. The distances $\mathrm{SW}_1$ and $\mathrm{RW}_1$ are CND metrics, and $\mathrm{RW}_1$ gives rise to a new positive definite kernel, stable with respect to $d_1$. For $p>1$, neither $\mathrm{SW}_p$ nor $\mathrm{RW}_p$ satisfies the triangle inequality, and rigorous interval-arithmetic certificates prove that $\mathrm{SW}_2$ and $\mathrm{RW}_2$ are not CND; numerical experiments suggest that the CND property fails for every $p$ above some $p^*\in(1,1.2]$. On diagrams with at most $N$ points, $\mathrm{SW}_p$ and $\mathrm{RW}_p$ are metrically equivalent to $d_p$ and $d_p^r$, with sharp upper constants. On an orbit recognition benchmark, the kernels built from $\mathrm{SW}_1$ and $\mathrm{RW}_1$ reach about 85% accuracy, outperform the persistence scale space kernel, perform on par with the persistence weighted Gaussian kernel, and are about one percentage point below the kernel built from $\mathrm{SW}_2$, which is not positive definite.

Does RRT* Work on Uneven Terrain Meshes?

from arXiv: Computational Geometry

Authors: Otobong Jerome, Tiago Nascimento

The optimal rapidly exploring random tree (RRT*) is proven asymptotically optimal on open subsets of Euclidean space with straight-line local planning. We ask whether it works on a triangle-mesh terrain, a two-dimensional surface in R^3. First, the original RRT* cannot run on a mesh: the surface has zero volume, so uniform sampling never hits it, no clearance ball fits in it, and chords across a fold leave it. Second, the surface reformulation we call Mesh RRT*, with its naive edge, the chord costed by its length, solved every run on heightfields with slopes up to 75 degrees but minimises the wrong objective: its cost, the straight-line length of a polyline whose vertices lie on the surface while its segments may leave it by a tolerance tau, is the length of no route on the surface and lies strictly below the geodesic length on any curved mesh. In numeric simulation and after 30,000 iterations the reported cost is on average up to 2.7% below the exact geodesic, a cost no path attains, and the returned route on average 0.1-0.5% longer than it. Rewiring keeps the biased edges, the bias decays only linearly with the step on polyhedral surfaces, and shrinking tau trades bias for connectivity. Third, replacing each chord by a path on the surface above it, built from projected points joined across the mesh edges they straddle and costed by its own length, makes every edge cost the exact length of a feasible path; with no other change the exact length of the returned routes is on average within 0.07-0.16% of the geodesic at every roughness tested, two to 3.4 times closer than with chords on the rougher terrains. The planner is in MeshNav3D 0.2.0 (release pending).

Authors: Otobong Jerome, Tiago Nascimento

The optimal rapidly exploring random tree (RRT*) is proven asymptotically optimal on open subsets of Euclidean space with straight-line local planning. We ask whether it works on a triangle-mesh terrain, a two-dimensional surface in R^3. First, the original RRT* cannot run on a mesh: the surface has zero volume, so uniform sampling never hits it, no clearance ball fits in it, and chords across a fold leave it. Second, the surface reformulation we call Mesh RRT*, with its naive edge, the chord costed by its length, solved every run on heightfields with slopes up to 75 degrees but minimises the wrong objective: its cost, the straight-line length of a polyline whose vertices lie on the surface while its segments may leave it by a tolerance tau, is the length of no route on the surface and lies strictly below the geodesic length on any curved mesh. In numeric simulation and after 30,000 iterations the reported cost is on average up to 2.7% below the exact geodesic, a cost no path attains, and the returned route on average 0.1-0.5% longer than it. Rewiring keeps the biased edges, the bias decays only linearly with the step on polyhedral surfaces, and shrinking tau trades bias for connectivity. Third, replacing each chord by a path on the surface above it, built from projected points joined across the mesh edges they straddle and costed by its own length, makes every edge cost the exact length of a feasible path; with no other change the exact length of the returned routes is on average within 0.07-0.16% of the geodesic at every roughness tested, two to 3.4 times closer than with chords on the rougher terrains. The planner is in MeshNav3D 0.2.0 (release pending).

Softmax Attention with Any Temperature in Fixed Dimension: Matching the Best Known Maximum-Inner-Product Bounds

from arXiv: Data Structures and Algorithms

Authors: Zixuan Huang, Guozhang Chen

Attention is central to modern AI models for language, vision, and scientific modeling. However, direct evaluation compares every query with every key, requiring quadratic time in the context length, which limits its scalability to long contexts. Consequently, characterizing the complexity bounds of attention is of fundamental importance. For every fixed dimension $d$, we give an algorithm for $m$ queries, $n$ key-value pairs, and additive error $\varepsilon$, with expected time $ \widetilde{\mathrm{O}}\bigl((mn)^{k/(k+1)}+m+n\bigr),\qquad k=\lfloor d/2\rfloor. $ For fixed dimension and a fixed number of value channels, soft-O hides factors of the form $\operatorname{poly}_d\bigl(b,\log(m+n),\log B,\log(1/\varepsilon)\bigr),$ where $b$ bounds input encoding lengths and $B$ bounds coordinate and value magnitudes. In particular, the dependence on $B$ and $1/\varepsilon$ is polylogarithmic, which allows arbitrary positive temperature. When $m=n$ and $d\ge2$, the bound is $\widetilde{\mathrm{O}}(n^{2-2/(\lfloor d/2\rfloor+1)})$: near-linear in dimensions two and three, with exponent $4/3$ in dimensions four and five and $3/2$ in dimensions six and seven. It improves the previous $\widetilde{\mathrm{O}}(n^{2-1/d})$ upper bound and matches the best known batch maximum-inner-product bound in each fixed dimension. Additionally, we reduce integer batch maximum inner product computation to one attention call with the same dimension and input sizes $m,n$.

Authors: Zixuan Huang, Guozhang Chen

Attention is central to modern AI models for language, vision, and scientific modeling. However, direct evaluation compares every query with every key, requiring quadratic time in the context length, which limits its scalability to long contexts. Consequently, characterizing the complexity bounds of attention is of fundamental importance. For every fixed dimension $d$, we give an algorithm for $m$ queries, $n$ key-value pairs, and additive error $\varepsilon$, with expected time $ \widetilde{\mathrm{O}}\bigl((mn)^{k/(k+1)}+m+n\bigr),\qquad k=\lfloor d/2\rfloor. $ For fixed dimension and a fixed number of value channels, soft-O hides factors of the form $\operatorname{poly}_d\bigl(b,\log(m+n),\log B,\log(1/\varepsilon)\bigr),$ where $b$ bounds input encoding lengths and $B$ bounds coordinate and value magnitudes. In particular, the dependence on $B$ and $1/\varepsilon$ is polylogarithmic, which allows arbitrary positive temperature. When $m=n$ and $d\ge2$, the bound is $\widetilde{\mathrm{O}}(n^{2-2/(\lfloor d/2\rfloor+1)})$: near-linear in dimensions two and three, with exponent $4/3$ in dimensions four and five and $3/2$ in dimensions six and seven. It improves the previous $\widetilde{\mathrm{O}}(n^{2-1/d})$ upper bound and matches the best known batch maximum-inner-product bound in each fixed dimension. Additionally, we reduce integer batch maximum inner product computation to one attention call with the same dimension and input sizes $m,n$.

Towards the Query Complexity of Sampling under Strong Log-Concavity and Log-Sobolev Inequalities

from arXiv: Data Structures and Algorithms

Authors: Huanjian Zhou, Taiji Suzuki

We study the query complexity of sampling from smooth distributions on $\mathbb R^d$ using first-order oracles. For $d$-dimensional strongly log-concave targets with condition number $κ=2$, we prove that any randomized sampler requires $\exp({Ω(\log^{1/3} d)})$ queries even from a constant warm start. This rules out polylogarithmic dimension dependence and strictly separates strongly log-concave sampling from Gaussian sampling. For targets with $L$-smooth potentials satisfying an $α$-log-Sobolev inequality with $κ= L/α$, we prove an $Ω(κ)$ query lower bound for $d\gtrsim\log^2κ$. This shows that log-Sobolev sampling is strictly harder than strongly log-concave sampling. Our proofs construct instance-dependent tests that target samples typically pass but outputs of algorithms making few queries typically fail. For strongly log-concave targets, we embed a sequence of hidden Gaussian into high-degree tensor polynomial with smooth gates, forcing any algorithm to uncover the layers sequentially. The log-Sobolev construction hides a constant fraction of the target mass in angular caps while controlling the log-Sobolev constant via delicate estimates.

Authors: Huanjian Zhou, Taiji Suzuki

We study the query complexity of sampling from smooth distributions on $\mathbb R^d$ using first-order oracles. For $d$-dimensional strongly log-concave targets with condition number $κ=2$, we prove that any randomized sampler requires $\exp({Ω(\log^{1/3} d)})$ queries even from a constant warm start. This rules out polylogarithmic dimension dependence and strictly separates strongly log-concave sampling from Gaussian sampling. For targets with $L$-smooth potentials satisfying an $α$-log-Sobolev inequality with $κ= L/α$, we prove an $Ω(κ)$ query lower bound for $d\gtrsim\log^2κ$. This shows that log-Sobolev sampling is strictly harder than strongly log-concave sampling. Our proofs construct instance-dependent tests that target samples typically pass but outputs of algorithms making few queries typically fail. For strongly log-concave targets, we embed a sequence of hidden Gaussian into high-degree tensor polynomial with smooth gates, forcing any algorithm to uncover the layers sequentially. The log-Sobolev construction hides a constant fraction of the target mass in angular caps while controlling the log-Sobolev constant via delicate estimates.

Improved lower bounds on the approximability of Multiway Cut using Sherali-Adams relaxations

from arXiv: Data Structures and Algorithms

Authors: Joshua Brakensiek, Neng Huang, Aaron Potechin, Uri Zwick

The Multiway Cut problem is among the most well-studied problems in the area of combinatorial approximation algorithms. The currently best approximation ratio for the problem is 1.2787, obtained recently by the authors (2026). The best lower bound on the approximation ratio that can be achieved for the problem in polynomial time, under the Unique Games Conjecture (UGC), is 1.20016, obtained by Bérczi, Chandrasekaran, Király, and Madan (2020), building on and slightly improving a bound of $\frac65=1.2$, obtained by Angelidakis, Makarychev, and Manurangsi (2017) by introducing the important framework of non-opposite multiway cuts. We obtain a substantially improved lower bound of 1.22232 on the approximation ratio of the Multiway Cut problem. Manokaran, Naor, Raghavendra, and Schwartz (2008) showed that under UGC, the approximation ratio of Multiway Cut is equal to the integrality ratio of a simple LP relaxation, known as the CKR relaxation, introduced by Călinescu, Karloff, and Rabani (2000). Thus, stronger LP relaxations are not required for obtaining an optimal approximation algorithm for the problem. We show, however, that relying on the framework of Angelidakis, Makarychev, and Manurangsi (2017), stronger LP relaxations from the Sherali-Adams hierarchy for non-opposite multiway cuts can be used to obtain improved integrality ratios for the CKR relaxation, and hence improved lower bounds on the approximation ratio of the Multiway Cut problem. One of the novel features of our analysis is considering a continuous limit of discrete LPs defined on a simplex. Our proof of the new 1.22232 bound is completely analytic. We also present strong numerical evidence that suggests that our techniques can be used to improve the lower bound to at least 1.23003.

Authors: Joshua Brakensiek, Neng Huang, Aaron Potechin, Uri Zwick

The Multiway Cut problem is among the most well-studied problems in the area of combinatorial approximation algorithms. The currently best approximation ratio for the problem is 1.2787, obtained recently by the authors (2026). The best lower bound on the approximation ratio that can be achieved for the problem in polynomial time, under the Unique Games Conjecture (UGC), is 1.20016, obtained by Bérczi, Chandrasekaran, Király, and Madan (2020), building on and slightly improving a bound of $\frac65=1.2$, obtained by Angelidakis, Makarychev, and Manurangsi (2017) by introducing the important framework of non-opposite multiway cuts. We obtain a substantially improved lower bound of 1.22232 on the approximation ratio of the Multiway Cut problem. Manokaran, Naor, Raghavendra, and Schwartz (2008) showed that under UGC, the approximation ratio of Multiway Cut is equal to the integrality ratio of a simple LP relaxation, known as the CKR relaxation, introduced by Călinescu, Karloff, and Rabani (2000). Thus, stronger LP relaxations are not required for obtaining an optimal approximation algorithm for the problem. We show, however, that relying on the framework of Angelidakis, Makarychev, and Manurangsi (2017), stronger LP relaxations from the Sherali-Adams hierarchy for non-opposite multiway cuts can be used to obtain improved integrality ratios for the CKR relaxation, and hence improved lower bounds on the approximation ratio of the Multiway Cut problem. One of the novel features of our analysis is considering a continuous limit of discrete LPs defined on a simplex. Our proof of the new 1.22232 bound is completely analytic. We also present strong numerical evidence that suggests that our techniques can be used to improve the lower bound to at least 1.23003.

Constant-Round Massively Parallel Algorithms for Approximate MSTs in Metric Spaces

from arXiv: Data Structures and Algorithms

Authors: Eunjin Oh, Hyeonjun Shin

We develop a unified algorithmic framework for computing approximate minimum spanning trees (MSTs) in metric spaces in the fully scalable Massively Parallel Computation (MPC) model. Using this framework, we obtain the following results for any fixed constant $σ\in(0,1)$: - General metrics: Given all pairwise distances, we compute a $(1+\varepsilon)$-approximate MST in $O(\log(1/\varepsilon))$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^2)$ total space. - Euclidean spaces: Given $n$ points in $\mathbb{R}^d$, we compute an $O(1/σ^4)$-approximate MST in $O(1)$ rounds using $(nd)^σ$ local space per machine and $O(nd+n^{1+σ})$ total space. - Doubling metrics: Given $n$ points in a metric of doubling dimension $\mathrm{ddim}=O(σ\log n)$, we compute an $O(1/σ)$-approximate MST in $O(\mathrm{ddim}/σ^4)$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^{1+σ})\cdot 2^{O(\mathrm{ddim})}$ total space. For any fixed constant $\varepsilon\in(0,1)$, these results give the first constant-round MPC algorithms achieving $(1+\varepsilon)$-approximation in general metrics and $O(1)$-approximation in a high-dimensional Euclidean space, respectively. They improve upon the $O(\log\log n+\log(1/\varepsilon))$-round algorithm for general metrics [SODA'25] and the $O(\log\log n\cdot\log\log\log n)$-round algorithm for Euclidean spaces [SODA'24], respectively. Moreover, our round complexity for general metrics is asymptotically optimal in its dependence on $\varepsilon$, matching the $Ω(\log(1/\varepsilon))$ lower bound of [SODA'25] under the 1-vs-2-cycles conjecture. Our result also answers an open question raised by [STOC'23].

Authors: Eunjin Oh, Hyeonjun Shin

We develop a unified algorithmic framework for computing approximate minimum spanning trees (MSTs) in metric spaces in the fully scalable Massively Parallel Computation (MPC) model. Using this framework, we obtain the following results for any fixed constant $σ\in(0,1)$: - General metrics: Given all pairwise distances, we compute a $(1+\varepsilon)$-approximate MST in $O(\log(1/\varepsilon))$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^2)$ total space. - Euclidean spaces: Given $n$ points in $\mathbb{R}^d$, we compute an $O(1/σ^4)$-approximate MST in $O(1)$ rounds using $(nd)^σ$ local space per machine and $O(nd+n^{1+σ})$ total space. - Doubling metrics: Given $n$ points in a metric of doubling dimension $\mathrm{ddim}=O(σ\log n)$, we compute an $O(1/σ)$-approximate MST in $O(\mathrm{ddim}/σ^4)$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^{1+σ})\cdot 2^{O(\mathrm{ddim})}$ total space. For any fixed constant $\varepsilon\in(0,1)$, these results give the first constant-round MPC algorithms achieving $(1+\varepsilon)$-approximation in general metrics and $O(1)$-approximation in a high-dimensional Euclidean space, respectively. They improve upon the $O(\log\log n+\log(1/\varepsilon))$-round algorithm for general metrics [SODA'25] and the $O(\log\log n\cdot\log\log\log n)$-round algorithm for Euclidean spaces [SODA'24], respectively. Moreover, our round complexity for general metrics is asymptotically optimal in its dependence on $\varepsilon$, matching the $Ω(\log(1/\varepsilon))$ lower bound of [SODA'25] under the 1-vs-2-cycles conjecture. Our result also answers an open question raised by [STOC'23].

A Simple Subquadratic Cut Query Protocol for Directed Reachability

from arXiv: Data Structures and Algorithms

Authors: Henry Fleischmann, Sagnik Mukhopadhyay, Swagato Sanyal

We study $s$--$t$ reachability in an unweighted directed graph on $n$ vertices, accessed through a directed cut oracle: given $S\subseteq V$, the oracle returns the number of edges directed from $S$ to $V\setminus S$. Directed cut functions are submodular, and reachability asks whether the minimum $s$--$t$ cut has positive value. This connects the problem to the query complexity of submodular function minimisation. Whether or not reachability can be solved in truly subquadratic query complexity has been an outstanding open question for a decade. We give a simple algorithmic blueprint with two randomised implementations. The first uses $\widetilde O(n^{9/5})$ outgoing-cut queries; the second improves this bound to $\widetilde O(n^{5/3})$. The common idea is to construct an alternative orientation with the same underlying undirected graph and vertex-wise degree imbalances, thereby preserving reachability. We first approximate the target imbalances using an averaging and rounding procedure, then correct the remaining error with a flow. An earlier manuscript, made available in August 2026 in an online AI repository, reports an $\widetilde O(n^{3/2})$ query bound. Our contribution is a simple and comprehensible alternative approach that already achieves a polynomial saving over quadratic query complexity.

Authors: Henry Fleischmann, Sagnik Mukhopadhyay, Swagato Sanyal

We study $s$--$t$ reachability in an unweighted directed graph on $n$ vertices, accessed through a directed cut oracle: given $S\subseteq V$, the oracle returns the number of edges directed from $S$ to $V\setminus S$. Directed cut functions are submodular, and reachability asks whether the minimum $s$--$t$ cut has positive value. This connects the problem to the query complexity of submodular function minimisation. Whether or not reachability can be solved in truly subquadratic query complexity has been an outstanding open question for a decade. We give a simple algorithmic blueprint with two randomised implementations. The first uses $\widetilde O(n^{9/5})$ outgoing-cut queries; the second improves this bound to $\widetilde O(n^{5/3})$. The common idea is to construct an alternative orientation with the same underlying undirected graph and vertex-wise degree imbalances, thereby preserving reachability. We first approximate the target imbalances using an averaging and rounding procedure, then correct the remaining error with a flow. An earlier manuscript, made available in August 2026 in an online AI repository, reports an $\widetilde O(n^{3/2})$ query bound. Our contribution is a simple and comprehensible alternative approach that already achieves a polynomial saving over quadratic query complexity.

Online Edge-Weighted Matching Beyond 0.55

from arXiv: Data Structures and Algorithms

Authors: Alireza Farhadi, Mohammad Hajiaghayi

We give a $0.556195$-competitive randomized algorithm for edge-weighted online bipartite matching with free disposal, improving the $0.5368$ guarantee of Blanc and Charikar. The result holds for arbitrary nonnegative edge weights and any input sequence fixed independently of the algorithm's randomness. Our algorithm builds on online correlated selection, introduced by Fahrbach, Huang, Tao, and Zadimoghaddam, and its multiway extension due to Blanc and Charikar. Our main contribution is a state-dependent primal--dual analysis of correlated selection. At each weight threshold, we track the state of the selection process jointly with an auxiliary failure event. The analysis accounts for arrivals below the threshold, which change the state without improving the matching value at that threshold. A potential function bounds the dual increase in both cases. We instantiate the analysis with an explicit win map that assigns a common score to an initial range of states. Its selection inequalities are verified by a finite computer-assisted proof covering every required candidate multiplicity and every real threshold; the verifiers are available in an accompanying repository. Triangular unit-weight inputs asymptotically attain the guarantee of the underlying ideal algorithm, proving that its analysis is tight.

Authors: Alireza Farhadi, Mohammad Hajiaghayi

We give a $0.556195$-competitive randomized algorithm for edge-weighted online bipartite matching with free disposal, improving the $0.5368$ guarantee of Blanc and Charikar. The result holds for arbitrary nonnegative edge weights and any input sequence fixed independently of the algorithm's randomness. Our algorithm builds on online correlated selection, introduced by Fahrbach, Huang, Tao, and Zadimoghaddam, and its multiway extension due to Blanc and Charikar. Our main contribution is a state-dependent primal--dual analysis of correlated selection. At each weight threshold, we track the state of the selection process jointly with an auxiliary failure event. The analysis accounts for arrivals below the threshold, which change the state without improving the matching value at that threshold. A potential function bounds the dual increase in both cases. We instantiate the analysis with an explicit win map that assigns a common score to an initial range of states. Its selection inequalities are verified by a finite computer-assisted proof covering every required candidate multiplicity and every real threshold; the verifiers are available in an accompanying repository. Triangular unit-weight inputs asymptotically attain the guarantee of the underlying ideal algorithm, proving that its analysis is tight.

Nonuniform Buy-at-Bulk Network Design: Improved Approximation and LP Integrality Gap

from arXiv: Data Structures and Algorithms

Authors: Iman Gholami, MohammadTaghi Hajiaghayi

In nonuniform buy-at-bulk network design, we are given an undirected graph and $h$ terminal pairs with positive demands, and we must route every demand at minimum total cost when each network resource has two independent parameters: a fixed buying price paid once if the resource is used, and a per-unit rental cost (or physical length) paid proportionally to the traffic it carries. Unlike the uniform-cost or single-sink variants, the general multicommodity problem allows unrelated buying and rental metrics across arbitrary source--sink pairs, arbitrary demands, and both edge and node resources. We give a deterministic polynomial-time $O(\log^2 h / (\log\log h)^2)$-approximation algorithm for undirected nonuniform buy-at-bulk network design. The guarantee holds for edge costs, node costs, and their combination, and applies to arbitrary positive rational demands with no dependence on the number of vertices or the numerical demand ratio. This improves upon the $O(\log^4 h)$ bound of Chekuri, Hajiaghayi, Kortsarz, and Salavatipour (2010) for unrestricted instances. Our result also establishes the same $O(\log^2 h / (\log\log h)^2)$ upper bound on the integrality gap of the natural affine multicommodity flow LP, bounding integral buying and rental costs separately against their fractional counterparts and preserving zero rental budgets exactly.

Authors: Iman Gholami, MohammadTaghi Hajiaghayi

In nonuniform buy-at-bulk network design, we are given an undirected graph and $h$ terminal pairs with positive demands, and we must route every demand at minimum total cost when each network resource has two independent parameters: a fixed buying price paid once if the resource is used, and a per-unit rental cost (or physical length) paid proportionally to the traffic it carries. Unlike the uniform-cost or single-sink variants, the general multicommodity problem allows unrelated buying and rental metrics across arbitrary source--sink pairs, arbitrary demands, and both edge and node resources. We give a deterministic polynomial-time $O(\log^2 h / (\log\log h)^2)$-approximation algorithm for undirected nonuniform buy-at-bulk network design. The guarantee holds for edge costs, node costs, and their combination, and applies to arbitrary positive rational demands with no dependence on the number of vertices or the numerical demand ratio. This improves upon the $O(\log^4 h)$ bound of Chekuri, Hajiaghayi, Kortsarz, and Salavatipour (2010) for unrestricted instances. Our result also establishes the same $O(\log^2 h / (\log\log h)^2)$ upper bound on the integrality gap of the natural affine multicommodity flow LP, bounding integral buying and rental costs separately against their fractional counterparts and preserving zero rental budgets exactly.

Prophet Secretary for Matroids and XOS Valuations: A Strict Separation at $1-1/e$

from arXiv: Data Structures and Algorithms

Authors: Hossein Esfandiari, MohammadTaghi Hajiaghayi

The matroid prophet-secretary problem asks an online algorithm to select an independent set of elements whose nonnegative values are drawn independently and revealed in uniformly random order. Its combinatorial-auction counterpart asks the algorithm to allocate disjoint bundles to arriving buyers with independently drawn XOS valuation functions. In both problems the distributions are known, and the benchmark is the expected offline optimum. Ehsani, Hajiaghayi, Kesselheim, and Singla (SODA 2018) give $(1-1/e)$-competitive algorithms for both settings. In this work we show that these two guarantees have different limits. For every matroid, we give a randomized $(1-1/e+\varepsilon_0)$-competitive algorithm, where $\varepsilon_0=10^{-6}$. The guarantee holds for independent nonnegative values with finite expected optimum. For explicitly given finite distributions with rational values and probabilities, the algorithm has polynomial worst-case running time in the matroid independence-oracle model. The oracle tests whether a queried set is independent. The algorithm makes $O(N^4)$ such queries in expectation and $O(N^4\log(N+2))$ in the worst case, where $N$ is the number of elements. For combinatorial auctions with XOS valuations, however, we show that $1-1/e$ is the optimal competitive ratio. We give an explicit family of finite instances showing that no online algorithm can guarantee a larger competitive ratio than $1-1/e$, even when the buyers' valuations are independent and identically distributed, every additive clause has coefficients in $\{0,1\}$, and computation is unrestricted. Thus the matroid improvement exploits structure absent from general allocation with XOS valuations.

Authors: Hossein Esfandiari, MohammadTaghi Hajiaghayi

The matroid prophet-secretary problem asks an online algorithm to select an independent set of elements whose nonnegative values are drawn independently and revealed in uniformly random order. Its combinatorial-auction counterpart asks the algorithm to allocate disjoint bundles to arriving buyers with independently drawn XOS valuation functions. In both problems the distributions are known, and the benchmark is the expected offline optimum. Ehsani, Hajiaghayi, Kesselheim, and Singla (SODA 2018) give $(1-1/e)$-competitive algorithms for both settings. In this work we show that these two guarantees have different limits. For every matroid, we give a randomized $(1-1/e+\varepsilon_0)$-competitive algorithm, where $\varepsilon_0=10^{-6}$. The guarantee holds for independent nonnegative values with finite expected optimum. For explicitly given finite distributions with rational values and probabilities, the algorithm has polynomial worst-case running time in the matroid independence-oracle model. The oracle tests whether a queried set is independent. The algorithm makes $O(N^4)$ such queries in expectation and $O(N^4\log(N+2))$ in the worst case, where $N$ is the number of elements. For combinatorial auctions with XOS valuations, however, we show that $1-1/e$ is the optimal competitive ratio. We give an explicit family of finite instances showing that no online algorithm can guarantee a larger competitive ratio than $1-1/e$, even when the buyers' valuations are independent and identically distributed, every additive clause has coefficients in $\{0,1\}$, and computation is unrestricted. Thus the matroid improvement exploits structure absent from general allocation with XOS valuations.

Deterministic Fault-Tolerant Connectivity Oracles with Subquadratic Space

from arXiv: Data Structures and Algorithms

Authors: Merav Parter, Elad Tzalik

A fault-tolerant (FT) connectivity oracle for an $n$-vertex graph $G=(V,E)$ determines whether $G- F$ is connected for any query set $F\subseteq E$. When the number of failures is unrestricted, prior constructions exhibited a sharp contrast between randomization and determinism: randomized oracles use only $\widetilde{O}(n)$ space [Duan and Pettie, SICOMP 2020], whereas no deterministic construction was known to improve upon storing $G$ explicitly, requiring $Θ(n^2)$ bits in the worst case. The promise that $F\subseteq E$ is essential for subquadratic space: allowing arbitrary queries $F\subseteq V\times V$ forces quadratic space. For queries containing at most $f$ failures, Khanna, Konrad, and Putterman [SODA 2027] recently constructed deterministic oracles using $\widetilde{O}(\sqrt{f}n)$ space that also report approximate distances with polylogarithmic stretch. Their result improves upon the classical $O(fn)$ space bound for deterministic connectivity oracles [Pătraşcu and Thorup, FOCS 2007], but offers no space savings over storing $G$ when $f=Θ(n^2)$. We present the first deterministic connectivity oracles with subquadratic space for failure sets of unrestricted size, using $\widetilde{O}(n^{7/4})$ space. We also establish the first polynomial separation between deterministic and randomized connectivity oracles, proving that every deterministic oracle requires $\widetildeΩ(n^{3/2})$ bits in the worst case. For queries containing at most $f$ failures, we obtain deterministic connectivity oracles using $\widetilde{O}(f^{2/5}n)$ space, improving the dependence on $f$ in the bound of Khanna et al., and prove a lower bound of $\widetildeΩ(f^{1/4}n)$ bits.

Authors: Merav Parter, Elad Tzalik

A fault-tolerant (FT) connectivity oracle for an $n$-vertex graph $G=(V,E)$ determines whether $G- F$ is connected for any query set $F\subseteq E$. When the number of failures is unrestricted, prior constructions exhibited a sharp contrast between randomization and determinism: randomized oracles use only $\widetilde{O}(n)$ space [Duan and Pettie, SICOMP 2020], whereas no deterministic construction was known to improve upon storing $G$ explicitly, requiring $Θ(n^2)$ bits in the worst case. The promise that $F\subseteq E$ is essential for subquadratic space: allowing arbitrary queries $F\subseteq V\times V$ forces quadratic space. For queries containing at most $f$ failures, Khanna, Konrad, and Putterman [SODA 2027] recently constructed deterministic oracles using $\widetilde{O}(\sqrt{f}n)$ space that also report approximate distances with polylogarithmic stretch. Their result improves upon the classical $O(fn)$ space bound for deterministic connectivity oracles [Pătraşcu and Thorup, FOCS 2007], but offers no space savings over storing $G$ when $f=Θ(n^2)$. We present the first deterministic connectivity oracles with subquadratic space for failure sets of unrestricted size, using $\widetilde{O}(n^{7/4})$ space. We also establish the first polynomial separation between deterministic and randomized connectivity oracles, proving that every deterministic oracle requires $\widetildeΩ(n^{3/2})$ bits in the worst case. For queries containing at most $f$ failures, we obtain deterministic connectivity oracles using $\widetilde{O}(f^{2/5}n)$ space, improving the dependence on $f$ in the bound of Khanna et al., and prove a lower bound of $\widetildeΩ(f^{1/4}n)$ bits.

The 3-Uniform Hyperclique Hypothesis is False

from arXiv: Data Structures and Algorithms

Authors: Timothy M. Chan, Reina Itakura, Zhiyuan Ma, Yuancheng Yu

We present truly subquartic algorithms for detecting or counting 4-hypercliques in a 3-uniform hypergraph with $n$ vertices, disproving a popular hypothesis in fine-grained complexity. The result immediately implies a faster algorithm for finding a $k$-hyperclique in a 3-uniform hypergraph for all constant $k\ge 4$, and a faster exponential-time algorithm for Max-3-SAT.

Authors: Timothy M. Chan, Reina Itakura, Zhiyuan Ma, Yuancheng Yu

We present truly subquartic algorithms for detecting or counting 4-hypercliques in a 3-uniform hypergraph with $n$ vertices, disproving a popular hypothesis in fine-grained complexity. The result immediately implies a faster algorithm for finding a $k$-hyperclique in a 3-uniform hypergraph for all constant $k\ge 4$, and a faster exponential-time algorithm for Max-3-SAT.

Monadically Dependent Graph Classes Have Almost Bounded Merge-Width

from arXiv: Data Structures and Algorithms

Authors: Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk

A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.

Authors: Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk

A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.

Finding full immersions is fixed-parameter tractable in labeled tour graphs

from arXiv: Data Structures and Algorithms

Authors: Hadas Barabash, Jim Geelen, Rose McCarty, Caleb McFarland, Paul Wollan

We give an $O_H(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $H$. A labeled tour graph is a connected $4$-regular graph whose edges are labeled by binary vectors of fixed length. To obtain a full immersion, a vertex may be removed only by pairing up its four incident edges and "splitting off" both pairs, summing the labels within each pair. Even for one-bit labels, no polynomial-time algorithm was previously known for arbitrary fixed $H$. Our main tool is of independent interest: we give an algorithmic min-max theorem that, in particular, characterizes the maximum number of odd-length circuits in a rooted circuit decomposition of an Eulerian graph. Our results have implications for vertex-minors, whose study is motivated by applications in quantum networking, connections with graph minor theory, and the goal of developing an irrelevant vertex technique for dense graphs. For vertex-minors, circle graphs play the same role as planar graphs do for graph minors. As a corollary of our main theorem, we obtain a fixed-parameter tractable algorithm for apex-vertex-minor testing in $k$-apex circle graphs. We also prove an irrelevant vertex theorem for such graphs of large rank-width. More broadly, our results are essential to forthcoming work of three of the authors proving a structure theorem for graphs with a forbidden vertex-minor. There, our min-max theorem plays the role of both the algorithmic $A$-paths and Two Paths Theorems.

Authors: Hadas Barabash, Jim Geelen, Rose McCarty, Caleb McFarland, Paul Wollan

We give an $O_H(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $H$. A labeled tour graph is a connected $4$-regular graph whose edges are labeled by binary vectors of fixed length. To obtain a full immersion, a vertex may be removed only by pairing up its four incident edges and "splitting off" both pairs, summing the labels within each pair. Even for one-bit labels, no polynomial-time algorithm was previously known for arbitrary fixed $H$. Our main tool is of independent interest: we give an algorithmic min-max theorem that, in particular, characterizes the maximum number of odd-length circuits in a rooted circuit decomposition of an Eulerian graph. Our results have implications for vertex-minors, whose study is motivated by applications in quantum networking, connections with graph minor theory, and the goal of developing an irrelevant vertex technique for dense graphs. For vertex-minors, circle graphs play the same role as planar graphs do for graph minors. As a corollary of our main theorem, we obtain a fixed-parameter tractable algorithm for apex-vertex-minor testing in $k$-apex circle graphs. We also prove an irrelevant vertex theorem for such graphs of large rank-width. More broadly, our results are essential to forthcoming work of three of the authors proving a structure theorem for graphs with a forbidden vertex-minor. There, our min-max theorem plays the role of both the algorithmic $A$-paths and Two Paths Theorems.

Approximation and Hardness for the Data Validation Problem

from arXiv: Data Structures and Algorithms

Authors: Antoine Tinguely, G. Wessel van der Heijden

Finnah, Gönsch and Otto [Eur. J. Oper. Res. 2025] recently introduced the Data Validation Problem (DVP): Given a transitive directed acyclic graph $G$ on $n$ vertices (resp. tasks) and $m$ edges (resp. precedences) and a probing budget, a policy can successively pay a probing fee and probe an edge in $G$ as long as the total probing fees do not exceed the budget. If a probe succeeds, the targeted edge $e$ is removed from $G$ and we receive a reward; if the probe fails, the edge stays in $G$ and cannot be probed again. We seek a policy with maximum expected cumulative reward. Crucially, $G$ must stay transitive throughout the probing process, so a policy can only challenge the precedence between consecutive tasks, that is, only probe edges in the transitive reduction of the current precedence graph. We show that DVP is strongly NP-hard and W[1]-hard with the probing budget $T$ as a parameter, even in the deterministic case (when every probe has probability $0$ or $1$ to succeed) and with unit probing costs and rewards. For the same setting, we give an approximation-preserving reduction from the Densest $k$-Subgraph (D$k$S) Problem, with the consequence that there is no polynomial-time $2^{(\log n)^{1-δ}}$-approximation algorithm and no FPT $\sqrt{T}$-approximation algorithm for DVP, assuming the Almost-Exponential Time Hypothesis and the Strongish Planted Clique Hypothesis, respectively. We also present the first known policies with provable worst-case guarantees for DVP, namely an $m$-approximation algorithm for instances with uniform probabilities and probing costs, a $2T/p_{\min}^T$-approximation algorithm for general instances, where $p_{\min}$ is the minimum probing success probability, and an exact FPT algorithm on tournament graphs, in the deterministic setting.

Authors: Antoine Tinguely, G. Wessel van der Heijden

Finnah, Gönsch and Otto [Eur. J. Oper. Res. 2025] recently introduced the Data Validation Problem (DVP): Given a transitive directed acyclic graph $G$ on $n$ vertices (resp. tasks) and $m$ edges (resp. precedences) and a probing budget, a policy can successively pay a probing fee and probe an edge in $G$ as long as the total probing fees do not exceed the budget. If a probe succeeds, the targeted edge $e$ is removed from $G$ and we receive a reward; if the probe fails, the edge stays in $G$ and cannot be probed again. We seek a policy with maximum expected cumulative reward. Crucially, $G$ must stay transitive throughout the probing process, so a policy can only challenge the precedence between consecutive tasks, that is, only probe edges in the transitive reduction of the current precedence graph. We show that DVP is strongly NP-hard and W[1]-hard with the probing budget $T$ as a parameter, even in the deterministic case (when every probe has probability $0$ or $1$ to succeed) and with unit probing costs and rewards. For the same setting, we give an approximation-preserving reduction from the Densest $k$-Subgraph (D$k$S) Problem, with the consequence that there is no polynomial-time $2^{(\log n)^{1-δ}}$-approximation algorithm and no FPT $\sqrt{T}$-approximation algorithm for DVP, assuming the Almost-Exponential Time Hypothesis and the Strongish Planted Clique Hypothesis, respectively. We also present the first known policies with provable worst-case guarantees for DVP, namely an $m$-approximation algorithm for instances with uniform probabilities and probing costs, a $2T/p_{\min}^T$-approximation algorithm for general instances, where $p_{\min}$ is the minimum probing success probability, and an exact FPT algorithm on tournament graphs, in the deterministic setting.

A Quantum Lower Bound for Triangle Finding

from arXiv: Data Structures and Algorithms

Authors: Arjan Cornelissen, Amin Shiraz Gilani, Xingyu Zhou

Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.

Authors: Arjan Cornelissen, Amin Shiraz Gilani, Xingyu Zhou

Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.

Generalized Sorting via Four-Cycles

from arXiv: Data Structures and Algorithms

Authors: Soheil Behnezhad, Shane Ferrante

We study the generalized sorting problem where only a subset of the pairs can be compared. Equivalently, we are given an $n$-vertex $m$-edge graph $G=(V, E)$ where the vertex set $V$ is to be sorted, and the edge set $E$ indicates possible comparisons. The goal is to sort $V$ with as few (adaptive) queries to $E$ as possible. When $G$ is the complete graph, the problem is equivalent to standard sorting for which $Θ(n \log n)$ queries are sufficient and necessary. All classical sorting algorithms crucially rely on the ability to compare any pair and do not imply an $o(n^2)$-query algorithm in the general case. This was improved 15 years ago by Huang, Kannan, and Khanna [FOCS'11] who achieved a bound of $\widetilde{O}(n\sqrt{n})$. Kuszmaul and Narayanan [FOCS'21] later gave a bound of $\widetilde{O}(\sqrt{m n})$, which is more efficient for sparse graphs, but matches the $\widetilde{O}(n\sqrt{n})$ barrier for dense graphs. In this paper, we present an algorithm for generalized sorting that makes at most $\widetilde{O}(n^{4/3})$ queries, therefore breaking the longstanding $\widetilde{O}(n\sqrt{n})$ barrier for all graphs. The key to our result is to query edges that belong to many (specific types of) four-cycles. We show that querying such edges allows the algorithm to verify the direction of many other edges (specifically, their opposing edges in the four-cycles) and thus achieve a much better amortized query cost.

Authors: Soheil Behnezhad, Shane Ferrante

We study the generalized sorting problem where only a subset of the pairs can be compared. Equivalently, we are given an $n$-vertex $m$-edge graph $G=(V, E)$ where the vertex set $V$ is to be sorted, and the edge set $E$ indicates possible comparisons. The goal is to sort $V$ with as few (adaptive) queries to $E$ as possible. When $G$ is the complete graph, the problem is equivalent to standard sorting for which $Θ(n \log n)$ queries are sufficient and necessary. All classical sorting algorithms crucially rely on the ability to compare any pair and do not imply an $o(n^2)$-query algorithm in the general case. This was improved 15 years ago by Huang, Kannan, and Khanna [FOCS'11] who achieved a bound of $\widetilde{O}(n\sqrt{n})$. Kuszmaul and Narayanan [FOCS'21] later gave a bound of $\widetilde{O}(\sqrt{m n})$, which is more efficient for sparse graphs, but matches the $\widetilde{O}(n\sqrt{n})$ barrier for dense graphs. In this paper, we present an algorithm for generalized sorting that makes at most $\widetilde{O}(n^{4/3})$ queries, therefore breaking the longstanding $\widetilde{O}(n\sqrt{n})$ barrier for all graphs. The key to our result is to query edges that belong to many (specific types of) four-cycles. We show that querying such edges allows the algorithm to verify the direction of many other edges (specifically, their opposing edges in the four-cycles) and thus achieve a much better amortized query cost.

Combinatorial Boolean Matrix Multiplication in $n^3 / 2^{Ω(\sqrt{\log n})}$ Time

from arXiv: Data Structures and Algorithms

Authors: Piaoyang Cui, Yiming Liu, David P. Woodruff

For over fifty years, combinatorial algorithms for Boolean matrix multiplication (BMM) and triangle detection saved only polylogarithmic factors over cubic time. At STOC 2024, Abboud, Fischer, Kelley, Lovett, and Meka broke this barrier with a deterministic $O(n^3 / 2^{Ω((\log n)^{1/7})})$-time algorithm, and asked whether $n^3 / 2^{Ω(\sqrt{\log n})}$ time is achievable. We answer this question in the affirmative, giving deterministic combinatorial algorithms for both BMM and triangle detection running in $O(n^3 / 2^{Ω(\sqrt{\log n})})$ time. Following Abboud et al., we decompose $(A,B)$ into pairs of submatrices $(A_k,B_k)$, called \emph{pieces}, each of which is either sparse or pseudorandom in a sense that forces $A_k\star B_k$ to be nearly all-ones on its vertex sets. Whereas their decomposition produces $2^{O(d^7)}$ pieces at regularity parameter $d$, we obtain $2^{O(d^2)}$ pieces, directly improving the runtime exponent from $(\log n)^{1/7}$ to $\sqrt{\log n}$. Prior work found pseudorandom pieces by repeatedly restricting to denser submatrices, paying worst-case vertex losses at each step that compounded across nested recursion levels, together with an expensive derandomization step. We eliminate the extra powers of $d$ by matching the vertex loss at each step to the size of its density jump so losses telescope across scales, derandomizing witness search via $L_d$-norm estimation so the sampler length stays $2^{O(d)}$, and bounding overlap across the recursion with a fractional density-weighted volume budget rather than raw submatrix volume.

Authors: Piaoyang Cui, Yiming Liu, David P. Woodruff

For over fifty years, combinatorial algorithms for Boolean matrix multiplication (BMM) and triangle detection saved only polylogarithmic factors over cubic time. At STOC 2024, Abboud, Fischer, Kelley, Lovett, and Meka broke this barrier with a deterministic $O(n^3 / 2^{Ω((\log n)^{1/7})})$-time algorithm, and asked whether $n^3 / 2^{Ω(\sqrt{\log n})}$ time is achievable. We answer this question in the affirmative, giving deterministic combinatorial algorithms for both BMM and triangle detection running in $O(n^3 / 2^{Ω(\sqrt{\log n})})$ time. Following Abboud et al., we decompose $(A,B)$ into pairs of submatrices $(A_k,B_k)$, called \emph{pieces}, each of which is either sparse or pseudorandom in a sense that forces $A_k\star B_k$ to be nearly all-ones on its vertex sets. Whereas their decomposition produces $2^{O(d^7)}$ pieces at regularity parameter $d$, we obtain $2^{O(d^2)}$ pieces, directly improving the runtime exponent from $(\log n)^{1/7}$ to $\sqrt{\log n}$. Prior work found pseudorandom pieces by repeatedly restricting to denser submatrices, paying worst-case vertex losses at each step that compounded across nested recursion levels, together with an expensive derandomization step. We eliminate the extra powers of $d$ by matching the vertex loss at each step to the size of its density jump so losses telescope across scales, derandomizing witness search via $L_d$-norm estimation so the sampler length stays $2^{O(d)}$, and bounding overlap across the recursion with a fractional density-weighted volume budget rather than raw submatrix volume.

The Online $k$-Taxi Problem Is Competitive on All Metrics

from arXiv: Data Structures and Algorithms

Authors: Christian Coester, Yichen Huang, Tze-Yang Poon

The online $k$-taxi problem is a natural generalization of the $k$-server problem: $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of points $(r,s)$, and the algorithm serves the request by first moving a taxi to $r$ (to pick up a passenger) and then to $s$ (to drop off the passenger). The cost is the total distance traveled by the taxis without a passenger. Previous bounds for general $k$ all depended on the number of points $n$ or the aspect ratio $Δ$ of the metric space, both of which can be infinite on general metrics, and competitive algorithms were previously known on general metrics only for $k \le 3$. In this work, we give a $9^{k-1}$-competitive randomized algorithm against adaptive online adversaries and, as a corollary, an $81^{k-1}$-competitive deterministic algorithm for the online $k$-taxi problem.

Authors: Christian Coester, Yichen Huang, Tze-Yang Poon

The online $k$-taxi problem is a natural generalization of the $k$-server problem: $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of points $(r,s)$, and the algorithm serves the request by first moving a taxi to $r$ (to pick up a passenger) and then to $s$ (to drop off the passenger). The cost is the total distance traveled by the taxis without a passenger. Previous bounds for general $k$ all depended on the number of points $n$ or the aspect ratio $Δ$ of the metric space, both of which can be infinite on general metrics, and competitive algorithms were previously known on general metrics only for $k \le 3$. In this work, we give a $9^{k-1}$-competitive randomized algorithm against adaptive online adversaries and, as a corollary, an $81^{k-1}$-competitive deterministic algorithm for the online $k$-taxi problem.

Polynomial-Time Differentially Private Synthetic Graph Release with Nearly-Linear Additive Error

from arXiv: Data Structures and Algorithms

Authors: Marek Eliáš, Gautam Kamath, Janardhan Kulkarni, George Z. Li, Jingcheng Liu, Sepideh Mahabadi, Pan Peng, Jakub Tarnawski, Hangyu Xu, Zongrui Zou

We give a polynomial-time $(\varepsilon, δ)$-edge-differentially private algorithm that releases a synthetic graph approximately preserving all cut values of a nonnegative weighted input graph $G$. Our algorithm outputs a nonnegative weighted graph $H$ such that, with probability at least $1-β$, simultaneously for all $S\subseteq V$, \[ \left|w_G(S,V\setminus S)-w_H(S,V\setminus S)\right| \leq γw_G(S,V\setminus S) + O\left(\frac{n}{γ\varepsilon} \log^7 \frac{n}{δβγ\varepsilon}\right). \] For fixed $γ, \varepsilon$ and $δ= β= n^{-10}$, the additive error is $n \operatorname{polylog} n$, improving upon the previous efficient bound of $n^{13/12 + o(1)}$. Furthermore, we establish an $Ω(n/γ\varepsilon)$ lower bound for the relevant parameter regime, showing that our result is tight up to polylogarithmic factors. Our algorithm uses a log-det-regularized SDP to generate fractional queries for multiplicative-weights updates and certify accuracy for all cuts. The privacy analysis charges privacy loss only fractionally per mechanism call, based on the amount of mass on edge coordinates retired from subsequent iterations. Our synthetic graph release yields improved error bounds for several downstream tasks in private graph analysis including correlation clustering, Max-Cut, Max-$k$-Cut, constrained cut problems and hierarchical clustering. Most notably, for unweighted complete-graph correlation clustering, we achieve the optimal $\widetilde O(n)$ additive error while reducing the best previously known approximation ratio for this additive error from several thousand to $1.387$. We also provide a formalization of our results in Lean.

Authors: Marek Eliáš, Gautam Kamath, Janardhan Kulkarni, George Z. Li, Jingcheng Liu, Sepideh Mahabadi, Pan Peng, Jakub Tarnawski, Hangyu Xu, Zongrui Zou

We give a polynomial-time $(\varepsilon, δ)$-edge-differentially private algorithm that releases a synthetic graph approximately preserving all cut values of a nonnegative weighted input graph $G$. Our algorithm outputs a nonnegative weighted graph $H$ such that, with probability at least $1-β$, simultaneously for all $S\subseteq V$, \[ \left|w_G(S,V\setminus S)-w_H(S,V\setminus S)\right| \leq γw_G(S,V\setminus S) + O\left(\frac{n}{γ\varepsilon} \log^7 \frac{n}{δβγ\varepsilon}\right). \] For fixed $γ, \varepsilon$ and $δ= β= n^{-10}$, the additive error is $n \operatorname{polylog} n$, improving upon the previous efficient bound of $n^{13/12 + o(1)}$. Furthermore, we establish an $Ω(n/γ\varepsilon)$ lower bound for the relevant parameter regime, showing that our result is tight up to polylogarithmic factors. Our algorithm uses a log-det-regularized SDP to generate fractional queries for multiplicative-weights updates and certify accuracy for all cuts. The privacy analysis charges privacy loss only fractionally per mechanism call, based on the amount of mass on edge coordinates retired from subsequent iterations. Our synthetic graph release yields improved error bounds for several downstream tasks in private graph analysis including correlation clustering, Max-Cut, Max-$k$-Cut, constrained cut problems and hierarchical clustering. Most notably, for unweighted complete-graph correlation clustering, we achieve the optimal $\widetilde O(n)$ additive error while reducing the best previously known approximation ratio for this additive error from several thousand to $1.387$. We also provide a formalization of our results in Lean.

Approximating #BIS in $\exp(\widetilde O(n^{1/3}))$ Time

from arXiv: Data Structures and Algorithms

Authors: Jingcheng Liu, Yitong Yin

We give randomized algorithms for approximately counting and nearly uniformly sampling independent sets in arbitrary bipartite graphs (#BIS). At fixed accuracy, both algorithms run in $\exp(O(n^{1/3}\log^{4/3}n))$ time on $n$-vertex graphs. More generally, the same running-time dependence on the number of variables holds for approximate counting and sampling with efficiently evaluable positive log-supermodular weights on Boolean variables. A weighted Sinclair-Jerrum walk on the self-reduction tree reduces these tasks to coarse lower estimates of residual log-partition functions. We obtain these estimates by optimizing a concave relaxation of an exact variational principle on the tree; threshold smoothing certifies that the low-degree search suffices.

Authors: Jingcheng Liu, Yitong Yin

We give randomized algorithms for approximately counting and nearly uniformly sampling independent sets in arbitrary bipartite graphs (#BIS). At fixed accuracy, both algorithms run in $\exp(O(n^{1/3}\log^{4/3}n))$ time on $n$-vertex graphs. More generally, the same running-time dependence on the number of variables holds for approximate counting and sampling with efficiently evaluable positive log-supermodular weights on Boolean variables. A weighted Sinclair-Jerrum walk on the self-reduction tree reduces these tasks to coarse lower estimates of residual log-partition functions. We obtain these estimates by optimizing a concave relaxation of an exact variational principle on the tree; threshold smoothing certifies that the low-degree search suffices.

Deterministic Connectivity Sketches Under Edge Deletions

from arXiv: Data Structures and Algorithms

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

How much information about a graph is needed to determine its connectivity after arbitrary valid edge deletions? This question is at the heart of designing deterministic dynamic streaming algorithms for graph connectivity. Towards answering this question, we study the deterministic one-way communication problem in which Alice holds a graph $G_A = (V, E_A)$ on $n$ vertices, Bob holds a graph $G_B = (V, E_B)$ with $E_B \subseteq E_A$, and Alice sends a message $M \in \{0,1\}^m$ that allows Bob to decide whether $G_A - G_B$ is connected. Randomized sketches solve this problem using $\tilde O(n)$ bits, whereas deterministic recovery of even one surviving edge requires $Ω(n^2)$ bits. We prove an upper bound of $m = \widetilde{O}(n^{7/4})$ bits and a lower bound of $m = \widetildeΩ(n^{3/2})$ bits for deciding connectivity. Thus, connectivity admits a subquadratic deterministic representation despite the quadratic cost of edge recovery, but remains polynomially harder than its randomized counterpart. The upper bound immediately yields an $\widetilde{O}(n^{7/4})$-bit deterministic $n^2$-fault tolerant connectivity oracle, which reports the connected components of a graph $G$ after any valid set of edge deletions. We also obtain an $\tilde Ω(n f^{1/4})$-bit lower bound for deterministic $f$-fault tolerant connectivity oracles for every $f \in [1,n^2]$. Finally, we show that the separation between connectivity and edge recovery persists in deterministic single-pass dynamic streams with bounded update multiplicity. For every fixed $ε\in (0,1/3)$, our algorithm reports the connected components of the final simple graph using $\widetilde{O}(n^{15/8 + 3ε/8})$ bits (subquadratic space), provided each edge undergoes at most $n^ε$ insertion or deletion updates in total.

Authors: Sanjeev Khanna, Aaron Putterman, Madhu Sudan

How much information about a graph is needed to determine its connectivity after arbitrary valid edge deletions? This question is at the heart of designing deterministic dynamic streaming algorithms for graph connectivity. Towards answering this question, we study the deterministic one-way communication problem in which Alice holds a graph $G_A = (V, E_A)$ on $n$ vertices, Bob holds a graph $G_B = (V, E_B)$ with $E_B \subseteq E_A$, and Alice sends a message $M \in \{0,1\}^m$ that allows Bob to decide whether $G_A - G_B$ is connected. Randomized sketches solve this problem using $\tilde O(n)$ bits, whereas deterministic recovery of even one surviving edge requires $Ω(n^2)$ bits. We prove an upper bound of $m = \widetilde{O}(n^{7/4})$ bits and a lower bound of $m = \widetildeΩ(n^{3/2})$ bits for deciding connectivity. Thus, connectivity admits a subquadratic deterministic representation despite the quadratic cost of edge recovery, but remains polynomially harder than its randomized counterpart. The upper bound immediately yields an $\widetilde{O}(n^{7/4})$-bit deterministic $n^2$-fault tolerant connectivity oracle, which reports the connected components of a graph $G$ after any valid set of edge deletions. We also obtain an $\tilde Ω(n f^{1/4})$-bit lower bound for deterministic $f$-fault tolerant connectivity oracles for every $f \in [1,n^2]$. Finally, we show that the separation between connectivity and edge recovery persists in deterministic single-pass dynamic streams with bounded update multiplicity. For every fixed $ε\in (0,1/3)$, our algorithm reports the connected components of the final simple graph using $\widetilde{O}(n^{15/8 + 3ε/8})$ bits (subquadratic space), provided each edge undergoes at most $n^ε$ insertion or deletion updates in total.

Dimension-Free Private Mean Estimation is Impossible Without the Covariance

from arXiv: Data Structures and Algorithms

Authors: Renaud Gaucher, Ephraim Linder, Adam Smith, Jonathan Ullman, Lydia Zakynthinou

We study the limits of differentially private Gaussian mean estimation when the covariance matrix is unknown. Prior work shows that when the covariance $Σ$ is known, the optimal sample complexity can depend only on its spectrum, rather than on the ambient dimension $d$. We establish new lower bounds for Gaussian mean estimation with Euclidean error which show that private estimators cannot achieve similar guarantees when the covariance is unknown. First, when the sample size $n$ is smaller than the effective dimension $\mathrm{trace}(Σ)/\|Σ\|_{\text{op}}$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d/n}$. Second, when the sample size is larger than the effective dimension, even for diagonal covariance matrices with effective dimension bounded by $\log d$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d}/n^{3/2}$. Together, these results show that in the unknown covariance setting, exploiting anisotropic structure under differential privacy can require a dimension-dependent error or sample complexity.

Authors: Renaud Gaucher, Ephraim Linder, Adam Smith, Jonathan Ullman, Lydia Zakynthinou

We study the limits of differentially private Gaussian mean estimation when the covariance matrix is unknown. Prior work shows that when the covariance $Σ$ is known, the optimal sample complexity can depend only on its spectrum, rather than on the ambient dimension $d$. We establish new lower bounds for Gaussian mean estimation with Euclidean error which show that private estimators cannot achieve similar guarantees when the covariance is unknown. First, when the sample size $n$ is smaller than the effective dimension $\mathrm{trace}(Σ)/\|Σ\|_{\text{op}}$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d/n}$. Second, when the sample size is larger than the effective dimension, even for diagonal covariance matrices with effective dimension bounded by $\log d$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d}/n^{3/2}$. Together, these results show that in the unknown covariance setting, exploiting anisotropic structure under differential privacy can require a dimension-dependent error or sample complexity.

Constant-Time Dynamic Edge Coloring via Dynamic Splitting and LLL

from arXiv: Data Structures and Algorithms

Authors: Mohsen Ghaffari, Sijin Peng

We give a dynamic algorithm that maintains a near-optimal edge coloring in constant expected time per update. For every fixed $ε>0$, the algorithm maintains a $(1+ε)Δ$ edge coloring under edge insertions and deletions in $\mathrm{poly}(1/ε)$ worst-case expected update time against an oblivious adversary, provided $Δ\geΔ_ε=\widetildeΘ(1/ε^2)$. The previous constant-time algorithm of Bhattacharya, Costa, Panski, and Solomon (BCPS) required $Δ\ge(\log n/ε^4)^{Θ(\log(1/ε)/ε)}$, where $n$ is the number of vertices. We remove this dependence on $n$, answering their open question. Our approach differs substantially from that of BCPS, which relied on coloring in locally tree-like subgraphs. Instead, we draw on distributed algorithms that organize partial coloring through successive applications of the Lovász Local Lemma (LLL). We develop a dynamic LLL solver with worst-case expected update bounds and a dynamic splitting algorithm that partitions edges into low-degree subgraphs. A key technical contribution is a bottom-up method for iterative splitting, which avoids the logarithmic overhead of maintaining successive levels. It achieves constant expected update time by adaptively choosing which levels to inspect and repair, rather than inspecting every level after each update. We believe these dynamic splitting and LLL tools can be of interest beyond edge coloring.

Authors: Mohsen Ghaffari, Sijin Peng

We give a dynamic algorithm that maintains a near-optimal edge coloring in constant expected time per update. For every fixed $ε>0$, the algorithm maintains a $(1+ε)Δ$ edge coloring under edge insertions and deletions in $\mathrm{poly}(1/ε)$ worst-case expected update time against an oblivious adversary, provided $Δ\geΔ_ε=\widetildeΘ(1/ε^2)$. The previous constant-time algorithm of Bhattacharya, Costa, Panski, and Solomon (BCPS) required $Δ\ge(\log n/ε^4)^{Θ(\log(1/ε)/ε)}$, where $n$ is the number of vertices. We remove this dependence on $n$, answering their open question. Our approach differs substantially from that of BCPS, which relied on coloring in locally tree-like subgraphs. Instead, we draw on distributed algorithms that organize partial coloring through successive applications of the Lovász Local Lemma (LLL). We develop a dynamic LLL solver with worst-case expected update bounds and a dynamic splitting algorithm that partitions edges into low-degree subgraphs. A key technical contribution is a bottom-up method for iterative splitting, which avoids the logarithmic overhead of maintaining successive levels. It achieves constant expected update time by adaptively choosing which levels to inspect and repair, rather than inspecting every level after each update. We believe these dynamic splitting and LLL tools can be of interest beyond edge coloring.

A Sub-$3^n$ Upper Bound on the Number of Stable Matchings

from arXiv: Data Structures and Algorithms

Authors: Ken-ichi Kawarabayashi, Hikaru Wada, Masataka Yoneda

We study the problem of determining the maximum number of stable matchings. This classical counting problem was posed by Knuth and has been studied for five decades. We obtain a new $2.999^n$ upper bound on the number of stable matchings with $n$ men and $n$ women. This improves the best-known previous result by Palmer and Pálvölgyi. This result gets much closer to the well-known lower bound $\frac{2.28^n}{\text{poly}(n)}$. We refine their entropy-based framework by capturing broader structural relationships among stable matchings. To capture these relationships, we introduce new frameworks, including cones and supports. This establishes the $3.332^n$ bound. We also exploit relationships among local structures and use a charging argument to improve the bound to $2.999^n$.

Authors: Ken-ichi Kawarabayashi, Hikaru Wada, Masataka Yoneda

We study the problem of determining the maximum number of stable matchings. This classical counting problem was posed by Knuth and has been studied for five decades. We obtain a new $2.999^n$ upper bound on the number of stable matchings with $n$ men and $n$ women. This improves the best-known previous result by Palmer and Pálvölgyi. This result gets much closer to the well-known lower bound $\frac{2.28^n}{\text{poly}(n)}$. We refine their entropy-based framework by capturing broader structural relationships among stable matchings. To capture these relationships, we introduce new frameworks, including cones and supports. This establishes the $3.332^n$ bound. We also exploit relationships among local structures and use a charging argument to improve the bound to $2.999^n$.

A $2.81^n$ upper bound on the number of stable matchings

from arXiv: Data Structures and Algorithms

Authors: Ebrahim Ghorbani, Matthias Mnich

We study the maximum number ${\rm SM}(n)$ of stable matchings in an instance with $n$ men and $n$ women. Karlin, Oveis Gharan, and Weber (STOC 2018) proved the first single-exponential upper bound; their argument gives ${\rm SM}(n)\le 2^{17n}$ for sufficiently large $n$. Palmer and Pálvölgyi (FOCS 2021) improved this bound to $3.55^n+O(1)$. We prove that ${\rm SM}(n)<2.81^n$. This brings the exponential base closer to $2.28$, the base of the best known lower bound. More precisely, we bound the number of stable matchings in each instance in terms of each person's number of stable partners, where a stable partner is someone matched to that person in at least one stable matching. Our proof assigns each person a weight determined by this number and uses these weights to bound the number of stable matchings by induction.

Authors: Ebrahim Ghorbani, Matthias Mnich

We study the maximum number ${\rm SM}(n)$ of stable matchings in an instance with $n$ men and $n$ women. Karlin, Oveis Gharan, and Weber (STOC 2018) proved the first single-exponential upper bound; their argument gives ${\rm SM}(n)\le 2^{17n}$ for sufficiently large $n$. Palmer and Pálvölgyi (FOCS 2021) improved this bound to $3.55^n+O(1)$. We prove that ${\rm SM}(n)<2.81^n$. This brings the exponential base closer to $2.28$, the base of the best known lower bound. More precisely, we bound the number of stable matchings in each instance in terms of each person's number of stable partners, where a stable partner is someone matched to that person in at least one stable matching. Our proof assigns each person a weight determined by this number and uses these weights to bound the number of stable matchings by induction.

Computing Forest Degree Realizations with Minimum Cardinality Domination

from arXiv: Data Structures and Algorithms

Authors: Amotz Bar-Noy, Omri Kosary, David Peleg, Dror Rawitz

In the Degree Realization problem with respect to a family $\mathcal{P}$ of graphs the input is a non-increasing sequence $d = (d_1,\ldots,d_n)$ of positive integers, and the goal is to decide whether there exists a simple undirected graph $G \in \mathcal{P}$, whose degrees correspond to $d$, i.e., such that $deg(G) = d$. In this paper we consider the version of Degree Realization in which the realization is required to be a forest (i.e., $\mathcal{P}$ is the family of forests). We consider \emph{optimized} Degree Realization in which the goal is to obtain a realization that minimizes an objective function $f$. That is, the goal is to find a realization $G$ that minimizes $f(G)$ among the realizations of the given input sequence. More specifically, we focus on the following functions: the size of an optimal \emph{vertex cover} and the size of an optimal \emph{dominating set}. We also consider the \emph{total} and \emph{paired} versions of both Min Vertex Cover and Min Dominating Set. We provide characterizations and linear time realization algorithms for all the above-mentioned problems. Since Min Paired Vertex Cover is closely related to Min Edge Dominating Set, our results for Degree Realization with respect to the former extend to the latter.

Authors: Amotz Bar-Noy, Omri Kosary, David Peleg, Dror Rawitz

In the Degree Realization problem with respect to a family $\mathcal{P}$ of graphs the input is a non-increasing sequence $d = (d_1,\ldots,d_n)$ of positive integers, and the goal is to decide whether there exists a simple undirected graph $G \in \mathcal{P}$, whose degrees correspond to $d$, i.e., such that $deg(G) = d$. In this paper we consider the version of Degree Realization in which the realization is required to be a forest (i.e., $\mathcal{P}$ is the family of forests). We consider \emph{optimized} Degree Realization in which the goal is to obtain a realization that minimizes an objective function $f$. That is, the goal is to find a realization $G$ that minimizes $f(G)$ among the realizations of the given input sequence. More specifically, we focus on the following functions: the size of an optimal \emph{vertex cover} and the size of an optimal \emph{dominating set}. We also consider the \emph{total} and \emph{paired} versions of both Min Vertex Cover and Min Dominating Set. We provide characterizations and linear time realization algorithms for all the above-mentioned problems. Since Min Paired Vertex Cover is closely related to Min Edge Dominating Set, our results for Degree Realization with respect to the former extend to the latter.

Beyond Gomory-Hu: All-Pairs $\times$ All-Pairs Mincuts in $O(n)$ Space

from arXiv: Data Structures and Algorithms

Authors: Koustav Bhanja, Merav Parter, Asaf Petruschka

For over four decades, two classical structures have offered complementary views of mincuts. The Gomory-Hu tree encodes one $(x,y)$-mincut for every vertex pair $x,y$ in $O(n)$ space, but retains only one mincut per pair. In contrast, for a fixed pair $x,y$, the Picard-Queyranne representation admits an $O(n)$ space vertex ordering that determines whether some $(x,y)$-mincut separates any given pair $u,v$ and reports such a cut. Can these complementary guarantees hold simultaneously in $O(n)$ space? We answer affirmatively. For undirected graphs, we construct an $O(n)$ space structure encoding, for every four vertices $x,y,u,v$, an $(x,y)$-mincut separating $u,v$ whenever one exists. More precisely, given any terminal pair $x,y$, the structure recovers the Picard-Queyranne ordering in $O(n)$ time. Thereafter, separability queries take $O(1)$ time, and reporting a separating mincut $C$ takes $O(|C|)$ time. Thus, information for all $Θ(n^4)$ choices of $x,y,u,v$ is encoded in $O(n)$ space. This is essentially the strongest information of this form that fits in $O(n)$ space: if one additionally requires the mincut to place $u$ on side of $x$ and $v$ on side of $y$, then $Ω(n^2)$ space is necessary even for fixed $x,y$. As a consequence, we obtain an $O(n)$ space all-pairs mincut sensitivity oracle for a single edge failure in multigraphs, with optimal $O(n)$ time to report a new mincut. This resolves an open question of Baswana and Pandey [SODA'22] on achieving $o(n^2)$ space and $o(m)$ query time simultaneously. Our construction substantially generalizes the $O(n)$ space nearest-mincut-hierarchy [SODA'27], which reports only the nearest $(x,y)$-mincut. Recovering the entire Picard-Queyranne ordering for an arbitrary pair requires new structural properties of the Dinitz-Vainshtein [STOC'94] connectivity carcass, extending its use beyond the Steiner mincuts it represents.

Authors: Koustav Bhanja, Merav Parter, Asaf Petruschka

For over four decades, two classical structures have offered complementary views of mincuts. The Gomory-Hu tree encodes one $(x,y)$-mincut for every vertex pair $x,y$ in $O(n)$ space, but retains only one mincut per pair. In contrast, for a fixed pair $x,y$, the Picard-Queyranne representation admits an $O(n)$ space vertex ordering that determines whether some $(x,y)$-mincut separates any given pair $u,v$ and reports such a cut. Can these complementary guarantees hold simultaneously in $O(n)$ space? We answer affirmatively. For undirected graphs, we construct an $O(n)$ space structure encoding, for every four vertices $x,y,u,v$, an $(x,y)$-mincut separating $u,v$ whenever one exists. More precisely, given any terminal pair $x,y$, the structure recovers the Picard-Queyranne ordering in $O(n)$ time. Thereafter, separability queries take $O(1)$ time, and reporting a separating mincut $C$ takes $O(|C|)$ time. Thus, information for all $Θ(n^4)$ choices of $x,y,u,v$ is encoded in $O(n)$ space. This is essentially the strongest information of this form that fits in $O(n)$ space: if one additionally requires the mincut to place $u$ on side of $x$ and $v$ on side of $y$, then $Ω(n^2)$ space is necessary even for fixed $x,y$. As a consequence, we obtain an $O(n)$ space all-pairs mincut sensitivity oracle for a single edge failure in multigraphs, with optimal $O(n)$ time to report a new mincut. This resolves an open question of Baswana and Pandey [SODA'22] on achieving $o(n^2)$ space and $o(m)$ query time simultaneously. Our construction substantially generalizes the $O(n)$ space nearest-mincut-hierarchy [SODA'27], which reports only the nearest $(x,y)$-mincut. Recovering the entire Picard-Queyranne ordering for an arbitrary pair requires new structural properties of the Dinitz-Vainshtein [STOC'94] connectivity carcass, extending its use beyond the Steiner mincuts it represents.

Polynomial $(2α+\varepsilon)$- and Quadratic $4$-Approximation Algorithms for the Biased Traveling Salesman Problem in Bipartitioned Networks

from arXiv: Data Structures and Algorithms

Authors: Jingyang Zhao, Trung Q. Duong, Een-Kee Hong, Hyundong Shin

The traveling salesman problem (TSP) is one of the fundamental problems in combinatorial optimization that asks for a minimum-cost Hamiltonian cycle in a complete undirected graph with $N$ vertices. For the metric TSP, where edge costs satisfy the triangle inequality, a long line of work has produced constant-factor approximation algorithms, culminating in the best-known approximation ratio $α<3/2-10^{-36}$. The biased TSP (BTSP) generalizes the metric TSP to bipartitioned networks: the vertex set is partitioned into two subsets, edge costs satisfy the triangle inequality within each subset, while edges crossing the partition may violate it by a certain bias factor. This model captures heterogeneous routing scenarios in which distances are metric only locally but may be distorted across different regions or vertex types. Previously, the only known result for the BTSP was a $7/2$-approximation algorithm with running time $O(N^3)$. In this paper, we present two new approximation algorithms that significantly improve the state of the art. The first achieves an approximation ratio of $2α+\varepsilon$ (strictly less than $3$) in polynomial time $N^{O(1)}$ for any constant $\varepsilon>0$, where the exponent grows at least linearly with $1/\varepsilon$. The second achieves a $4$-approximation in only $O(N^2)$ running time, reducing the previous cubic-time bound by a full polynomial factor. These advances directly support emerging networked systems---including multi-tier 6G architectures and hybrid quantum-classical networks---where cross-region transitions incur biased and non-metric overheads.

Authors: Jingyang Zhao, Trung Q. Duong, Een-Kee Hong, Hyundong Shin

The traveling salesman problem (TSP) is one of the fundamental problems in combinatorial optimization that asks for a minimum-cost Hamiltonian cycle in a complete undirected graph with $N$ vertices. For the metric TSP, where edge costs satisfy the triangle inequality, a long line of work has produced constant-factor approximation algorithms, culminating in the best-known approximation ratio $α<3/2-10^{-36}$. The biased TSP (BTSP) generalizes the metric TSP to bipartitioned networks: the vertex set is partitioned into two subsets, edge costs satisfy the triangle inequality within each subset, while edges crossing the partition may violate it by a certain bias factor. This model captures heterogeneous routing scenarios in which distances are metric only locally but may be distorted across different regions or vertex types. Previously, the only known result for the BTSP was a $7/2$-approximation algorithm with running time $O(N^3)$. In this paper, we present two new approximation algorithms that significantly improve the state of the art. The first achieves an approximation ratio of $2α+\varepsilon$ (strictly less than $3$) in polynomial time $N^{O(1)}$ for any constant $\varepsilon>0$, where the exponent grows at least linearly with $1/\varepsilon$. The second achieves a $4$-approximation in only $O(N^2)$ running time, reducing the previous cubic-time bound by a full polynomial factor. These advances directly support emerging networked systems---including multi-tier 6G architectures and hybrid quantum-classical networks---where cross-region transitions incur biased and non-metric overheads.

Partially Dynamic Algorithms for Single-Source Temporal Paths

from arXiv: Data Structures and Algorithms

Authors: Davide Bilò, Bartłomiej Bosek, Jadwiga Czyżewska, Evangelos Kipouridis, Karol Węgrzycki, Anna Zych-Pawlewicz

We present efficient partially dynamic algorithms for maintaining earliest arrival times from a fixed source in temporal digraphs with $n$ vertices. For the incremental scenario where updates are arc insertions, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the total number of arcs added to the arc-less graph. For the decremental scenario where updates are arc removals, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the number of arcs in the graph upon its initialization. We also consider earliest-arrival queries with an arbitrary departure-time threshold and develop an incremental algorithm with $O(n+Δ_s m\log m)$ total update time, $O(\log m)$ query time, and $O(n+m)$ space, where $Δ_s$ is the number of distinct departure times of arcs leaving the source. All of our algorithms are also capable of reporting an earliest arrival path from the source to the queried vertex $v$ in constant time per edge once the earliest arrival time at $v$ is known. We complement our positive results with conditional lower bounds based on the Online Boolean Matrix-Vector Multiplication conjecture (a.k.a. OMv conjecture). For both the incremental and the decremental scenarios, we show the non-existence of an algorithm with $O(n^{1-\varepsilon})$ amortized update time and $O(n^{2-\varepsilon})$ amortized query time, even if the algorithm is randomized and latencies on the arcs are uniform. For arbitrary departure-time queries, we also establish an OMv-based conditional lower bound showing that our incremental algorithm is optimal up to logarithmic factors on dense temporal graphs. Thus, efficient total running time is essentially the best outcome we can get. Our conditional lower bounds hold also for the case of undirected temporal graphs.

Authors: Davide Bilò, Bartłomiej Bosek, Jadwiga Czyżewska, Evangelos Kipouridis, Karol Węgrzycki, Anna Zych-Pawlewicz

We present efficient partially dynamic algorithms for maintaining earliest arrival times from a fixed source in temporal digraphs with $n$ vertices. For the incremental scenario where updates are arc insertions, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the total number of arcs added to the arc-less graph. For the decremental scenario where updates are arc removals, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the number of arcs in the graph upon its initialization. We also consider earliest-arrival queries with an arbitrary departure-time threshold and develop an incremental algorithm with $O(n+Δ_s m\log m)$ total update time, $O(\log m)$ query time, and $O(n+m)$ space, where $Δ_s$ is the number of distinct departure times of arcs leaving the source. All of our algorithms are also capable of reporting an earliest arrival path from the source to the queried vertex $v$ in constant time per edge once the earliest arrival time at $v$ is known. We complement our positive results with conditional lower bounds based on the Online Boolean Matrix-Vector Multiplication conjecture (a.k.a. OMv conjecture). For both the incremental and the decremental scenarios, we show the non-existence of an algorithm with $O(n^{1-\varepsilon})$ amortized update time and $O(n^{2-\varepsilon})$ amortized query time, even if the algorithm is randomized and latencies on the arcs are uniform. For arbitrary departure-time queries, we also establish an OMv-based conditional lower bound showing that our incremental algorithm is optimal up to logarithmic factors on dense temporal graphs. Thus, efficient total running time is essentially the best outcome we can get. Our conditional lower bounds hold also for the case of undirected temporal graphs.

Estimating the Volume of a Union of Convex Bodies

from arXiv: Data Structures and Algorithms

Authors: Eeshan Banerjee, Aranya Kumar Bal, Debabrota Basu, Sourav Chakraborty, Debarshi Chanda, Uddalok Sarkar

The \textit{Union Volume Estimation problem}, i.e. yielding an $(\varepsilon, δ)$-approximation of the volume of the union of convex bodies, is a classical problem in computational geometry and formal methods. In this paper, we first detect where and why the efficient union volume estimators with approximate sampling and PAC volume estimation oracles fail. Specifically, we show that having only TV-distance guarantees on the approximate sampler does not suffice, and demonstrate this bottleneck by constructing the sunflower families of convex bodies. In this context, we propose the concept of \textit{tail-controlled samplers} that can yield with high-probability almost-$\ell_{\infty}$ closeness. Secondly, we leverage this concept to propose a generic algorithmic framework \texttt{Alpona} that properly combines a PAC volume computation and approximate point sampling oracles with tail-controls to yield an $(\varepsilon, δ)$-estimate of the union volume for any collection of convex bodies. We show that \texttt{Alpona} needs only linear (in number of convex bodies) oracle calls to be $(\varepsilon, δ)$-correct. Finally, we provide a tail-controlled analysis of the Hit-and-Run sampler~\citep{lovasz2006fast}. We further provide a novel geometric analysis to improve the required walk length of Hit-and-Run. We further instantiate \texttt{Alpona}'s correctness for the sparse frequency streams of convex bodies: a special case of the union volume estimation problem where no convex body intersects more than $τ$ many other bodies.

Authors: Eeshan Banerjee, Aranya Kumar Bal, Debabrota Basu, Sourav Chakraborty, Debarshi Chanda, Uddalok Sarkar

The \textit{Union Volume Estimation problem}, i.e. yielding an $(\varepsilon, δ)$-approximation of the volume of the union of convex bodies, is a classical problem in computational geometry and formal methods. In this paper, we first detect where and why the efficient union volume estimators with approximate sampling and PAC volume estimation oracles fail. Specifically, we show that having only TV-distance guarantees on the approximate sampler does not suffice, and demonstrate this bottleneck by constructing the sunflower families of convex bodies. In this context, we propose the concept of \textit{tail-controlled samplers} that can yield with high-probability almost-$\ell_{\infty}$ closeness. Secondly, we leverage this concept to propose a generic algorithmic framework \texttt{Alpona} that properly combines a PAC volume computation and approximate point sampling oracles with tail-controls to yield an $(\varepsilon, δ)$-estimate of the union volume for any collection of convex bodies. We show that \texttt{Alpona} needs only linear (in number of convex bodies) oracle calls to be $(\varepsilon, δ)$-correct. Finally, we provide a tail-controlled analysis of the Hit-and-Run sampler~\citep{lovasz2006fast}. We further provide a novel geometric analysis to improve the required walk length of Hit-and-Run. We further instantiate \texttt{Alpona}'s correctness for the sparse frequency streams of convex bodies: a special case of the union volume estimation problem where no convex body intersects more than $τ$ many other bodies.

A Polynomial Erdős-Pósa bound for Eulerian digraphs

from arXiv: Data Structures and Algorithms

Authors: Esther Galby, Roohani Sharma, Bart van der Steenhoven

In [Combinatorica 1996], Reed, Robertson, Seymour and Thomas proved Younger's conjecture dating back to 1973 that states that directed cycles have the Erdős-Pósa property. Specifically, they showed that every directed graph $D$ contains either $k$ arc-disjoint cycles or a set of at most $f(k)$ arcs that intersect every directed cycle. The function $f$, depending only on $k$, is however notoriously large--in fact, not even elementary. The first improvement on this function was only very recently provided by Hatzel, Kreutzer, Milani, and Muzi [FOCS 2024] who showed that $f$ can be taken to be an exponential tower of height $8$. In this work, we improve this function to $O(k^4)$ in the case where the digraph is Eulerian. This also answers an open question by Wahlström [Springer Monographs in Mathematics 2018] who asked explicitly when Eulerian digraphs have a polynomial Erdős-Pósa property. The only other known result about Erdős-Pósa on Eulerian digraphs is a result from Seymour [Combinatorica 1996] who showed that $f(k)=k$ when the Eulerian digraph admits a linkless embedding in 3-space, and such an equality is not possible for general Eulerian digraphs. We also complement our positive result, by showing that the result of Seymour cannot be extended to give a linear bound ($O(k)$) on general Eulerian digraphs, by showing that even on Eulerian digraphs the lower bound for the Erdős-Pósa property is $Ω(k \log k)$.

Authors: Esther Galby, Roohani Sharma, Bart van der Steenhoven

In [Combinatorica 1996], Reed, Robertson, Seymour and Thomas proved Younger's conjecture dating back to 1973 that states that directed cycles have the Erdős-Pósa property. Specifically, they showed that every directed graph $D$ contains either $k$ arc-disjoint cycles or a set of at most $f(k)$ arcs that intersect every directed cycle. The function $f$, depending only on $k$, is however notoriously large--in fact, not even elementary. The first improvement on this function was only very recently provided by Hatzel, Kreutzer, Milani, and Muzi [FOCS 2024] who showed that $f$ can be taken to be an exponential tower of height $8$. In this work, we improve this function to $O(k^4)$ in the case where the digraph is Eulerian. This also answers an open question by Wahlström [Springer Monographs in Mathematics 2018] who asked explicitly when Eulerian digraphs have a polynomial Erdős-Pósa property. The only other known result about Erdős-Pósa on Eulerian digraphs is a result from Seymour [Combinatorica 1996] who showed that $f(k)=k$ when the Eulerian digraph admits a linkless embedding in 3-space, and such an equality is not possible for general Eulerian digraphs. We also complement our positive result, by showing that the result of Seymour cannot be extended to give a linear bound ($O(k)$) on general Eulerian digraphs, by showing that even on Eulerian digraphs the lower bound for the Erdős-Pósa property is $Ω(k \log k)$.

A Dichotomy for Terminal Cyclability

from arXiv: Data Structures and Algorithms

Authors: Matthias Bentert, Dario Cavallaro, Sebastian Wiederrecht

Given an undirected graph $G$ and a set $R \subseteq V(G)$ of terminals, Terminal Cyclability asks whether $G$ contains a simple cycle visiting every vertex of $R$. We establish a complexity dichotomy for this problem on red-minor-closed classes of annotated graphs. In particular, we show that Terminal Cyclability is polynomial-time solvable on every such class excluding any annotated planar graph. The problem is NP-complete whenever the class contains all annotated planar graphs. More precisely, we give an $|G|^{2^{\mathbf{poly}(b)}}$-time algorithm, where $b$ is the bidimensionality of $(G,R)$. Our main technical contribution is the introduction of a weak irrelevant vertex rule. Rather than removing irrelevant vertices until the treewidth becomes constant, we only remove irrelevant vertices until the treewidth is logarithmic in the graph size. Such a weakening is necessary as a parameter-only treewidth threshold for an irrelevant vertex rule in this setting is known to be impossible to realise. Combined with a single-exponential algorithm for Terminal Cyclability parameterized by treewidth, this yields the polynomial-time algorithm for our dichotomy.

Authors: Matthias Bentert, Dario Cavallaro, Sebastian Wiederrecht

Given an undirected graph $G$ and a set $R \subseteq V(G)$ of terminals, Terminal Cyclability asks whether $G$ contains a simple cycle visiting every vertex of $R$. We establish a complexity dichotomy for this problem on red-minor-closed classes of annotated graphs. In particular, we show that Terminal Cyclability is polynomial-time solvable on every such class excluding any annotated planar graph. The problem is NP-complete whenever the class contains all annotated planar graphs. More precisely, we give an $|G|^{2^{\mathbf{poly}(b)}}$-time algorithm, where $b$ is the bidimensionality of $(G,R)$. Our main technical contribution is the introduction of a weak irrelevant vertex rule. Rather than removing irrelevant vertices until the treewidth becomes constant, we only remove irrelevant vertices until the treewidth is logarithmic in the graph size. Such a weakening is necessary as a parameter-only treewidth threshold for an irrelevant vertex rule in this setting is known to be impossible to realise. Combined with a single-exponential algorithm for Terminal Cyclability parameterized by treewidth, this yields the polynomial-time algorithm for our dichotomy.

Consistent Submodular Maximization with Full Foresight

from arXiv: Data Structures and Algorithms

Authors: Eric Balkanski, Paul Dütting, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Ola Svensson, Morteza Zadimoghaddam

Online algorithms with bounded recourse face three intertwined obstacles: uncertainty about the future, computational limits, and a limited budget for changes. We ask a fundamental question that separates consistency from the other two: what does solution stability cost when the entire future is known and computation is unlimited? We answer this question for monotone submodular maximization, where an algorithm must maintain a solution of at most $k$ arrived elements, with a constant number of changes per step. We determine the optimal approximation thresholds: $3/4$ for deterministic algorithms and $ξ^\star \approx 0.7942$ for randomized algorithms. For every $\varepsilon>0$, we get within $\varepsilon$ of these thresholds with worst-case recourse polynomial in $1/\varepsilon$, independent of $k$ and the stream length. We prove matching impossibility results even for coverage functions. We attain the sharp bounds by optimally balancing two opposing forces: preparing for future arrivals and protecting the current solution. For randomized algorithms, we use addition robustness against prefixes of the known future, which we analyze tightly via an interval-selection problem. For deterministic algorithms, we introduce certificates, paths in a fractional relaxation that maintain high value, which we round into an update schedule.

Authors: Eric Balkanski, Paul Dütting, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Ola Svensson, Morteza Zadimoghaddam

Online algorithms with bounded recourse face three intertwined obstacles: uncertainty about the future, computational limits, and a limited budget for changes. We ask a fundamental question that separates consistency from the other two: what does solution stability cost when the entire future is known and computation is unlimited? We answer this question for monotone submodular maximization, where an algorithm must maintain a solution of at most $k$ arrived elements, with a constant number of changes per step. We determine the optimal approximation thresholds: $3/4$ for deterministic algorithms and $ξ^\star \approx 0.7942$ for randomized algorithms. For every $\varepsilon>0$, we get within $\varepsilon$ of these thresholds with worst-case recourse polynomial in $1/\varepsilon$, independent of $k$ and the stream length. We prove matching impossibility results even for coverage functions. We attain the sharp bounds by optimally balancing two opposing forces: preparing for future arrivals and protecting the current solution. For randomized algorithms, we use addition robustness against prefixes of the known future, which we analyze tightly via an interval-selection problem. For deterministic algorithms, we introduce certificates, paths in a fractional relaxation that maintain high value, which we round into an update schedule.

Sublinear Approximations for Spanning Tree Congestion

from arXiv: Data Structures and Algorithms

Authors: Sunny Atalig, Marek Chrobak, Petr Kolman, Hans Raj Tiwary

The spanning tree congestion of a connected graph \(G\) is the minimum, over all spanning trees \(T\) of \(G\), of the maximum number of edges of \(G\) crossing a cut induced by deleting an edge of \(T\). In the spanning tree congestion problem we are asked to compute a spanning tree with minimum congestion. We describe two sublinear-ratio approximation algorithms for the spanning tree congestion problem (STC): a simple $\Ot(n^{8/9})$-approximation, and a more involved $\Ot(n^{2/3})$-approximation, where the notation $\Ot$ hides factors polynomial in $\log n$.. Until now, no algorithm with sublinear ratio was known. We also prove that for any fixed $0<\varepsilon<1$, approximation of the spanning tree congestion within $2-\varepsilon$ is NP-hard. This improves the previous best known lower bound $6/5$.

Authors: Sunny Atalig, Marek Chrobak, Petr Kolman, Hans Raj Tiwary

The spanning tree congestion of a connected graph \(G\) is the minimum, over all spanning trees \(T\) of \(G\), of the maximum number of edges of \(G\) crossing a cut induced by deleting an edge of \(T\). In the spanning tree congestion problem we are asked to compute a spanning tree with minimum congestion. We describe two sublinear-ratio approximation algorithms for the spanning tree congestion problem (STC): a simple $\Ot(n^{8/9})$-approximation, and a more involved $\Ot(n^{2/3})$-approximation, where the notation $\Ot$ hides factors polynomial in $\log n$.. Until now, no algorithm with sublinear ratio was known. We also prove that for any fixed $0<\varepsilon<1$, approximation of the spanning tree congestion within $2-\varepsilon$ is NP-hard. This improves the previous best known lower bound $6/5$.

Efficient Stable Word-RAM Simulation by Population Protocols

from arXiv: Data Structures and Algorithms

Authors: Leszek Gąsieniec, Tytus Grodzicki, Grzegorz Stachowiak

Population protocols model computation by indistinguishable agents interacting in uniformly random pairs, with unary inputs and outputs encoded by state counts. Efficiency requires polylogarithmically many states and polylogarithmic stabilisation time, measured as interactions divided by population size $n$. Stability requires eventual stabilisation to the correct output with probability $1$; silence requires that no agent changes state after stabilisation. Fast stable leader election and majority in leaderless populations require growing state spaces. Despite progress on individual problems, the general computational power of efficient stable protocols remains largely unexplored. We show that it extends beyond semilinear predicates, which characterise constant-state stable computation. We give an efficient, stable, and silent simulation of every $\mathcal{P}$-program: a word-RAM program with running time and number of registers polynomial in $w=Θ(\log n)$. Stabilisation takes polylogarithmic time both in expectation and with high probability. The framework supports general word-oriented computations, including multiple outputs. The construction extends MC$+$ protocols, fast Monte Carlo protocols whose failures are eventually detected as anomalies, and the system of weights. MC$+$ majority comparisons against the weights convert unary inputs into binary registers, with one agent per bit. MC$+$ operations on individual bits emulate RAM instructions, after which outputs are converted back to unary. A detected anomaly triggers a new universal rank-based backup that stabilises correctly and silently. These results open a systematic study of computational power and resource trade-offs, including extensions to external or spatial data.

Authors: Leszek Gąsieniec, Tytus Grodzicki, Grzegorz Stachowiak

Population protocols model computation by indistinguishable agents interacting in uniformly random pairs, with unary inputs and outputs encoded by state counts. Efficiency requires polylogarithmically many states and polylogarithmic stabilisation time, measured as interactions divided by population size $n$. Stability requires eventual stabilisation to the correct output with probability $1$; silence requires that no agent changes state after stabilisation. Fast stable leader election and majority in leaderless populations require growing state spaces. Despite progress on individual problems, the general computational power of efficient stable protocols remains largely unexplored. We show that it extends beyond semilinear predicates, which characterise constant-state stable computation. We give an efficient, stable, and silent simulation of every $\mathcal{P}$-program: a word-RAM program with running time and number of registers polynomial in $w=Θ(\log n)$. Stabilisation takes polylogarithmic time both in expectation and with high probability. The framework supports general word-oriented computations, including multiple outputs. The construction extends MC$+$ protocols, fast Monte Carlo protocols whose failures are eventually detected as anomalies, and the system of weights. MC$+$ majority comparisons against the weights convert unary inputs into binary registers, with one agent per bit. MC$+$ operations on individual bits emulate RAM instructions, after which outputs are converted back to unary. A detected anomaly triggers a new universal rank-based backup that stabilises correctly and silently. These results open a systematic study of computational power and resource trade-offs, including extensions to external or spatial data.

Optimal LCA and Sublinear Estimation for Bipartite Matching in $Δ^{O(1/\varepsilon)}$ Queries

from arXiv: Data Structures and Algorithms

Authors: Edith Cohen, Mohammad Roghani

A local computation algorithm (LCA) for matching answers ``who is the mate of $v$?'' by inspecting only a small part of the graph, with all answers consistent with one matching. For $n$-vertex graphs of maximum degree $Δ\ge2$, Levi et al. [Algorithmica'17] give a $(1-\varepsilon)$-approximate matching LCA using $(Δ/\varepsilon)^{O(1/\varepsilon^2)} \operatorname{polylog}(n)$ queries per invocation, while Yoshida et al. [SIAM J. Comput.'12] estimate the maximum matching size within additive error $\varepsilon n$ using $Δ^{O(1/\varepsilon^2)}$ queries. Behnezhad, Roghani, and Rubinstein [FOCS'23] proved a $Δ^{Ω(1/\varepsilon)}$ lower bound for both tasks, even on bipartite graphs and even when the LCA's matching guarantee is relaxed to additive $\varepsilon n$ error. This leaves a gap between linear and quadratic dependence on $1/\varepsilon$ in the exponent. We close this gap on bipartite graphs. We give an LCA that answers each mate query with $Δ^{O(1/\varepsilon)}$ queries for a $(1-\varepsilon)$-approximate matching, and an estimator for the maximum matching size within $\varepsilon n$ using $Δ^{O(1/\varepsilon)}$ queries. On general graphs, the same algorithm gives a half-integral fractional matching and estimates the maximum fractional matching size. Our approach runs $O(\varepsilon^{-1}\logΔ)$ rounds of a deterministic local optimization method in which vertices propose to neighbors and update prices according to incoming loads. We design its updates to admit exact recursive sampling with constant expected branching, obtaining the improved randomized bound without exploring full dependency neighborhoods. To our knowledge, this is the first LCA construction based on exact recursive sampling of the iterates of a deterministic continuous optimization method. We expect this approach to have applications beyond matching.

Authors: Edith Cohen, Mohammad Roghani

A local computation algorithm (LCA) for matching answers ``who is the mate of $v$?'' by inspecting only a small part of the graph, with all answers consistent with one matching. For $n$-vertex graphs of maximum degree $Δ\ge2$, Levi et al. [Algorithmica'17] give a $(1-\varepsilon)$-approximate matching LCA using $(Δ/\varepsilon)^{O(1/\varepsilon^2)} \operatorname{polylog}(n)$ queries per invocation, while Yoshida et al. [SIAM J. Comput.'12] estimate the maximum matching size within additive error $\varepsilon n$ using $Δ^{O(1/\varepsilon^2)}$ queries. Behnezhad, Roghani, and Rubinstein [FOCS'23] proved a $Δ^{Ω(1/\varepsilon)}$ lower bound for both tasks, even on bipartite graphs and even when the LCA's matching guarantee is relaxed to additive $\varepsilon n$ error. This leaves a gap between linear and quadratic dependence on $1/\varepsilon$ in the exponent. We close this gap on bipartite graphs. We give an LCA that answers each mate query with $Δ^{O(1/\varepsilon)}$ queries for a $(1-\varepsilon)$-approximate matching, and an estimator for the maximum matching size within $\varepsilon n$ using $Δ^{O(1/\varepsilon)}$ queries. On general graphs, the same algorithm gives a half-integral fractional matching and estimates the maximum fractional matching size. Our approach runs $O(\varepsilon^{-1}\logΔ)$ rounds of a deterministic local optimization method in which vertices propose to neighbors and update prices according to incoming loads. We design its updates to admit exact recursive sampling with constant expected branching, obtaining the improved randomized bound without exploring full dependency neighborhoods. To our knowledge, this is the first LCA construction based on exact recursive sampling of the iterates of a deterministic continuous optimization method. We expect this approach to have applications beyond matching.

Faster Cluster Decomposition with Stronger Bi-Criteria Guarantee

from arXiv: Data Structures and Algorithms

Authors: He Sun, Yong Zhang, Yacong Zhou

Partitioning the vertex set of a graph $G=(V,E)$ into clusters is a classical problem in computer science. As vertices within clusters should be well connected and cuts between clusters sparse, Oveis Gharan and Trevisan (SODA 2014, [OGT14]) proposed to measure the partition quality using inner and outer conductance, i.e., the bi-criteria guarantee. Specifically, when partitioning $V$ into vertex-disjoint subsets $P_1,\ldots, P_k$, every subgraph induced by $P_i$ should have high conductance, while the conductance of $P_i$ in $G$ should be low. They studied the existence of such a partition and designed a polynomial-time algorithm achieving this guarantee. For more than a decade, their algorithm remained the only one achieving this guarantee on its output clusters. In this paper, we study faster cluster decomposition algorithms under the same setup as [OGT14] and prove three results: 1. We make their existential result algorithmic: Our algorithms run in nearly-linear time, and achieve the same bi-criteria guarantee as the one in [OGT14] up to constant factors, including the asymptotically tight dependence on $k$. 2. We prove that their existential result holds algorithmically under a much weaker condition on the initial disjoint subsets. To achieve this, we develop a general algorithm framework that boosts the approximation guarantee of a solution: given an $α$-approximate solution to the $k$-way conductance problem, if $ρ_G(k+1)/ρ_G(k)=Ω\left(α/\varepsilon\right)$, then a $(1+\varepsilon)$-approximate solution can be obtained using only $O(k(1+\logα))$ minimum $s$-$t$ cut computations. 3. We present a nearly-linear time algorithm that finds clusters with the bi-criteria guarantee. Our algorithm has the same guarantee as the one in [OGT14], which, for any unweighted graph of $n$ vertices and $m$ edges, runs in $\widetilde{O}(knm^2)$ time.

Authors: He Sun, Yong Zhang, Yacong Zhou

Partitioning the vertex set of a graph $G=(V,E)$ into clusters is a classical problem in computer science. As vertices within clusters should be well connected and cuts between clusters sparse, Oveis Gharan and Trevisan (SODA 2014, [OGT14]) proposed to measure the partition quality using inner and outer conductance, i.e., the bi-criteria guarantee. Specifically, when partitioning $V$ into vertex-disjoint subsets $P_1,\ldots, P_k$, every subgraph induced by $P_i$ should have high conductance, while the conductance of $P_i$ in $G$ should be low. They studied the existence of such a partition and designed a polynomial-time algorithm achieving this guarantee. For more than a decade, their algorithm remained the only one achieving this guarantee on its output clusters. In this paper, we study faster cluster decomposition algorithms under the same setup as [OGT14] and prove three results: 1. We make their existential result algorithmic: Our algorithms run in nearly-linear time, and achieve the same bi-criteria guarantee as the one in [OGT14] up to constant factors, including the asymptotically tight dependence on $k$. 2. We prove that their existential result holds algorithmically under a much weaker condition on the initial disjoint subsets. To achieve this, we develop a general algorithm framework that boosts the approximation guarantee of a solution: given an $α$-approximate solution to the $k$-way conductance problem, if $ρ_G(k+1)/ρ_G(k)=Ω\left(α/\varepsilon\right)$, then a $(1+\varepsilon)$-approximate solution can be obtained using only $O(k(1+\logα))$ minimum $s$-$t$ cut computations. 3. We present a nearly-linear time algorithm that finds clusters with the bi-criteria guarantee. Our algorithm has the same guarantee as the one in [OGT14], which, for any unweighted graph of $n$ vertices and $m$ edges, runs in $\widetilde{O}(knm^2)$ time.

Single-Exponential FPT Algorithms for Minimum Bisection and Matching Cut via a New Unbreakable Decomposition

from arXiv: Data Structures and Algorithms

Authors: Soh Kumabe

Minimum Bisection asks whether the vertices of a graph can be partitioned into two parts whose sizes differ by at most one and with at most k edges between the two parts. Cygan, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saurabh (SICOMP 2019) gave the first fixed-parameter algorithm, which runs in O*(2^(O(k^3))) time, by introducing an unbreakable decomposition. Cygan, Komosa, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saurabh, and Wahlström (TALG 2021) strengthened the decomposition and improved the running time to O*(2^(O(k log k))). However, whether the running time could be improved to single-exponential remained open. In this paper, we give a deterministic O*(2^(O(k)))-time algorithm for Minimum Bisection. Our algorithm also counts all feasible bisections. To obtain an O*(2^(O(k)))-time algorithm, we introduce two new techniques. First, we give a new algorithm that computes a (2k,k)-edge-unbreakable decomposition with adhesion at most 2k in O*(2^(O(k))) time by relaxing the unbreakability and adhesion guarantees of the decomposition in (TALG 2021). Second, we improve the running time of the dynamic program over the decomposition from O*(2^(O(k log k))) to O*(2^(O(k))) using important cuts, inclusion-exclusion, and formal logarithms and exponentials. Together with color coding, the same techniques give a deterministic O*(2^(O(k)))-time algorithm for Matching Cut, which asks for a cut whose crossing edges form a matching. This improves the O*(2^(O(k log k)))-time algorithm of Aravind and Saxena (IWOCA 2021).

Authors: Soh Kumabe

Minimum Bisection asks whether the vertices of a graph can be partitioned into two parts whose sizes differ by at most one and with at most k edges between the two parts. Cygan, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saurabh (SICOMP 2019) gave the first fixed-parameter algorithm, which runs in O*(2^(O(k^3))) time, by introducing an unbreakable decomposition. Cygan, Komosa, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saurabh, and Wahlström (TALG 2021) strengthened the decomposition and improved the running time to O*(2^(O(k log k))). However, whether the running time could be improved to single-exponential remained open. In this paper, we give a deterministic O*(2^(O(k)))-time algorithm for Minimum Bisection. Our algorithm also counts all feasible bisections. To obtain an O*(2^(O(k)))-time algorithm, we introduce two new techniques. First, we give a new algorithm that computes a (2k,k)-edge-unbreakable decomposition with adhesion at most 2k in O*(2^(O(k))) time by relaxing the unbreakability and adhesion guarantees of the decomposition in (TALG 2021). Second, we improve the running time of the dynamic program over the decomposition from O*(2^(O(k log k))) to O*(2^(O(k))) using important cuts, inclusion-exclusion, and formal logarithms and exponentials. Together with color coding, the same techniques give a deterministic O*(2^(O(k)))-time algorithm for Matching Cut, which asks for a cut whose crossing edges form a matching. This improves the O*(2^(O(k log k)))-time algorithm of Aravind and Saxena (IWOCA 2021).

Single-Variable Connectivity in Random $k$-SAT

from arXiv: Data Structures and Algorithms

Authors: Chunyang Wang, Yitong Yin, Yixiao Yu, Yiyao Zhang

We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.

Authors: Chunyang Wang, Yitong Yin, Yixiao Yu, Yiyao Zhang

We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.

Faster Dynamic Maximal Independent Set against Adaptive Adversaries

from arXiv: Data Structures and Algorithms

Authors: Hongyi Chen, Shuyan Fan, Tianyi Zhang

We study the fully dynamic maximal independent set (MIS) problem: maintaining an MIS in an undirected graph on $n$ vertices with at most $m$ edges under edge insertions and deletions. Against an oblivious adversary, which fixes the edge update sequence in advance, the randomized algorithms of Behnezhad et al. (2019) and Chechik and Zhang (2019) achieve polylogarithmic update time. Against an adaptive adversary, which may choose each update after observing the previously reported MISs, the best previously known update bound was $O(m^{2/3})$, given by the deterministic algorithm of Gupta and Khan (2018, 2021). Assuming the combinatorial Boolean matrix multiplication hypothesis, Bernstein et al. (2026) proved that any combinatorial algorithm against an adaptive adversary requires at least $m^{1/2-o(1)}$ update time even under edge insertions alone, leaving a polynomial gap between upper and lower bounds. We present a randomized fully dynamic MIS algorithm with $\widetilde O(m^{5/8})=\widetilde O(m^{2/3-1/24})$ update time against adaptive adversaries. This gives the first polynomial improvement over the $m^{2/3}$ bound for adaptive fully dynamic MIS.

Authors: Hongyi Chen, Shuyan Fan, Tianyi Zhang

We study the fully dynamic maximal independent set (MIS) problem: maintaining an MIS in an undirected graph on $n$ vertices with at most $m$ edges under edge insertions and deletions. Against an oblivious adversary, which fixes the edge update sequence in advance, the randomized algorithms of Behnezhad et al. (2019) and Chechik and Zhang (2019) achieve polylogarithmic update time. Against an adaptive adversary, which may choose each update after observing the previously reported MISs, the best previously known update bound was $O(m^{2/3})$, given by the deterministic algorithm of Gupta and Khan (2018, 2021). Assuming the combinatorial Boolean matrix multiplication hypothesis, Bernstein et al. (2026) proved that any combinatorial algorithm against an adaptive adversary requires at least $m^{1/2-o(1)}$ update time even under edge insertions alone, leaving a polynomial gap between upper and lower bounds. We present a randomized fully dynamic MIS algorithm with $\widetilde O(m^{5/8})=\widetilde O(m^{2/3-1/24})$ update time against adaptive adversaries. This gives the first polynomial improvement over the $m^{2/3}$ bound for adaptive fully dynamic MIS.

Faster Isomorphism for Groups of Class 2

from arXiv: Data Structures and Algorithms

Authors: Xiaorui Sun

The finite-group isomorphism problem asks whether two groups, given by their Cayley tables, are isomorphic. Despite decades of study, obtaining an $N^{o(\log N)}$-time algorithm for general groups of order $N$ remains a major open challenge. Groups of nilpotency class $2$ have long been a central obstacle to achieving this goal. Recent breakthroughs have established $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ with odd $p$ (Sun, STOC '23; Ivanyos-Mendoza-Qiao-Sun-Zhang, FOCS '24; Grochow-Qiao-Stange-Sun, STOC '25) and for groups of Frattini class $2$ (Grochow-Ivanyos-Qiao-Sun, FOCS '26). However, the problem remained open for general groups of class $2$. In this paper, we present an algorithm for testing isomorphism of groups of class $2$, with running time $N^{O((\log N)^{25/26} \log \log N)}$. Our algorithm combines a tensor-based restriction of the search space, an encoding of group relations by alternating matrix tuples, and an orbit computation to recover the remaining central power information.

Authors: Xiaorui Sun

The finite-group isomorphism problem asks whether two groups, given by their Cayley tables, are isomorphic. Despite decades of study, obtaining an $N^{o(\log N)}$-time algorithm for general groups of order $N$ remains a major open challenge. Groups of nilpotency class $2$ have long been a central obstacle to achieving this goal. Recent breakthroughs have established $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ with odd $p$ (Sun, STOC '23; Ivanyos-Mendoza-Qiao-Sun-Zhang, FOCS '24; Grochow-Qiao-Stange-Sun, STOC '25) and for groups of Frattini class $2$ (Grochow-Ivanyos-Qiao-Sun, FOCS '26). However, the problem remained open for general groups of class $2$. In this paper, we present an algorithm for testing isomorphism of groups of class $2$, with running time $N^{O((\log N)^{25/26} \log \log N)}$. Our algorithm combines a tensor-based restriction of the search space, an encoding of group relations by alternating matrix tuples, and an orbit computation to recover the remaining central power information.

Improved approximation algorithms for capacitated cluster editing

from arXiv: Data Structures and Algorithms

Authors: Cong Ma, Guangting Chen, Yong Chen, Guohui Lin, Jiaxuan Ma, Lusheng Wang

We study the minimum-disagreement cluster editing problem that, from a complete edge-signed graph, where a positive edge represents two vertices agreeing to each other while a negative represents a disagreement, constructs disjoint cliques of upper bounded size such that the total number of negative edges inside cliques and positive edges outside cliques is minimized. When the sizes of the candidate cliques are unbounded, the problem is NP-hard and APX-hard, and many approximation algorithms with improving ratios have been proposed in the literature and the state-of-the-art approximation ratio of $α= 1.3865 + ε$. The capacitated variant, in which the candidate cliques have sizes no greater than a given $k$, is less studied and the best approximation ratio is $5.37$. In this paper, we study the latter problem. We first present a downsizing procedure that achieves an approximation ratio of $α_k = 2 α+ 1$, based on any $α$-approximation algorithm for the uncapacitated problem. Secondly, we propose a hybrid of cluster-LP sampling and pivoting and show that for every fractional capacitated cluster-LP solution of cost $L$, the hybrid using ordinary pivoting has an expected cost at most $\frac 95 L + \frac 1{10} \OPT_k$, and the hybrid using conditional pivoting has an expected cost at most $\frac {306}{203} L + \frac {50}{203} \OPT_k$. A cardinality-constrained adaptation of the approximate dual-separation framework of García-Soriano and Schohn supplies the required fractional solution and yields a randomized $(1.9 + ε)$-approximation algorithm and a randomized $(1.7537 + ε)$-approximation algorithm, respectively.

Authors: Cong Ma, Guangting Chen, Yong Chen, Guohui Lin, Jiaxuan Ma, Lusheng Wang

We study the minimum-disagreement cluster editing problem that, from a complete edge-signed graph, where a positive edge represents two vertices agreeing to each other while a negative represents a disagreement, constructs disjoint cliques of upper bounded size such that the total number of negative edges inside cliques and positive edges outside cliques is minimized. When the sizes of the candidate cliques are unbounded, the problem is NP-hard and APX-hard, and many approximation algorithms with improving ratios have been proposed in the literature and the state-of-the-art approximation ratio of $α= 1.3865 + ε$. The capacitated variant, in which the candidate cliques have sizes no greater than a given $k$, is less studied and the best approximation ratio is $5.37$. In this paper, we study the latter problem. We first present a downsizing procedure that achieves an approximation ratio of $α_k = 2 α+ 1$, based on any $α$-approximation algorithm for the uncapacitated problem. Secondly, we propose a hybrid of cluster-LP sampling and pivoting and show that for every fractional capacitated cluster-LP solution of cost $L$, the hybrid using ordinary pivoting has an expected cost at most $\frac 95 L + \frac 1{10} \OPT_k$, and the hybrid using conditional pivoting has an expected cost at most $\frac {306}{203} L + \frac {50}{203} \OPT_k$. A cardinality-constrained adaptation of the approximate dual-separation framework of García-Soriano and Schohn supplies the required fractional solution and yields a randomized $(1.9 + ε)$-approximation algorithm and a randomized $(1.7537 + ε)$-approximation algorithm, respectively.

Improved Bounds on Distance Preservers

from arXiv: Data Structures and Algorithms

Authors: Yanyu Chen, Yu Chen, Zihan Tan, Mingyang Yang

Given an undirected unweighted graph $G$ on $n$ vertices and a set $P$ of $p$ pairs of its vertices, a distance preserver is a sparse subgraph of $G$ that exactly preserves the shortest-path distances between all pairs in $P$. In this paper, we prove a new upper bound of $O(n^{8/9} p^{4/9})$ for distance preservers when $p\in [n^{1/2}, n]$. Our result improves the previous bound of $O\big(n^{2/3}p^{2/3}+np^{1/3}\big)$ by Bodwin and Vassilevska Williams and $O(\min\{n+n^{1/2}p,\,np^{1/2}\})$ by Coppersmith and Elkin, particularly in the regime $p\in [n^{7/10}, n]$, and narrows the gap toward the conjectured $O(n^{2/3}p^{2/3}+n)$ bound. Using the techniques developed for distance preservers, we are also able to obtain pairwise additive $+2$ spanners with $O(n+n^{9/10}p^{2/5})$ edges, pairwise additive $+4$ spanners with $O(n+n^{24/25}p^{8/25})$ edges, and new bounds for sourcewise spanners, improving upon previous results in certain parameter regimes.

Authors: Yanyu Chen, Yu Chen, Zihan Tan, Mingyang Yang

Given an undirected unweighted graph $G$ on $n$ vertices and a set $P$ of $p$ pairs of its vertices, a distance preserver is a sparse subgraph of $G$ that exactly preserves the shortest-path distances between all pairs in $P$. In this paper, we prove a new upper bound of $O(n^{8/9} p^{4/9})$ for distance preservers when $p\in [n^{1/2}, n]$. Our result improves the previous bound of $O\big(n^{2/3}p^{2/3}+np^{1/3}\big)$ by Bodwin and Vassilevska Williams and $O(\min\{n+n^{1/2}p,\,np^{1/2}\})$ by Coppersmith and Elkin, particularly in the regime $p\in [n^{7/10}, n]$, and narrows the gap toward the conjectured $O(n^{2/3}p^{2/3}+n)$ bound. Using the techniques developed for distance preservers, we are also able to obtain pairwise additive $+2$ spanners with $O(n+n^{9/10}p^{2/5})$ edges, pairwise additive $+4$ spanners with $O(n+n^{24/25}p^{8/25})$ edges, and new bounds for sourcewise spanners, improving upon previous results in certain parameter regimes.

Makespan Bounds for Synchronous Pebble Motion on Graphs

from arXiv: Data Structures and Algorithms

Authors: Lucy McManamon, Jingjin Yu

We study the largest minimum makespan of labeled pebble motion on graphs under a model that permits synchronous pebble movement. While classical pebble motion on graphs allows only one pebble to move at a time, our model permits any number of pebbles to move to an adjacent vertex, provided that no two pebbles collide. At full occupancy every vertex is assigned a pebble and a legal round consists of disjoint cycle rotations. We prove a series of makespan bounds for this problem on graphs with varying structure and pebble density. For cycles with a chord, we extend a known quadratic bound at span two to every fixed span. For random cubic graphs, we convert permutation routing via matchings into legal schedules in two ways, applying cycle-cover batching at full occupancy and disjoint vacancy-delivery paths below full occupancy. For simple 3-connected planar cubic graphs, we prove a tight linear class maximum at every pebble count. We extend the results to increasing degree, common-neighbor conditions, and vertex connectivity. The results describe how graph connectivity, cycle structure, expansion, planarity, and occupancy affect worst-case routing time.

Authors: Lucy McManamon, Jingjin Yu

We study the largest minimum makespan of labeled pebble motion on graphs under a model that permits synchronous pebble movement. While classical pebble motion on graphs allows only one pebble to move at a time, our model permits any number of pebbles to move to an adjacent vertex, provided that no two pebbles collide. At full occupancy every vertex is assigned a pebble and a legal round consists of disjoint cycle rotations. We prove a series of makespan bounds for this problem on graphs with varying structure and pebble density. For cycles with a chord, we extend a known quadratic bound at span two to every fixed span. For random cubic graphs, we convert permutation routing via matchings into legal schedules in two ways, applying cycle-cover batching at full occupancy and disjoint vacancy-delivery paths below full occupancy. For simple 3-connected planar cubic graphs, we prove a tight linear class maximum at every pebble count. We extend the results to increasing degree, common-neighbor conditions, and vertex connectivity. The results describe how graph connectivity, cycle structure, expansion, planarity, and occupancy affect worst-case routing time.

Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums

from arXiv: Data Structures and Algorithms

Authors: Qihao Zhou

We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\varepsilon\le c_0L^2/(nρ)$ is $\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\varepsilon\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\tilde O(n+\sqrt{n}LΔ\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.

Authors: Qihao Zhou

We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\varepsilon\le c_0L^2/(nρ)$ is $\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\varepsilon\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\tilde O(n+\sqrt{n}LΔ\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.

Near linear mixing of random-cluster Glauber dynamics on random regular graphs

from arXiv: Data Structures and Algorithms

Authors: Zhangsong Li

We study the random-cluster model on locally treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p \in (0,1)$ and a clustering weight $q \in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, at parameter $p$ below the tree uniqueness threshold $p_{\mathsf u}(q,d)$, it is expected that sampling is easy and local Markov chains mix rapidly on all graphs with degree bounded by $d$. On typical graphs, the same is predicted at $p>p_{\mathsf s}(q,d)$, where $p_{\mathsf s}(q,d)$ is a second uniqueness transition point on the $d$-regular wired tree. Our result confirms this prediction for random regular graphs, showing that the random-cluster Glauber dynamics mix in time $O(n^{1+o(1)})$ on a random $d$-regular graph for all fixed $q>2$ and $p>p_{\mathsf s}$. For comparison, the best prior sampling results for these models that held for all $p>p_{\mathsf s}$ required $q\gtrsim \log d$ \cite{BGPZ26}. The key difficulty is that simply utilizing the contraction of cavity messages on $d$-regular trees does not suffice to cover the entire regime $q>2$. To address this issue, motivated by \cite{LS26}, we develop a careful bootstrap argument to show the shrinkage of disagreement under grand coupling. The core ideas of the proof were generated using GPT-6 and the Pharos system; the proof was then refined and simplified by the author.

Authors: Zhangsong Li

We study the random-cluster model on locally treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p \in (0,1)$ and a clustering weight $q \in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, at parameter $p$ below the tree uniqueness threshold $p_{\mathsf u}(q,d)$, it is expected that sampling is easy and local Markov chains mix rapidly on all graphs with degree bounded by $d$. On typical graphs, the same is predicted at $p>p_{\mathsf s}(q,d)$, where $p_{\mathsf s}(q,d)$ is a second uniqueness transition point on the $d$-regular wired tree. Our result confirms this prediction for random regular graphs, showing that the random-cluster Glauber dynamics mix in time $O(n^{1+o(1)})$ on a random $d$-regular graph for all fixed $q>2$ and $p>p_{\mathsf s}$. For comparison, the best prior sampling results for these models that held for all $p>p_{\mathsf s}$ required $q\gtrsim \log d$ \cite{BGPZ26}. The key difficulty is that simply utilizing the contraction of cavity messages on $d$-regular trees does not suffice to cover the entire regime $q>2$. To address this issue, motivated by \cite{LS26}, we develop a careful bootstrap argument to show the shrinkage of disagreement under grand coupling. The core ideas of the proof were generated using GPT-6 and the Pharos system; the proof was then refined and simplified by the author.

Sublinear Metric Distortion for Matching

from arXiv: Data Structures and Algorithms

Authors: Moses Charikar, Alireza Haqi, Jabari Hastings, Yichen Huang, Prasanna Ramakrishnan

We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.

Authors: Moses Charikar, Alireza Haqi, Jabari Hastings, Yichen Huang, Prasanna Ramakrishnan

We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.

Noncrossing is All You Need: Beyond Identical Delay Functions in Online Matching with Delays on General Metrics

from arXiv: Data Structures and Algorithms

Authors: Yossi Azar, Junhao Gan, Shahar Lewkowicz, Xiao Sun, Seeun William Umboh

Online optimization with delays has emerged as a powerful paradigm with numerous applica- tions, including matching, caching, server management, network design, and set cover. In most of these problems, competitive algorithms are known for arbitrary monotone delay functions, even when different requests are associated with different delay functions. The notable exception is online matching with delays, where all previously known algorithms achieving constant or polylogarithmic competitive ratios require the delay functions to be identical, either linear or concave, and known in advance. We study the online minimum-cost perfect matching with delays problem on general metrics. We introduce the broad class of noncrossing delay functions, which strictly generalizes the classic setting of identical concave delay functions while naturally modeling scenarios in which different requests may experience different delay costs. We present the first deterministic $O(1)$-competitive algorithm for the single-location online matching problem with noncrossing delay functions in the non-clairvoyant setting as well as a randomized $O(lognlogΔ+log^2Δ)$-competitive algorithm of an arbitrary metric space where $n$ is the number of points of the metric space and $Δ$ is its aspect ratio. Our result allows each request to have an arbitrary noncrossing delay function, thereby simultaneously removing the assumptions of identical delay functions, concavity, and clairvoyance that were required by all previous constant or polylogarithmic-competitive algorithms.

Authors: Yossi Azar, Junhao Gan, Shahar Lewkowicz, Xiao Sun, Seeun William Umboh

Online optimization with delays has emerged as a powerful paradigm with numerous applica- tions, including matching, caching, server management, network design, and set cover. In most of these problems, competitive algorithms are known for arbitrary monotone delay functions, even when different requests are associated with different delay functions. The notable exception is online matching with delays, where all previously known algorithms achieving constant or polylogarithmic competitive ratios require the delay functions to be identical, either linear or concave, and known in advance. We study the online minimum-cost perfect matching with delays problem on general metrics. We introduce the broad class of noncrossing delay functions, which strictly generalizes the classic setting of identical concave delay functions while naturally modeling scenarios in which different requests may experience different delay costs. We present the first deterministic $O(1)$-competitive algorithm for the single-location online matching problem with noncrossing delay functions in the non-clairvoyant setting as well as a randomized $O(lognlogΔ+log^2Δ)$-competitive algorithm of an arbitrary metric space where $n$ is the number of points of the metric space and $Δ$ is its aspect ratio. Our result allows each request to have an arbitrary noncrossing delay function, thereby simultaneously removing the assumptions of identical delay functions, concavity, and clairvoyance that were required by all previous constant or polylogarithmic-competitive algorithms.

Flow-Optimal Separators for Minor-Free and Induced-Minor-Free Graphs in Linear Time via Follow the Regularized Leader

from arXiv: Data Structures and Algorithms

Authors: Hung Le, Jack Spalding-Jamieson

We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

Authors: Hung Le, Jack Spalding-Jamieson

We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

Do Language Models Know Which Constraints Belong Together?

from arXiv: Data Structures and Algorithms

Authors: Junbo Jacob Lian, Kaixuan Chen, Hanzhang Qin, Huiling Chen, Chaoyu Zhang, Chung-Piaw Teo

Dantzig-Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying a chosen set of constraint families, and every candidate costs a column generation run. The choice is hard when families strengthen the bound only together, because a search that adds one family at a time then stops at the linear relaxation. The classical inequalities of lot sizing, network design and facility location rest on such zero-marginal groups. We prove that when each evaluation convexifies at most $s$ of $k$ families, any method that observes only bounds needs a covering number of evaluations, of order $k^w$, to find a hidden group of $w$ families, also on actual Dantzig-Wolfe relaxations. A prediction that misses $e$ members reduces the cost to $Θ(k^e)$, through one staged search that does not need to know $e$, so the evaluations a prediction saves measure what the predictor knows. We study two sources of predictions. A one-pass lookahead at an optimal LP dual is provably blind to groups whose families share no variable, and a language model reads the names and text of the model. JointDW verifies the prediction of the language model with the staged search and interleaves it with the lookahead at a bounded loss. Groups that strengthen the bound only jointly appear in a quarter of the MIPLIB-NL models whose bound can be improved. There the strongest language models identify them about as often as the lookahead, their predictions rest on the names for textbook models, and reasoning about doubles the accuracy of the same model. On larger models with distracting families, the lookahead finds the group in one evaluation of three families. Language models given only the text seldom name it, while one reasoning model given every coefficient names it in all but one answer and then needs fewer evaluations than the lookahead for pairs of families.

Authors: Junbo Jacob Lian, Kaixuan Chen, Hanzhang Qin, Huiling Chen, Chaoyu Zhang, Chung-Piaw Teo

Dantzig-Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying a chosen set of constraint families, and every candidate costs a column generation run. The choice is hard when families strengthen the bound only together, because a search that adds one family at a time then stops at the linear relaxation. The classical inequalities of lot sizing, network design and facility location rest on such zero-marginal groups. We prove that when each evaluation convexifies at most $s$ of $k$ families, any method that observes only bounds needs a covering number of evaluations, of order $k^w$, to find a hidden group of $w$ families, also on actual Dantzig-Wolfe relaxations. A prediction that misses $e$ members reduces the cost to $Θ(k^e)$, through one staged search that does not need to know $e$, so the evaluations a prediction saves measure what the predictor knows. We study two sources of predictions. A one-pass lookahead at an optimal LP dual is provably blind to groups whose families share no variable, and a language model reads the names and text of the model. JointDW verifies the prediction of the language model with the staged search and interleaves it with the lookahead at a bounded loss. Groups that strengthen the bound only jointly appear in a quarter of the MIPLIB-NL models whose bound can be improved. There the strongest language models identify them about as often as the lookahead, their predictions rest on the names for textbook models, and reasoning about doubles the accuracy of the same model. On larger models with distracting families, the lookahead finds the group in one evaluation of three families. Language models given only the text seldom name it, while one reasoning model given every coefficient names it in all but one answer and then needs fewer evaluations than the lookahead for pairs of families.

Faster Approximate Text-to-Pattern Hamming Distances via Near-Optimal Sumset Estimation

from arXiv: Data Structures and Algorithms

Authors: Nick Fischer, Yael Kirkpatrick, Vasileios Nakos

\emph{Text-to-Pattern Hamming Distances} is one of the most studied problems in string algorithms: given a text $T$ of length~$n$ and a pattern~$P$, the task is to compute the Hamming distance between~$P$ and every length-$|P|$ substring of~$T$. As exact algorithms have been stuck at time~$\tilde O(n^{3/2})$ for decades, a long line of work has turned to the $(1+ε)$-approximate version of the problem. Milestones include Karloff's $\tilde O(ε^{-2} n)$-time algorithm [IPL '93], followed by randomized improvements to $\tilde O(ε^{-1} n)$ by Kopelowitz and Porat [FOCS '15] and recently to $\tilde O(ε^{-0.915} n)$ by Chan, Jin, Vassilevska W., and Xu [FOCS '23]. In this work, we significantly improve this bound to $\tilde O(ε^{-2/3} n)$, using a considerably simpler algorithm. Our algorithm follows the framework of Chan, Jin, Vassilevska W., and Xu, which reduces the problem to a variant of 3SUM that we call \emph{Sumset Estimation}. We solve Sumset Estimation optimally (up to logarithmic factors) by replacing their matrix multiplication techniques with ideas from sparse recovery. Specifically, we adapt the classic Count Sketch by replacing its random signs with random complex roots of unity---a simple idea that may find further applications for other additive problems. Finally, we give evidence that our running time might in fact be optimal: we show that the $ε^{-2/3}$ dependence is \emph{best possible} for a certain natural class of algorithms, including ours and all previous algorithms.

Authors: Nick Fischer, Yael Kirkpatrick, Vasileios Nakos

\emph{Text-to-Pattern Hamming Distances} is one of the most studied problems in string algorithms: given a text $T$ of length~$n$ and a pattern~$P$, the task is to compute the Hamming distance between~$P$ and every length-$|P|$ substring of~$T$. As exact algorithms have been stuck at time~$\tilde O(n^{3/2})$ for decades, a long line of work has turned to the $(1+ε)$-approximate version of the problem. Milestones include Karloff's $\tilde O(ε^{-2} n)$-time algorithm [IPL '93], followed by randomized improvements to $\tilde O(ε^{-1} n)$ by Kopelowitz and Porat [FOCS '15] and recently to $\tilde O(ε^{-0.915} n)$ by Chan, Jin, Vassilevska W., and Xu [FOCS '23]. In this work, we significantly improve this bound to $\tilde O(ε^{-2/3} n)$, using a considerably simpler algorithm. Our algorithm follows the framework of Chan, Jin, Vassilevska W., and Xu, which reduces the problem to a variant of 3SUM that we call \emph{Sumset Estimation}. We solve Sumset Estimation optimally (up to logarithmic factors) by replacing their matrix multiplication techniques with ideas from sparse recovery. Specifically, we adapt the classic Count Sketch by replacing its random signs with random complex roots of unity---a simple idea that may find further applications for other additive problems. Finally, we give evidence that our running time might in fact be optimal: we show that the $ε^{-2/3}$ dependence is \emph{best possible} for a certain natural class of algorithms, including ours and all previous algorithms.

Incremental Minimum Spanning Trees on a Napkin

from arXiv: Data Structures and Algorithms

Authors: Lucian Bicsi, Adrian Budău

In the incremental minimum spanning forest problem, weighted edges arrive one at a time. After each insertion we must report the edge that leaves the minimum spanning forest, and queries ask for the bottleneck distance of two vertices: the smallest possible maximum edge cost on a path between them. Link-cut trees solve both in $O(\log n)$ amortized time. In 2021 we wrote the folklore BottleneckTree, a short competitive-programming structure that stores one parent pointer and one cost per vertex, and conjectured that it also runs in $O(\log n)$ amortized time. We show that this conjecture is false: on a family of inputs, its amortized cost per operation is $Ω(\sqrt{n})$. We then present a structure with the same state, two arrays and a single loop, and prove that it runs in $O(\log n)$ amortized time per insertion and per query. Unlike the AM-tree of Ding, Gu and Sun, which attains the same bound, it stores no subtree sizes and maintains no balance invariant. For a connected graph with distinct costs, the structure can be viewed as an implicit link-cut tree on the Kruskal reconstruction tree: each rotation corresponds to exactly one preferred-child change, and each skip acts on a stale path-parent pointer. In experiments it is 1.3-2.1 times faster than the folklore BottleneckTree on insertions, up to 480 times faster on its adversarial input, and 6.3-16 times faster than an explicit link-cut tree.

Authors: Lucian Bicsi, Adrian Budău

In the incremental minimum spanning forest problem, weighted edges arrive one at a time. After each insertion we must report the edge that leaves the minimum spanning forest, and queries ask for the bottleneck distance of two vertices: the smallest possible maximum edge cost on a path between them. Link-cut trees solve both in $O(\log n)$ amortized time. In 2021 we wrote the folklore BottleneckTree, a short competitive-programming structure that stores one parent pointer and one cost per vertex, and conjectured that it also runs in $O(\log n)$ amortized time. We show that this conjecture is false: on a family of inputs, its amortized cost per operation is $Ω(\sqrt{n})$. We then present a structure with the same state, two arrays and a single loop, and prove that it runs in $O(\log n)$ amortized time per insertion and per query. Unlike the AM-tree of Ding, Gu and Sun, which attains the same bound, it stores no subtree sizes and maintains no balance invariant. For a connected graph with distinct costs, the structure can be viewed as an implicit link-cut tree on the Kruskal reconstruction tree: each rotation corresponds to exactly one preferred-child change, and each skip acts on a stale path-parent pointer. In experiments it is 1.3-2.1 times faster than the folklore BottleneckTree on insertions, up to 480 times faster on its adversarial input, and 6.3-16 times faster than an explicit link-cut tree.

Reachability Diameter: Beyond the Square-Root Barrier, with Polynomial Hardness

from arXiv: Data Structures and Algorithms

Authors: Hector Tierno, Shuang Yang

The reachability diameter of a directed graph is the maximum finite shortest-path distance across all pairs of vertices. It has received recent attention under this name, although the underlying quantity has long played a central role in shortcut set and parallel reachability. Notably, Bals et al. establish hardness for every polynomial-factor approximation in nearly-linear time for weighted digraphs. Yet they give $n^{1/2+o(1)}$-approximation in the unweighted regime with nearly-linear running time, and their conditional lower bounds rule out certain constants only, leaving open whether polynomial loss is necessary. We give a randomized algorithm that estimates the reachability diameter of an unweighted digraph within a factor $n^{1/3+o(1)}$, with high probability, in $\widetilde{O}(m+n)$ time. We complement our algorithm with a lower bound conditional on the 3-uniform Hyperclique hypothesis, ruling out an $n^{2/15-\varepsilon}$-approximation in $\widetilde{O}(m+n)$ time for every fixed $\varepsilon\in(0,2/15)$, establishing the necessity of polynomial loss for nearly-linear time algorithms.

Authors: Hector Tierno, Shuang Yang

The reachability diameter of a directed graph is the maximum finite shortest-path distance across all pairs of vertices. It has received recent attention under this name, although the underlying quantity has long played a central role in shortcut set and parallel reachability. Notably, Bals et al. establish hardness for every polynomial-factor approximation in nearly-linear time for weighted digraphs. Yet they give $n^{1/2+o(1)}$-approximation in the unweighted regime with nearly-linear running time, and their conditional lower bounds rule out certain constants only, leaving open whether polynomial loss is necessary. We give a randomized algorithm that estimates the reachability diameter of an unweighted digraph within a factor $n^{1/3+o(1)}$, with high probability, in $\widetilde{O}(m+n)$ time. We complement our algorithm with a lower bound conditional on the 3-uniform Hyperclique hypothesis, ruling out an $n^{2/15-\varepsilon}$-approximation in $\widetilde{O}(m+n)$ time for every fixed $\varepsilon\in(0,2/15)$, establishing the necessity of polynomial loss for nearly-linear time algorithms.

The Harmonic Online $k$-Server Algorithm Is Poly(k)-Competitive

from arXiv: Data Structures and Algorithms

Authors: Yichen Huang

The online $k$-server problem is one of the most fundamental problems in online algorithms. In this problem, $k$ servers are located in a metric space and serve a sequence of requests at points in the metric space by moving a server there. The Harmonic algorithm is a very natural algorithm: it chooses a server with probability inversely proportional to its distance from the request. It is also highly efficient, being memoryless and answering each request in $O(k)$ time. However, its competitive ratio has remained unsettled between the $O(2^k \log k)$ upper bound (Bartal and Grove, JACM 2000) and the $k(k+1)/2$ lower bound (Raghavan and Snir, 1994, IBM J. Res. Dev.). We prove that Harmonic is $2k^4$-competitive on every metric space against adaptive online adversaries. Our analysis uses a potential combining corrected electrical energy with the distances of online servers to their assigned adversary locations.

Authors: Yichen Huang

The online $k$-server problem is one of the most fundamental problems in online algorithms. In this problem, $k$ servers are located in a metric space and serve a sequence of requests at points in the metric space by moving a server there. The Harmonic algorithm is a very natural algorithm: it chooses a server with probability inversely proportional to its distance from the request. It is also highly efficient, being memoryless and answering each request in $O(k)$ time. However, its competitive ratio has remained unsettled between the $O(2^k \log k)$ upper bound (Bartal and Grove, JACM 2000) and the $k(k+1)/2$ lower bound (Raghavan and Snir, 1994, IBM J. Res. Dev.). We prove that Harmonic is $2k^4$-competitive on every metric space against adaptive online adversaries. Our analysis uses a potential combining corrected electrical energy with the distances of online servers to their assigned adversary locations.

Private Online Release of Adaptive Linear Queries

from arXiv: Data Structures and Algorithms

Authors: Aleksandar Nikolov, Haohua Tang

In this paper we study answering counting and other linear queries online under differential privacy constraints. Recent work by Nikolov, Tang, and Ullman [STOC 2026] gave the first online algorithm that answers \emph{obliviously chosen} online linear queries privately, and achieves poly-logarithmic competitiveness against an optimal offline factorization mechanism. This also implied poly-logarithmic competitiveness against arbitrary private offline mechanisms when the private dataset is large enough. The error guarantees of their algorithm, however, can fail when the queries are adaptive. We give a new algorithm that also achieves poly-logarithmic competitiveness, while being able to handle \emph{adaptively chosen} queries. This implies that there is no large separation between the optimal error bounds for privately answering, respectively, offline linear queries, obliviously chosen online linear queries, and adaptively chosen online linear queries. Our algorithm combines the online factorization approach from prior work with a novel layered Euclidean iterative database construction, presenting a new way to use factorization in private algorithms. We also give a lower bound that shows that some poly-logarithmic factor in the competitive ratio is unavoidable. Our lower bound first shows that any interactive private mechanism implies a deterministic online factorization algorithm, and then gives a new lower bound for the competitive ratio of any deterministic online factorization algorithm.

Authors: Aleksandar Nikolov, Haohua Tang

In this paper we study answering counting and other linear queries online under differential privacy constraints. Recent work by Nikolov, Tang, and Ullman [STOC 2026] gave the first online algorithm that answers \emph{obliviously chosen} online linear queries privately, and achieves poly-logarithmic competitiveness against an optimal offline factorization mechanism. This also implied poly-logarithmic competitiveness against arbitrary private offline mechanisms when the private dataset is large enough. The error guarantees of their algorithm, however, can fail when the queries are adaptive. We give a new algorithm that also achieves poly-logarithmic competitiveness, while being able to handle \emph{adaptively chosen} queries. This implies that there is no large separation between the optimal error bounds for privately answering, respectively, offline linear queries, obliviously chosen online linear queries, and adaptively chosen online linear queries. Our algorithm combines the online factorization approach from prior work with a novel layered Euclidean iterative database construction, presenting a new way to use factorization in private algorithms. We also give a lower bound that shows that some poly-logarithmic factor in the competitive ratio is unavoidable. Our lower bound first shows that any interactive private mechanism implies a deterministic online factorization algorithm, and then gives a new lower bound for the competitive ratio of any deterministic online factorization algorithm.

Fast Quantum Byzantine Agreement with Nearly Linear Quantum Communication

from arXiv: Data Structures and Algorithms

Authors: Jan Olkowski

We consider synchronous binary Byzantine agreement among $n$ processes connected by authenticated point-to-point classical and quantum channels. The adversary is computationally unbounded, sees all classical information, adaptively corrupts up to $t\le(1/3-\varepsilon)n$ processes, and may suppress messages sent by a process in the round in which it is corrupted. We give a quantum algorithm that solves Byzantine agreement in this model with probability $1$. It uses $O(\log^4 n)$ rounds, $O(n\log^8 n)$ qubits, and $O(n^2\log^8 n)$ classical bits in expectation, with constants depending on $\varepsilon$. This reduces the quantum communication of the protocol of Ben-Or and Hassidim (STOC~2005), which achieves $O(1)$ expected rounds but whose Byzantine coin implementation requires $Ω(n^4\log n)$ qubits and classical bits. We also extend the quadratic message lower bound of Abraham et al.\ (PODC~2019) to quantum channels, obtaining an $Ω(t^2)$ lower bound on worst-case expected total communication. For $t=Θ(n)$, the communication and round complexity of our algorithm are therefore optimal up to polylogarithmic factors.

Authors: Jan Olkowski

We consider synchronous binary Byzantine agreement among $n$ processes connected by authenticated point-to-point classical and quantum channels. The adversary is computationally unbounded, sees all classical information, adaptively corrupts up to $t\le(1/3-\varepsilon)n$ processes, and may suppress messages sent by a process in the round in which it is corrupted. We give a quantum algorithm that solves Byzantine agreement in this model with probability $1$. It uses $O(\log^4 n)$ rounds, $O(n\log^8 n)$ qubits, and $O(n^2\log^8 n)$ classical bits in expectation, with constants depending on $\varepsilon$. This reduces the quantum communication of the protocol of Ben-Or and Hassidim (STOC~2005), which achieves $O(1)$ expected rounds but whose Byzantine coin implementation requires $Ω(n^4\log n)$ qubits and classical bits. We also extend the quadratic message lower bound of Abraham et al.\ (PODC~2019) to quantum channels, obtaining an $Ω(t^2)$ lower bound on worst-case expected total communication. For $t=Θ(n)$, the communication and round complexity of our algorithm are therefore optimal up to polylogarithmic factors.

Wednesday, November 04

TR26-302 | Tight Depth Lower Bounds for Stretch Amplification | Oliver Korten

from ECCC Papers
We investigate the depth overhead of stretch-amplification for pseudorandom generators in the black-box model. We show that any black box construction of a PRG $H: \{0,1\}^m \to \{0,1\}^M$ which makes $\mathrm{poly}(n)$ queries to a PRG $G: \{0,1\}^n \to \{0,1\}^N$ in $d$ adaptive phases during each evaluation (and with a black-box reduction to the security of $G$) must satisfy: $M \leq (m + O(\log n)) \Bigl( \frac{N}{n - O(\log n)} \Bigr)^d$. This implies that the standard amplification procedure [GM'84, GGM'86] based on repeated composition of $G$ is essentially optimal. In particular, amplifying $n \mapsto n+1$ to $m \mapsto 2m$ requires depth overhead $d = \Omega(\frac{n}{\log n})$, and amplifying $n \mapsto 2n$ to $m \mapsto m^2$ requires depth overhead $d = \Omega(\log n)$; each of these bounds is tight up to a constant factor. As an immediate corollary we obtain tight lower bounds on the depth complexity of black-box constructions of PRFs from PRGs. Both of these problems (pertaining to amplification and PRF construction) have been studied in various prior works [V'05, MV'11, BJP'11, MV'15, BMM'24, ADV'26]. Despite progress on special cases in which the structure of the reductions is constrained, it was previously unknown whether completely parallel constructions ($d=1$) existed in every stretch regime. The kinds of amplification procedures we investigate have more recently found use outside of cryptography in the study of range avoidance ("Avoid") [K'21]. We show an analogous result in this setting: any black-box $\mathrm{FP^{NP}}$ reduction from an Avoid instance $G: \{0,1\}^n \to \{0,1\}^N$ to a second Avoid instance $H: \{0,1\}^m \to \{0,1\}^M$ which has "depth $d$" in the above sense must satisfy the same bound relating $n,N,m,M,d$. Using this impossibility result for Avoid, we construct an oracle relative to which $\mathrm{E} \not\subseteq \mathrm{NC}/\mathrm{poly}$ but $\mathrm{E^{NP}} \subseteq \mathrm{P}/\mathrm{poly}$. Our oracle separation implies that in the relativized setting, the bounded arithmetic theory $S^1_2$ cannot prove the existence of functions outside $\mathrm{P}/\mathrm{poly}$, even given as an axiom the existence of functions outside $\mathrm{NC}/\mathrm{poly}$.
We investigate the depth overhead of stretch-amplification for pseudorandom generators in the black-box model. We show that any black box construction of a PRG $H: \{0,1\}^m \to \{0,1\}^M$ which makes $\mathrm{poly}(n)$ queries to a PRG $G: \{0,1\}^n \to \{0,1\}^N$ in $d$ adaptive phases during each evaluation (and with a black-box reduction to the security of $G$) must satisfy: $M \leq (m + O(\log n)) \Bigl( \frac{N}{n - O(\log n)} \Bigr)^d$. This implies that the standard amplification procedure [GM'84, GGM'86] based on repeated composition of $G$ is essentially optimal. In particular, amplifying $n \mapsto n+1$ to $m \mapsto 2m$ requires depth overhead $d = \Omega(\frac{n}{\log n})$, and amplifying $n \mapsto 2n$ to $m \mapsto m^2$ requires depth overhead $d = \Omega(\log n)$; each of these bounds is tight up to a constant factor. As an immediate corollary we obtain tight lower bounds on the depth complexity of black-box constructions of PRFs from PRGs. Both of these problems (pertaining to amplification and PRF construction) have been studied in various prior works [V'05, MV'11, BJP'11, MV'15, BMM'24, ADV'26]. Despite progress on special cases in which the structure of the reductions is constrained, it was previously unknown whether completely parallel constructions ($d=1$) existed in every stretch regime. The kinds of amplification procedures we investigate have more recently found use outside of cryptography in the study of range avoidance ("Avoid") [K'21]. We show an analogous result in this setting: any black-box $\mathrm{FP^{NP}}$ reduction from an Avoid instance $G: \{0,1\}^n \to \{0,1\}^N$ to a second Avoid instance $H: \{0,1\}^m \to \{0,1\}^M$ which has "depth $d$" in the above sense must satisfy the same bound relating $n,N,m,M,d$. Using this impossibility result for Avoid, we construct an oracle relative to which $\mathrm{E} \not\subseteq \mathrm{NC}/\mathrm{poly}$ but $\mathrm{E^{NP}} \subseteq \mathrm{P}/\mathrm{poly}$. Our oracle separation implies that in the relativized setting, the bounded arithmetic theory $S^1_2$ cannot prove the existence of functions outside $\mathrm{P}/\mathrm{poly}$, even given as an axiom the existence of functions outside $\mathrm{NC}/\mathrm{poly}$.

Less Popular Than Coal

from Ben Recht
On the datacenter referendums and the need for an AI reboot

Hi there, argmin readers! Today’s post is a late and brief reaction to a tiny subplot of yesterday’s historic election: the widespread rejection of AI datacenters. It was inspired by an insightful Dan Davies thread on bluesky that I have been unable to get out of my head.

Elections are a muddled measurement of “public opinion,” but they’re probably the only real measurement we get. And with every surprising measurement, a good rational agent must reflect on its beliefs. So what should we passive rational observers make of this historically large swing towards the Democratic Party in yesterday’s vote?

Obviously, this is a major repudiation of Donald Trump, his policies, and the Republican Party. However, you readers probably don’t come here for my political takes, and I don’t want to use this Substack for them anyway. Even if it wasn’t the biggest issue on the ballot, something closer to home is worth pondering. The American people are loud and clear that they hate AI.

Nearly fifty measures explicitly about building data centers appeared on ballots across the nation. All of them look like they will vote to limit further construction. A datacenter moratorium was part of Gina Hinojosa’s platform in her successful bid to become the first Democratic Party governor of Texas in over 30 years. The anti-datacenter position was undeniably the winning one. Poll results from this last summer showed people would rather have a coal plant in their backyard than a datacenter. Now elections confirm these sentiments were real.

We could try to convince ourselves that this sentiment is just about datacenters themselves. They are gross, gigantic buildings that employ no one, make a ton of noise, and are often powered with dirty energy. Who would want one of those? But datacenters are also a big, ugly, legible signal of a deeply unpopular industry.

At some point, we techy folks have to reckon with how much people hate what we build. We can point to revealed preferences all we want (people are adopting it! It’s making money!), but we did this during the social media boom of the 2010s (people spend a ton of time doomscrolling, therefore it’s good!). These economic numbers are also measurements, but we have to reconcile them with other data. Adoption and economic growth are so concentrated in so few hands that, while this props up our stock indices, it doesn’t lead to tangible benefits for everyone else.

Certainly, it doesn’t help that the industry has been selling itself as coming for everyone’s jobs. It doesn’t help that many prominent people working at these companies take to social media to gleefully exclaim that they think there’s more than a 10 percent chance that their company will kill everyone on earth. It doesn’t help that the big AI companies have been caught committing felonious cybercrimes all summer. It also doesn’t help that the leaders of these companies have been quite visibly embracing the Trump administration. Hugging a loser is a bad look, guys.

It’s a shame, because the technology is truly amazing. The AI true believers (accidentally) built something very cool that I would not have thought possible five years ago. But this amazing technology makes everyone miserable! It even immiserates its biggest boosters, who get sucked into productivity doom loops managing swarms of agents that speak in a unison chorus of Claudish. What’s our positive case here? The torment nexus of “superintelligence” ain’t it.

I have a lot of friends, colleagues, and alums who make a living building AI models. And I guess a bunch will be infinitely rich after the upcoming IPOs. But if you’re working for something more hated than the tobacco industry, maybe it’s way past time you stood up to your employer and demanded a better direction. And no, an open letter doesn’t count.

I’m not hopeful that this will happen. Ten years ago, I wrote a blog post shell-shocked by the election of Trump and disgusted with my industry. The cameo from Sam Altman explains a lot of my pessimism. I’ve been soul-searching for a decade. The leaders of the tech industry, instead of offering a positive vision and doing better, ended up sucking up to the new authoritarian government. They did it with pouting and feet dragging in 2016. They did it with their chests out in a celebratory embrace in 2024.

Our industry and field need a complete reboot. But that requires collective action. How do we fix it? What do we do next? I’d love to hear your thoughts on how the tech sector can clean up its image and make a positive, populist case for itself.

Subscribe now

By Ben Recht

Postdoc at UC Berkeley (apply by December 1, 2026)

from CCI: jobs
The Simons Institute at UC Berkeley invites applications for two new postdoctoral-level research fellowships for a two-year period on the intersections of AI and TCS. Postdocs will be associated with the Institute’s newly established Research Pod for AI and TCS. Website: simons.berkeley.edu/research-pods/ai-for-tcs-research-pod Email: cawinter@berkeley.edu

The Simons Institute at UC Berkeley invites applications for two new postdoctoral-level research fellowships for a two-year period on the intersections of AI and TCS. Postdocs will be associated with the Institute’s newly established Research Pod for AI and TCS.

Website: https://simons.berkeley.edu/research-pods/ai-for-tcs-research-pod
Email: cawinter@berkeley.edu

By shacharlovett

Another Step Towards Optiland

from Computational Complexity
When the Clay Mathematics Institute announced the Millennium Prize Problems back in 2000, we joked that if you proved P = NP, you wouldn't only get the $1 million prize for solving P v NP, but you would get $7 million since you could use P = NP to solve the rest.
In theory, if P = NP, then you could efficiently find the proof of any theorem, provided its formal proof was not too long. In practice, we don't have P = NP, and back then we didn't even have a good way to formalize a proof. Fast forward to today and we have a computer generated formalized proof of Navier-Stokes and a number of other important problems.
In my 2022 CACM article, I argued that we were heading towards Optiland, where we got all of the goodies of P = NP, without losing cryptography. But some challenges remained.
Machine learning seems to fail learning simple arithmetic—for example, summing up a large collection of numbers or multiplying large numbers. One could imagine combining machine learning with symbolic mathematical tools. While we’ve seen some impressive advances in theorem provers, we sit a long way from my dream task of taking one of my research papers, with its informal proofs, and having an AI system fill in the details and verify the proof.

I failed to imagine that in less than five years AI models would not just fill in the details of my proofs but find the proofs themselves.

Still plenty of math problems that AI hasn't solved, including P v NP itself. I've given up predicting what AI will prove next, though I still believe P v NP requires new techniques that AI won't come up with on its own.

I watch discussions on whether we could find some version of the ETH-refutation algorithm that would help solve SAT problems in practice. Spoiler alert: It will never beat CDCL.

We are running out of examples of real-world problems we still need P = NP to tackle. Outside of breaking codes, what could you do with P = NP that you couldn't do today using other means?

By Lance Fortnow

When the Clay Mathematics Institute announced the Millennium Prize Problems back in 2000, we joked that if you proved P = NP, you wouldn't only get the $1 million prize for solving P v NP, but you would get $7 million since you could use P = NP to solve the rest.

In theory, if P = NP, then you could efficiently find the proof of any theorem, provided its formal proof was not too long. In practice, we don't have P = NP, and back then we didn't even have a good way to formalize a proof. Fast forward to today and we have a computer generated formalized proof of Navier-Stokes and a number of other important problems.

In my 2022 CACM article, I argued that we were heading towards Optiland, where we got all of the goodies of P = NP, without losing cryptography. But some challenges remained.
Machine learning seems to fail learning simple arithmetic—for example, summing up a large collection of numbers or multiplying large numbers. One could imagine combining machine learning with symbolic mathematical tools. While we’ve seen some impressive advances in theorem provers, we sit a long way from my dream task of taking one of my research papers, with its informal proofs, and having an AI system fill in the details and verify the proof.

I failed to imagine that in less than five years AI models would not just fill in the details of my proofs but find the proofs themselves.

Still plenty of math problems that AI hasn't solved, including P v NP itself. I've given up predicting what AI will prove next, though I still believe P v NP requires new techniques that AI won't come up with on its own.

I watch discussions on whether we could find some version of the ETH-refutation algorithm that would help solve SAT problems in practice. Spoiler alert: It will never beat CDCL.

We are running out of examples of real-world problems we still need P = NP to tackle. Outside of breaking codes, what could you do with P = NP that you couldn't do today using other means?

By Lance Fortnow

STOC submissions are online, a dream finally come true!

from Emanuele Viola
I asked for this in one of the earliest posts on this blog, 12 years ago. The STOC committee has finally decided to post online the submissions, thus granting us all the option to follow what is happening in the greater theory community (see my blog post for more on my reasons). I guess that’s […]

I asked for this in one of the earliest posts on this blog, 12 years ago. The STOC committee has finally decided to post online the submissions, thus granting us all the option to follow what is happening in the greater theory community (see my blog post for more on my reasons). I guess that’s what it takes, an Alien Invasion.

By Manu

TR26-301 | Asymptotic non-negative rank can be exponentially smaller than non-negative rank | Pavel Hrubes

from ECCC Papers
We investigate the behavior of the non-negative rank of a matrix, $\hbox{rk}_+$, on Kronecker product, $\times$. Beasley et al. [2013] constructed matrices with $\hbox{rk}_+(M\times M)$ strictly smaller than $\hbox{rk}_+(M)^2$. We will show that i) there exists an $M$ with $\hbox{rk}_+(M\times M)\leq O(\hbox{rk}_+(M)\log(\hbox{rk}_+(M))$. Defining asymptotic non-negative rank as $\overline{\hbox{rk}}_+(M):= \lim_{k\rightarrow \infty} \sqrt[k]{{\hbox{rk}}_+(M^{(k)})}$, where $M^{(k)}$ is the $k$-th Kronecker power, we show that ii) there exists an $M$ with $\overline{\hbox{rk}}_+(M)\leq O(\log \hbox{rk}_+(M))$, iii) there exist $M,N$ with $\overline{\hbox{rk}}_+(M\times N)<\overline{\hbox{rk}}_+(M)\overline{\hbox{rk}}_+(N)$. We also refute a conjecture of Vandaele et al. [2016] that $\hbox{rk}_+(M\times M)\geq \Omega(\hbox{rk}_+(M)\hbox{rk}(M))$.
We investigate the behavior of the non-negative rank of a matrix, $\hbox{rk}_+$, on Kronecker product, $\times$. Beasley et al. [2013] constructed matrices with $\hbox{rk}_+(M\times M)$ strictly smaller than $\hbox{rk}_+(M)^2$. We will show that i) there exists an $M$ with $\hbox{rk}_+(M\times M)\leq O(\hbox{rk}_+(M)\log(\hbox{rk}_+(M))$. Defining asymptotic non-negative rank as $\overline{\hbox{rk}}_+(M):= \lim_{k\rightarrow \infty} \sqrt[k]{{\hbox{rk}}_+(M^{(k)})}$, where $M^{(k)}$ is the $k$-th Kronecker power, we show that ii) there exists an $M$ with $\overline{\hbox{rk}}_+(M)\leq O(\log \hbox{rk}_+(M))$, iii) there exist $M,N$ with $\overline{\hbox{rk}}_+(M\times N)<\overline{\hbox{rk}}_+(M)\overline{\hbox{rk}}_+(N)$. We also refute a conjecture of Vandaele et al. [2016] that $\hbox{rk}_+(M\times M)\geq \Omega(\hbox{rk}_+(M)\hbox{rk}(M))$.

TR26-300 | Factor Closure in Noncommutative Algebraic Complexity | Pushkar Joglekar, Vikraman Arvind

from ECCC Papers
We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant outer factors $a,b$, has a circuit of size $O(sD^6)$. The proof uses a small weighted nondeterministic finite automaton (NFA) with replacement labels. Given $f=gh$, its labels depend on a word of maximum degree in $h$ and the coefficients of its suffixes. A pairing of accepting paths proves that the induced linear map $T$ satisfies $T(gh)=g$. Matrix substitution implements $T$ in a circuit. Applying the same substitution to a Boolean-sum verifier proves $\VNPnc$ closure. The construction also proves formula closure. The automaton has one copying state and, after leaving it, uses only scalar replacements. This structure gives formulas of size $O(s^2)$ for one-sided factors and $O(s^4)$ for central factors of a size-$s$ formula. The same map proves that polynomial-size noncommutative algebraic branching programs are closed under taking factors. Polynomial-size ABP factorizations over finite fields are already known from~\cite{AJ25}. These are arithmetic size bounds, with arbitrary field constants at unit cost, including over $\Q$. Over finite fields, combining the formula construction with~\cite{AJ22} gives randomized polynomial-time complete factorization of formula inputs, with factors output as formulas. For polynomial-degree circuit inputs, we obtain polynomial-length succinct ABP descriptions with polynomial-time edge queries. We also give two algorithms that output complete ordered irreducible factorizations as polynomial-size circuits: a deterministic $\FP^{\SigTwo}$ algorithm and a zero-error randomized algorithm with an NP oracle and expected polynomial running time.
We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant outer factors $a,b$, has a circuit of size $O(sD^6)$. The proof uses a small weighted nondeterministic finite automaton (NFA) with replacement labels. Given $f=gh$, its labels depend on a word of maximum degree in $h$ and the coefficients of its suffixes. A pairing of accepting paths proves that the induced linear map $T$ satisfies $T(gh)=g$. Matrix substitution implements $T$ in a circuit. Applying the same substitution to a Boolean-sum verifier proves $\VNPnc$ closure. The construction also proves formula closure. The automaton has one copying state and, after leaving it, uses only scalar replacements. This structure gives formulas of size $O(s^2)$ for one-sided factors and $O(s^4)$ for central factors of a size-$s$ formula. The same map proves that polynomial-size noncommutative algebraic branching programs are closed under taking factors. Polynomial-size ABP factorizations over finite fields are already known from~\cite{AJ25}. These are arithmetic size bounds, with arbitrary field constants at unit cost, including over $\Q$. Over finite fields, combining the formula construction with~\cite{AJ22} gives randomized polynomial-time complete factorization of formula inputs, with factors output as formulas. For polynomial-degree circuit inputs, we obtain polynomial-length succinct ABP descriptions with polynomial-time edge queries. We also give two algorithms that output complete ordered irreducible factorizations as polynomial-size circuits: a deterministic $\FP^{\SigTwo}$ algorithm and a zero-error randomized algorithm with an NP oracle and expected polynomial running time.

TR26-299 | Approximate Polynomial Satisfiability and Integral Closure Membership in the Counting Hierarchy | Sanyam Agarwal, Archit Hari, Anurag Pandey, Himanshu Shukla

from ECCC Papers
Andrews, Garg, and Schost (FOCS~2026) recently, in a breakthrough result, placed Hilbert's Nullstellensatz (HN) in the Counting Hierarchy ($CH$) using constant-depth circuits for multivariate resultants. While HN deals with exact polynomial satisfiability, Guo, Saxena, and Sinhababu (CCC~2018; Theory of Computing~2019) introduced approximate polynomial satisfiability (APS) as a natural approximative analogue of polynomial system satisfiability. They proved its ${NP}$-hardness and a $PSPACE$ upper bound, and used APS to construct hitting sets for $\overline{{VP}}$ in polynomial space, substantially narrowing the GCT chasm in geometric complexity theory. We place APS in the Counting Hierarchy for rational polynomials given by arithmetic circuits. Our proof combines Noether normalization with a geometric restriction on how points in the image of a polynomial map approach the origin, reducing APS to a test on multivariate resultants which allows solvability in $CH$. It complements the independent and concurrent proofs of Balaji, Shirmohammadi, Tavenas, and Worrell, and Andrews, Garg, Oliveira, and Schost. Moreover, we also place integral closure membership testing in the Counting Hierarchy. Here, the question is whether a given polynomial $f$ is integral over $\mathbb{C}[f_1,\ldots,f_m]$. Our approach uses a global annihilator criterion and an effective degree bound to reduce non-integrality to APS in polynomial time. In fact, we show that the reduction holds both ways, and Integral Closure Membership Testing is equivalent to APS under polynomial-time reductions.
Andrews, Garg, and Schost (FOCS~2026) recently, in a breakthrough result, placed Hilbert's Nullstellensatz (HN) in the Counting Hierarchy ($CH$) using constant-depth circuits for multivariate resultants. While HN deals with exact polynomial satisfiability, Guo, Saxena, and Sinhababu (CCC~2018; Theory of Computing~2019) introduced approximate polynomial satisfiability (APS) as a natural approximative analogue of polynomial system satisfiability. They proved its ${NP}$-hardness and a $PSPACE$ upper bound, and used APS to construct hitting sets for $\overline{{VP}}$ in polynomial space, substantially narrowing the GCT chasm in geometric complexity theory. We place APS in the Counting Hierarchy for rational polynomials given by arithmetic circuits. Our proof combines Noether normalization with a geometric restriction on how points in the image of a polynomial map approach the origin, reducing APS to a test on multivariate resultants which allows solvability in $CH$. It complements the independent and concurrent proofs of Balaji, Shirmohammadi, Tavenas, and Worrell, and Andrews, Garg, Oliveira, and Schost. Moreover, we also place integral closure membership testing in the Counting Hierarchy. Here, the question is whether a given polynomial $f$ is integral over $\mathbb{C}[f_1,\ldots,f_m]$. Our approach uses a global annihilator criterion and an effective degree bound to reduce non-integrality to APS in polynomial time. In fact, we show that the reduction holds both ways, and Integral Closure Membership Testing is equivalent to APS under polynomial-time reductions.

TR26-298 | Semidirect sums of tensors and the matrix multiplication exponent | Baitian Li

from ECCC Papers
This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a generally applicable method for analyzing tensors. We also discuss further consequences of this method, as well as the relation between the OpenAI proof and previous work.
This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a generally applicable method for analyzing tensors. We also discuss further consequences of this method, as well as the relation between the OpenAI proof and previous work.

TR26-297 | An Exponential Top Fan-In Hierarchy for Border Depth-4 Circuits | Anakin Dey, Zeyu Guo, Pranjal Dutta

from ECCC Papers
We study depth-$4$ algebraic circuits with bounded top and bottom fan-in over fields of characteristic $0$. These circuits compute sums of at most $k$ products of polynomials of degree at most $\delta$. We denote this class of circuits by $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. Our hard polynomial for this class is $P_{k + 1,n}$, the sum of $k + 1$ monomials of degree $n$ on disjoint sets of variables. It can be computed by a depth-$3$ circuit of top fan-in $k + 1$ and size $O(kn)$. We give an impossibility result showing that for all constants $k$ and $\delta$ and all sufficiently large $n$, no $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuit computes $P_{k + 1,n}$. For $k \geq 2$ and $\delta = 2$, this follows implicitly from the rank bound for $\Sigma^{[k]}\Pi\Sigma\Pi^{[2]}$ identities of Garg, Oliveira, Sengupta, Shalmon, and Shpilka (CCC 2026). To the best of our knowledge, this impossibility result was not known even for $k \geq 2$ and $\delta = 3$. We also turn to border complexity, a notion central to geometric complexity theory, in which a polynomial only needs to be approximated arbitrarily well. Here impossibility fails over algebraically closed fields: Kumar (TOCT 2020) showed that $\overline{\Sigma^{[2]}\Pi\Sigma}$ circuits already approximate every polynomial, albeit with exponential size. In turn, we show that every $\overline{\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}}$ circuit computing $P_{k + 1,n}$ must have size $2^{\Omega(n)}$ when $k$ and $\delta$ are constants, in stark contrast to the exact depth-$3$ circuit of top fan-in $k + 1$ and size $O(kn)$. This lower bound gives an exponential top fan-in hierarchy for border depth-$4$ circuits of bounded bottom fan-in and answers the main open question of Dutta and Saxena (FOCS 2022, JACM 2026), who gave an analogous hierarchy for the case $\delta = 1$, i.e., depth $3$. Along the way, we develop tools for border depth-$4$ circuits that apply to arbitrary polynomials and may have further applications.
We study depth-$4$ algebraic circuits with bounded top and bottom fan-in over fields of characteristic $0$. These circuits compute sums of at most $k$ products of polynomials of degree at most $\delta$. We denote this class of circuits by $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$. Our hard polynomial for this class is $P_{k + 1,n}$, the sum of $k + 1$ monomials of degree $n$ on disjoint sets of variables. It can be computed by a depth-$3$ circuit of top fan-in $k + 1$ and size $O(kn)$. We give an impossibility result showing that for all constants $k$ and $\delta$ and all sufficiently large $n$, no $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuit computes $P_{k + 1,n}$. For $k \geq 2$ and $\delta = 2$, this follows implicitly from the rank bound for $\Sigma^{[k]}\Pi\Sigma\Pi^{[2]}$ identities of Garg, Oliveira, Sengupta, Shalmon, and Shpilka (CCC 2026). To the best of our knowledge, this impossibility result was not known even for $k \geq 2$ and $\delta = 3$. We also turn to border complexity, a notion central to geometric complexity theory, in which a polynomial only needs to be approximated arbitrarily well. Here impossibility fails over algebraically closed fields: Kumar (TOCT 2020) showed that $\overline{\Sigma^{[2]}\Pi\Sigma}$ circuits already approximate every polynomial, albeit with exponential size. In turn, we show that every $\overline{\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}}$ circuit computing $P_{k + 1,n}$ must have size $2^{\Omega(n)}$ when $k$ and $\delta$ are constants, in stark contrast to the exact depth-$3$ circuit of top fan-in $k + 1$ and size $O(kn)$. This lower bound gives an exponential top fan-in hierarchy for border depth-$4$ circuits of bounded bottom fan-in and answers the main open question of Dutta and Saxena (FOCS 2022, JACM 2026), who gave an analogous hierarchy for the case $\delta = 1$, i.e., depth $3$. Along the way, we develop tools for border depth-$4$ circuits that apply to arbitrary polynomials and may have further applications.

TFNP in the Random Oracle Model

from arXiv: Computational Complexity

Authors: Noah Fleming, Mika Göös, Ziyi Guan, Oliver Korten, Daniel Mitropolsky, Toniann Pitassi, Artur Riazanov, Dmitry Sokolov, Tianqi Yang, Weiqiang Yuan

We determine which major $\TFNP$ classes are hard in the random oracle model (ROM). Prior work establishes average-case hardness of highly structured subclasses of $\TFNP$ from strong cryptographic assumptions such as LWE and indistinguishability obfuscation. It is not known whether weaker assumptions such as one-way functions imply hardness of any of the standard $\TFNP$ subclasses, even in the worst case. The ROM, where all parties have query access to a random function, lies, morally, between these two regimes: a random function is one-way, but public-key cryptography cannot be built from it in a black-box way. We refer to this setting as \emph{Romantica}. Our first result settles a long-standing open problem: $\PPAD$, the class of total search problems reducible to finding a Nash equilibrium, is hard on average in Romantica. Combined with the known hardness of PLS (local search), this gives average-case hardness of $\EoPL = \PPAD \cap \PLS$. In contrast, we show that $\UEoPL$ can be solved in Romantica with polynomially many oracle queries by a computationally unbounded solver, a sharp distinction between $\EoPL$ and $\UEoPL$. This resolves the black-box hardness landscape of all major $\TFNP$ classes in Romantica. Finally, we initiate the study of separations between $\TFNP$ subclasses in Romantica: $\PWPP$ and $\PPA$ do not reduce to $\PLS$ under \emph{public-coin} reductions.

Authors: Noah Fleming, Mika Göös, Ziyi Guan, Oliver Korten, Daniel Mitropolsky, Toniann Pitassi, Artur Riazanov, Dmitry Sokolov, Tianqi Yang, Weiqiang Yuan

We determine which major $\TFNP$ classes are hard in the random oracle model (ROM). Prior work establishes average-case hardness of highly structured subclasses of $\TFNP$ from strong cryptographic assumptions such as LWE and indistinguishability obfuscation. It is not known whether weaker assumptions such as one-way functions imply hardness of any of the standard $\TFNP$ subclasses, even in the worst case. The ROM, where all parties have query access to a random function, lies, morally, between these two regimes: a random function is one-way, but public-key cryptography cannot be built from it in a black-box way. We refer to this setting as \emph{Romantica}. Our first result settles a long-standing open problem: $\PPAD$, the class of total search problems reducible to finding a Nash equilibrium, is hard on average in Romantica. Combined with the known hardness of PLS (local search), this gives average-case hardness of $\EoPL = \PPAD \cap \PLS$. In contrast, we show that $\UEoPL$ can be solved in Romantica with polynomially many oracle queries by a computationally unbounded solver, a sharp distinction between $\EoPL$ and $\UEoPL$. This resolves the black-box hardness landscape of all major $\TFNP$ classes in Romantica. Finally, we initiate the study of separations between $\TFNP$ subclasses in Romantica: $\PWPP$ and $\PPA$ do not reduce to $\PLS$ under \emph{public-coin} reductions.

Quantum Advantage has Few Complexity Implications

from arXiv: Computational Complexity

Authors: Eli Goldin, Alex Lombardi, Tomoyuki Morimae

We consider the complexity implications of proofs of quantumness and (quantum verifier) oblivious state preparation. We show that both of these objects can coexist with the easiness of nondeterministic computation. That is, we give evidence that proofs of quantumness can exist when $\mathsf{P} = \mathsf{NP}$ and that (quantum verifier) oblivious state preparation can exist when $\mathsf{BQP} = \mathsf{QCMA}$. In particular, we give a classical oracle relative to which (1) $\mathsf{P}^\mathcal{O} = \mathsf{NP}^\mathcal{O}$ (2) $\mathsf{BQP}^\mathcal{O} = \mathsf{PSPACE}^\mathcal{O}$ (3) Proofs of quantumness exist. In particular, there exists a language in $\mathsf{IP}^\mathcal{O} \subseteq \mathsf{BQP}^\mathcal{O}$ which is hard on average against $\mathsf{P}^\mathcal{O}/poly$. We give a unitary oracle such that (1) $\mathsf{BQP}^\mathcal{O} = \mathsf{QCMA}^\mathcal{O}$ (2) Quantum verifier oblivious state preparation exists. These oracle separations directly imply a number of related consequences. For example, our first result implies that proofs of quantumness cannot be used to construct classical or quantum cryptography with a relativizing proof. Our second result implies that quantum computation/classical communication oblivious transfer cannot be used to construct post-quantum one-way functions with a relativizing proof.

Authors: Eli Goldin, Alex Lombardi, Tomoyuki Morimae

We consider the complexity implications of proofs of quantumness and (quantum verifier) oblivious state preparation. We show that both of these objects can coexist with the easiness of nondeterministic computation. That is, we give evidence that proofs of quantumness can exist when $\mathsf{P} = \mathsf{NP}$ and that (quantum verifier) oblivious state preparation can exist when $\mathsf{BQP} = \mathsf{QCMA}$. In particular, we give a classical oracle relative to which (1) $\mathsf{P}^\mathcal{O} = \mathsf{NP}^\mathcal{O}$ (2) $\mathsf{BQP}^\mathcal{O} = \mathsf{PSPACE}^\mathcal{O}$ (3) Proofs of quantumness exist. In particular, there exists a language in $\mathsf{IP}^\mathcal{O} \subseteq \mathsf{BQP}^\mathcal{O}$ which is hard on average against $\mathsf{P}^\mathcal{O}/poly$. We give a unitary oracle such that (1) $\mathsf{BQP}^\mathcal{O} = \mathsf{QCMA}^\mathcal{O}$ (2) Quantum verifier oblivious state preparation exists. These oracle separations directly imply a number of related consequences. For example, our first result implies that proofs of quantumness cannot be used to construct classical or quantum cryptography with a relativizing proof. Our second result implies that quantum computation/classical communication oblivious transfer cannot be used to construct post-quantum one-way functions with a relativizing proof.

Universal Hardness From Masking Channels

from arXiv: Computational Complexity

Authors: Parikshit Gopalan, Manoj Prabhakaran, Pranay Tankala

We prove a broad learning-theoretic generalization of the celebrated hardcore lemma concerning the average-case complexity of computing a Boolean function (Impagliazzo, 1995). Our starting point is the observation that the classic notion of a hardcore measure, which reweights a function's input distribution, is equivalent to an input-dependent masking channel that injects randomness into the function's output distribution until it appears fully pseudorandom. The analogue of density of the measure is the fidelity of the channel. From this viewpoint, we develop a general framework for certifying hardness that allows for non-Boolean functions $f$ and a wide range of notions of hardness besides the classic 0-1 loss, by choosing the right notion of fidelity via convex duality. We show the difficulty of learning $f$ using a class $\mathcal{C}$ to minimize a convex loss is tightly characterized by the maximum fidelity possible for a masking channel, which measures the least amount of noise the channel must add to hide the structure of $f$ and make it $\mathcal{C}$-indistinguishable from random noise. Using our framework, we derive a number of multiclass generalizations of the hardcore lemma, and reveal some unexpected subtleties in going to larger alphabets. We construct a universal masking channel that applies to many natural notions of hardness simultaneously. Such channels are dual to omnipredictors, which optimize multiple losses at once (Gopalan, Kalai, Reingold, Sharan, and Wieder, 2022). We observe a symmetry between masking channels and generalizations of the hardcore lemma using pseudoentropy. We present a tight characterization of pseudoentropy using the MaxEnt algorithm from machine learning. A long line of research has shown that learning and hardness are two sides of the same coin; our framework gives a crisp and general formulation of this principle via convex duality.

Authors: Parikshit Gopalan, Manoj Prabhakaran, Pranay Tankala

We prove a broad learning-theoretic generalization of the celebrated hardcore lemma concerning the average-case complexity of computing a Boolean function (Impagliazzo, 1995). Our starting point is the observation that the classic notion of a hardcore measure, which reweights a function's input distribution, is equivalent to an input-dependent masking channel that injects randomness into the function's output distribution until it appears fully pseudorandom. The analogue of density of the measure is the fidelity of the channel. From this viewpoint, we develop a general framework for certifying hardness that allows for non-Boolean functions $f$ and a wide range of notions of hardness besides the classic 0-1 loss, by choosing the right notion of fidelity via convex duality. We show the difficulty of learning $f$ using a class $\mathcal{C}$ to minimize a convex loss is tightly characterized by the maximum fidelity possible for a masking channel, which measures the least amount of noise the channel must add to hide the structure of $f$ and make it $\mathcal{C}$-indistinguishable from random noise. Using our framework, we derive a number of multiclass generalizations of the hardcore lemma, and reveal some unexpected subtleties in going to larger alphabets. We construct a universal masking channel that applies to many natural notions of hardness simultaneously. Such channels are dual to omnipredictors, which optimize multiple losses at once (Gopalan, Kalai, Reingold, Sharan, and Wieder, 2022). We observe a symmetry between masking channels and generalizations of the hardcore lemma using pseudoentropy. We present a tight characterization of pseudoentropy using the MaxEnt algorithm from machine learning. A long line of research has shown that learning and hardness are two sides of the same coin; our framework gives a crisp and general formulation of this principle via convex duality.

Some algebraic and geometric computations in CH via Chow forms

from arXiv: Computational Complexity

Authors: Robert Andrews, Abhibhav Garg, Rafael Oliveira, Éric Schost

The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding (1) whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, (2) whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, (3) membership in a given equidimensional component of a given algebraic variety, and (4) membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.

Authors: Robert Andrews, Abhibhav Garg, Rafael Oliveira, Éric Schost

The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding (1) whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, (2) whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, (3) membership in a given equidimensional component of a given algebraic variety, and (4) membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.

The sum-of-squares complexity of linear-algebraic principles

from arXiv: Computational Complexity

Authors: Dmitriy Kunisky, Xifan Yu

We study the degree needed for sum-of-squares (SoS) proofs of several basic principles of real linear algebra. For any $n\times n$ matrix $A$ of rank greater than $r$, the Weak Rank Principle states that the equations $XY^\top=A$ with $n\times r$ matrix variables $X, Y$ have no solution. We show that this principle requires degree exactly $2r+2$ to prove. On the other hand, for $A=I_n$, setting $X = Y$ gives the Tight Frame Principle, which has a degree-$4$ proof. For $n\times n$ matrices, proving that a right inverse is also a left inverse requires degree exactly $2n+2$, while proving that orthonormal rows imply orthonormal columns again has a degree-4 proof. Using our lower bounds, we also derive related SoS degree lower bounds for polynomial systems encoding sign rank, matrix rigidity, small-width homogeneous non-commutative algebraic branching programs, and tensor CP rank. Lastly, we show that quantitative bounds on the Frobenius norms of the factors in the Weak Rank Principle can lower the degree required to refute $XY^\top=I_n$, but sufficiently large bounds still preserve the exact degree $2r+2$ from the unbounded case, giving a quantitative version of the Archimedean property arising in classical results on SoS proofs.

Authors: Dmitriy Kunisky, Xifan Yu

We study the degree needed for sum-of-squares (SoS) proofs of several basic principles of real linear algebra. For any $n\times n$ matrix $A$ of rank greater than $r$, the Weak Rank Principle states that the equations $XY^\top=A$ with $n\times r$ matrix variables $X, Y$ have no solution. We show that this principle requires degree exactly $2r+2$ to prove. On the other hand, for $A=I_n$, setting $X = Y$ gives the Tight Frame Principle, which has a degree-$4$ proof. For $n\times n$ matrices, proving that a right inverse is also a left inverse requires degree exactly $2n+2$, while proving that orthonormal rows imply orthonormal columns again has a degree-4 proof. Using our lower bounds, we also derive related SoS degree lower bounds for polynomial systems encoding sign rank, matrix rigidity, small-width homogeneous non-commutative algebraic branching programs, and tensor CP rank. Lastly, we show that quantitative bounds on the Frobenius norms of the factors in the Weak Rank Principle can lower the degree required to refute $XY^\top=I_n$, but sufficiently large bounds still preserve the exact degree $2r+2$ from the unbounded case, giving a quantitative version of the Archimedean property arising in classical results on SoS proofs.

Information Loss Is All You Need: $TC^0$ Pseudorandomness from Worst-Case Lattices

from arXiv: Computational Complexity

Authors: Chen Qian

Can deterministic information loss yield post-quantum pseudorandomness with $TC^0$ evaluation? We show that deterministic rounding achieves this under quantum worst-case lattice hardness, with fixed polynomial parameters and no predetermined sample bound. We introduce learning with algebraic rounding (LAR), which applies an explicit integer rounding map to blocks of secret-dependent linear measurements and encodes the result in a lattice quotient. For one fixed family with polynomially bounded modulus and dimensions, deterministic labels on exactly uniform, secret-independent public matrices are computationally indistinguishable from independent uniform labels for every polynomial number of classical random examples. Our classical reduction from Gaussian LWE, combined with the quantum worst-case reduction to LWE, bases LAR on GapSVP and SIVP over general lattices at approximation factor $\widetilde{O}(q)$, where $q$ is the modulus. The proof uses source-independent affine transport: public affine maps sampled independently of the hidden errors transport the Gaussian error law close to uniform on the rounding map's zero-label fiber while preserving exactly uniform public matrices. For every fixed inverse-polynomial accuracy, we give a polynomial-time sampler for the same public parameter family. This yields the first post-quantum weak pseudorandom function with joint key-and-input evaluation in $TC^0$, secure against uniform quantum polynomial-time distinguishers given any polynomial number of classical uniform random examples. We also prove hardness of learning polynomial-size two-hidden-layer ReLU networks from noiseless uniform random examples, even for quantum polynomial-time improper learners.

Authors: Chen Qian

Can deterministic information loss yield post-quantum pseudorandomness with $TC^0$ evaluation? We show that deterministic rounding achieves this under quantum worst-case lattice hardness, with fixed polynomial parameters and no predetermined sample bound. We introduce learning with algebraic rounding (LAR), which applies an explicit integer rounding map to blocks of secret-dependent linear measurements and encodes the result in a lattice quotient. For one fixed family with polynomially bounded modulus and dimensions, deterministic labels on exactly uniform, secret-independent public matrices are computationally indistinguishable from independent uniform labels for every polynomial number of classical random examples. Our classical reduction from Gaussian LWE, combined with the quantum worst-case reduction to LWE, bases LAR on GapSVP and SIVP over general lattices at approximation factor $\widetilde{O}(q)$, where $q$ is the modulus. The proof uses source-independent affine transport: public affine maps sampled independently of the hidden errors transport the Gaussian error law close to uniform on the rounding map's zero-label fiber while preserving exactly uniform public matrices. For every fixed inverse-polynomial accuracy, we give a polynomial-time sampler for the same public parameter family. This yields the first post-quantum weak pseudorandom function with joint key-and-input evaluation in $TC^0$, secure against uniform quantum polynomial-time distinguishers given any polynomial number of classical uniform random examples. We also prove hardness of learning polynomial-size two-hidden-layer ReLU networks from noiseless uniform random examples, even for quantum polynomial-time improper learners.

A Short Proof and Generalization of the Rainbow Cycle Theorem and Improved Bounds for Binary Even-Query LCCs

from arXiv: Computational Complexity

Authors: Andrew D. Lin

A \emph{rainbow cycle} in a list-edge-colored graph is a cycle where no color appears more than once. We give a short and simple proof of a result of [ABSZZ] showing that a properly edge-colored $n$-vertex graph with average degree $Ω(\log n\log\log n)$ always contains a rainbow cycle, and generalize it to weighted graphs with edges list colored with list size at most $\ell$ and where the total weight of all edges containing a given color $i$ incident to a vertex is at most $Δ$, showing that $Ω_\ell(Δ\log n\log\log n)$ average degree guarantees a rainbow cycle. Using the reduction of [AG26] and the new generalized rainbow cycle theorem, we prove that for $r\geq 3$, every binary linear $r$-query locally correctable code $E:\mathbb{F}_2^k\rightarrow \mathbb{F}_2^n$ satisfies $k\leq \tilde{O}(n^{1-2/(r-1)})$ by showing a rainbow analogue of the odd $r$-LDC lower bounds of [BHKL25,JM25], closing the gap between the best known lower bounds for odd- and even-query binary linear $r$-LCCs.

Authors: Andrew D. Lin

A \emph{rainbow cycle} in a list-edge-colored graph is a cycle where no color appears more than once. We give a short and simple proof of a result of [ABSZZ] showing that a properly edge-colored $n$-vertex graph with average degree $Ω(\log n\log\log n)$ always contains a rainbow cycle, and generalize it to weighted graphs with edges list colored with list size at most $\ell$ and where the total weight of all edges containing a given color $i$ incident to a vertex is at most $Δ$, showing that $Ω_\ell(Δ\log n\log\log n)$ average degree guarantees a rainbow cycle. Using the reduction of [AG26] and the new generalized rainbow cycle theorem, we prove that for $r\geq 3$, every binary linear $r$-query locally correctable code $E:\mathbb{F}_2^k\rightarrow \mathbb{F}_2^n$ satisfies $k\leq \tilde{O}(n^{1-2/(r-1)})$ by showing a rainbow analogue of the odd $r$-LDC lower bounds of [BHKL25,JM25], closing the gap between the best known lower bounds for odd- and even-query binary linear $r$-LCCs.

Hardness Self-Amplification in Linear Algebra and Fine-Grained Cryptography

from arXiv: Computational Complexity

Authors: Geoffroy Couteau, Nikolas Melissaris, Thejas Radhika Sajith

We study the possibility of basing fine-grained cryptography on the hardness of fundamental problems in linear algebra. Our main technical contribution is a hardness self-amplification method based on random low-rank updates: we show how to turn an algorithm that solves an inverse-polynomial fraction of random linear systems into a solver which succeeds for every right-hand side of almost every matrix. Our reduction increases the runtime only linearly in the inverse of the original success probability. In particular, this yields fine-grained one-way functions from the almost worst-case hardness of linear systems over large enough fields. Using the same approach, we also obtain tighter worst-case to average-case reductions for matrix multiplication and online matrix-vector multiplication, as well as variants of our reductions that recover exact solutions from approximate algorithms, whose expected coordinate agreement is only $1/q+\varepsilon$, with inverse-quadratic dependence on $\varepsilon$ up to logarithmic factors, over every finite field. Finally, we also use it to connect the existence of fine-grained key agreement to the problem of computing the rank of a matrix: over a suitable sequence of growing characteristic-two fields, the almost-everywhere average-case hardness of rank computation (for any of a large class of distributions) implies infinitely-often fine-grained key agreement.

Authors: Geoffroy Couteau, Nikolas Melissaris, Thejas Radhika Sajith

We study the possibility of basing fine-grained cryptography on the hardness of fundamental problems in linear algebra. Our main technical contribution is a hardness self-amplification method based on random low-rank updates: we show how to turn an algorithm that solves an inverse-polynomial fraction of random linear systems into a solver which succeeds for every right-hand side of almost every matrix. Our reduction increases the runtime only linearly in the inverse of the original success probability. In particular, this yields fine-grained one-way functions from the almost worst-case hardness of linear systems over large enough fields. Using the same approach, we also obtain tighter worst-case to average-case reductions for matrix multiplication and online matrix-vector multiplication, as well as variants of our reductions that recover exact solutions from approximate algorithms, whose expected coordinate agreement is only $1/q+\varepsilon$, with inverse-quadratic dependence on $\varepsilon$ up to logarithmic factors, over every finite field. Finally, we also use it to connect the existence of fine-grained key agreement to the problem of computing the rank of a matrix: over a suitable sequence of growing characteristic-two fields, the almost-everywhere average-case hardness of rank computation (for any of a large class of distributions) implies infinitely-often fine-grained key agreement.

Lower Bounds for Adaptive Relaxed Locally Decodable Codes via the Capped Horvitz--Thompson Estimator

from arXiv: Computational Complexity

Authors: Guy Goldberg, Tom Gur, Sidhant Saraogi

We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+Ω(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.

Authors: Guy Goldberg, Tom Gur, Sidhant Saraogi

We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+Ω(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.

On the Complexity of Exact Autobidding Equilibria

from arXiv: Computational Complexity

Authors: Yiyang Huang, Zhengyang Liu, Yonglei Yan

Autobidding equilibria link advertisers' value multipliers or participation probabilities through aggregate budget and return-on-investment (ROI) constraints. We establish the exact computational complexity of finding these equilibria across several canonical settings: exact equilibrium computation is $\mathsf{FIXP}$-complete for second-price ROI pacing, budget- or ROI-constrained generalized second-price (GSP) sponsored shopping, and second-price throttling. Furthermore, approximating the equilibrium control vector in distance to a true equilibrium is $\mathsf{FIXP}_{a}$-complete when the target accuracy is provided as input.

Authors: Yiyang Huang, Zhengyang Liu, Yonglei Yan

Autobidding equilibria link advertisers' value multipliers or participation probabilities through aggregate budget and return-on-investment (ROI) constraints. We establish the exact computational complexity of finding these equilibria across several canonical settings: exact equilibrium computation is $\mathsf{FIXP}$-complete for second-price ROI pacing, budget- or ROI-constrained generalized second-price (GSP) sponsored shopping, and second-price throttling. Furthermore, approximating the equilibrium control vector in distance to a true equilibrium is $\mathsf{FIXP}_{a}$-complete when the target accuracy is provided as input.

On the Hardness of Approximate Quantum Circuit Membership and Optimization

from arXiv: Computational Complexity

Authors: Adam Husted Kjelstrøm, Jaco van de Pol, Andreas Pavlogiannis

Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\{I\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\{I\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\{I\}$, or if $S$ contains only generalized permutation matrices, $I\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\varepsilon$. If $S=\{I\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.

Authors: Adam Husted Kjelstrøm, Jaco van de Pol, Andreas Pavlogiannis

Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\{I\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\{I\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\{I\}$, or if $S$ contains only generalized permutation matrices, $I\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\varepsilon$. If $S=\{I\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.

Monodromy Computation Model

from arXiv: Computational Complexity

Authors: Xiaoyang Liu

A closed path can return to its starting point while analytic continuation reaches another branch. We construct a finite computational calculus in which retained logarithmic branches supply memory, analytic observations select operations, and descent through a covering exposes deeper memory. Three branch coordinates suffice, and every finite observation uses the same cubic power covering. Each program defines a globally real analytic return map whose integer-lattice execution is realized by actual path lifting. Our first theorem simulates every finite-alphabet, finite-control Post--Turing machine, preserving complete configurations, halting, and divergence. Our second theorem identifies the partial numeric functions of the calculus, under fixed primitive-recursive input and output maps, with exactly the partial recursive functions. Deleting descent leaves a Forward language with a syntax-derived finite observation quotient and a primitive-recursive halting predicate. This yields a strict separation within the calculus and identifies access to deeper memory as its decisive additional operation. The constructions and semantic correspondences are formalized in Lean 4 with Mathlib.

Authors: Xiaoyang Liu

A closed path can return to its starting point while analytic continuation reaches another branch. We construct a finite computational calculus in which retained logarithmic branches supply memory, analytic observations select operations, and descent through a covering exposes deeper memory. Three branch coordinates suffice, and every finite observation uses the same cubic power covering. Each program defines a globally real analytic return map whose integer-lattice execution is realized by actual path lifting. Our first theorem simulates every finite-alphabet, finite-control Post--Turing machine, preserving complete configurations, halting, and divergence. Our second theorem identifies the partial numeric functions of the calculus, under fixed primitive-recursive input and output maps, with exactly the partial recursive functions. Deleting descent leaves a Forward language with a syntax-derived finite observation quotient and a primitive-recursive halting predicate. This yields a strict separation within the calculus and identifies access to deeper memory as its decisive additional operation. The constructions and semantic correspondences are formalized in Lean 4 with Mathlib.

On the Role of Geometry in Solving Variational Inequalities

from arXiv: Computational Complexity

Authors: Ioannis Anagnostides, Ioannis Panageas, Jingming Yan

The geometry of constraints in variational inequalities (VIs) plays a central role in applications ranging from resource allocation and equilibrium computation to constrained optimization. Yet, its effect on computational complexity has received surprisingly little attention beyond special cases. We examine this question when the constraint set is an $\ell_p$ ball. This family of problems encompasses two of the most extensively studied classes of problems in algorithmic game theory: symmetric bimatrix games for $p = 1$ and multi-player polymatrix games for $p = \infty$. Both of these problems are known to be PPAD-complete even for affine operators. We establish a comprehensive complexity landscape for VIs over $\ell_p$ balls across the full range $p\in[1,\infty]$. First, for any $p\neq 2$, we show that finding an inverse-polynomial approximate VI solution is PPAD-complete even for affine operators. In contrast, for $p=2$, we give a polynomial-time algorithm for affine VIs, extending a classic result of Ye (1992) beyond symmetric operators. On the other hand, once the operator becomes quadratic, we establish PPAD-completeness even at inverse polynomial precision. From a technical standpoint, we develop a new toolkit for exploiting the geometry of the constraint set and the structure of the operator to characterize the complexity of equilibrium and optimization problems, which is of independent interest.

Authors: Ioannis Anagnostides, Ioannis Panageas, Jingming Yan

The geometry of constraints in variational inequalities (VIs) plays a central role in applications ranging from resource allocation and equilibrium computation to constrained optimization. Yet, its effect on computational complexity has received surprisingly little attention beyond special cases. We examine this question when the constraint set is an $\ell_p$ ball. This family of problems encompasses two of the most extensively studied classes of problems in algorithmic game theory: symmetric bimatrix games for $p = 1$ and multi-player polymatrix games for $p = \infty$. Both of these problems are known to be PPAD-complete even for affine operators. We establish a comprehensive complexity landscape for VIs over $\ell_p$ balls across the full range $p\in[1,\infty]$. First, for any $p\neq 2$, we show that finding an inverse-polynomial approximate VI solution is PPAD-complete even for affine operators. In contrast, for $p=2$, we give a polynomial-time algorithm for affine VIs, extending a classic result of Ye (1992) beyond symmetric operators. On the other hand, once the operator becomes quadratic, we establish PPAD-completeness even at inverse polynomial precision. From a technical standpoint, we develop a new toolkit for exploiting the geometry of the constraint set and the structure of the operator to characterize the complexity of equilibrium and optimization problems, which is of independent interest.

On the (Quantum) Communication Complexity of Secure MPC from a Random Oracle

from arXiv: Computational Complexity

Authors: Alexander Hoover, Giuseppe Persiano, Kevin Yeo

Information-theoretic MPC among $n$ parties with $t$ corruptions can securely evaluate a circuit $C$ with $O(|C| \cdot t/(n-t))$ online communication in the preprocessing model and $O(|C| \cdot t/(n-2t))$ communication in the honest majority setting. Damgård, Larsen and Nielsen (CRYPTO 2019) showed that both communication bounds are tight for inner products. Fully homomorphic encryption and other assumptions known to imply public-key cryptography beat both communication bounds, but the communication complexity with symmetric-key cryptography alone is less understood. For MPC with preprocessing, we prove that the same $Ω(|C| \cdot t/(n-t))$ lower bound holds in the random oracle model, so a random oracle cannot reduce the online communication. For honest majority MPC relative to a random oracle, we present an $Ω(|C|)$ communication lower bound that is matched by a new construction for degree-2 polynomials (including inner products) when $n = O(\log λ)$, beating the information-theoretic bound. We also study quantum MPC. First, we prove that information-theoretic protocols with quantum communication (and preprocessing) cannot reduce communication compared to their classical counterparts. Furthermore, we extend our lower bound for MPC with preprocessing to protocols with a quantum-accessible random oracle, but use only classical communication. In contrast, with both quantum communication and a post-quantum one-way function, we present a simple two-party protocol that evaluates the two-party instance of these circuits with $\mathrm{poly}(λ, \log |C|)$ communication, beating these lower bounds. Finally, such communication savings do not come for free. We prove that there exist circuits $C$ such that every blackbox protocol with $o(|C|)$ quantum communication must perform $Ω(|C|)$ cryptographic operations.

Authors: Alexander Hoover, Giuseppe Persiano, Kevin Yeo

Information-theoretic MPC among $n$ parties with $t$ corruptions can securely evaluate a circuit $C$ with $O(|C| \cdot t/(n-t))$ online communication in the preprocessing model and $O(|C| \cdot t/(n-2t))$ communication in the honest majority setting. Damgård, Larsen and Nielsen (CRYPTO 2019) showed that both communication bounds are tight for inner products. Fully homomorphic encryption and other assumptions known to imply public-key cryptography beat both communication bounds, but the communication complexity with symmetric-key cryptography alone is less understood. For MPC with preprocessing, we prove that the same $Ω(|C| \cdot t/(n-t))$ lower bound holds in the random oracle model, so a random oracle cannot reduce the online communication. For honest majority MPC relative to a random oracle, we present an $Ω(|C|)$ communication lower bound that is matched by a new construction for degree-2 polynomials (including inner products) when $n = O(\log λ)$, beating the information-theoretic bound. We also study quantum MPC. First, we prove that information-theoretic protocols with quantum communication (and preprocessing) cannot reduce communication compared to their classical counterparts. Furthermore, we extend our lower bound for MPC with preprocessing to protocols with a quantum-accessible random oracle, but use only classical communication. In contrast, with both quantum communication and a post-quantum one-way function, we present a simple two-party protocol that evaluates the two-party instance of these circuits with $\mathrm{poly}(λ, \log |C|)$ communication, beating these lower bounds. Finally, such communication savings do not come for free. We prove that there exist circuits $C$ such that every blackbox protocol with $o(|C|)$ quantum communication must perform $Ω(|C|)$ cryptographic operations.

The Complexity of KKT Points of Quadratic Programs with Polynomially-Bounded Coefficients

from arXiv: Computational Complexity

Authors: Argyrios Deligkas, Michail Fasoulakis, Stavros D. Ioannidis

Computing a Karush-Kuhn-Tucker (KKT) point of a quadratic polynomial over the unit hypercube is known to be $CLS$-complete, when the coefficients of the polynomial are encoded in binary. We prove that the problem remains $CLS$-complete even when the coefficients are encoded in unary. This is the first problem proven to remain $CLS$-hard when the input is given in unary.

Authors: Argyrios Deligkas, Michail Fasoulakis, Stavros D. Ioannidis

Computing a Karush-Kuhn-Tucker (KKT) point of a quadratic polynomial over the unit hypercube is known to be $CLS$-complete, when the coefficients of the polynomial are encoded in binary. We prove that the problem remains $CLS$-complete even when the coefficients are encoded in unary. This is the first problem proven to remain $CLS$-hard when the input is given in unary.

Counterexamples to the Happy Edge Conjecture in Plane Spanning Trees

from arXiv: Computational Geometry

Authors: Oswin Aichholzer, Joseph Dorfer, Peter Kramer, Christian Rieck, Birgit Vogtenhuber

We study the problem of reconfiguring plane, straight-line spanning trees on point sets in the plane. A flip is a reconfiguration step in which one edge of a tree is replaced with another, yielding again a plane spanning tree. For a given point set $P$, the flip graph has a vertex for each plane spanning tree on $P$, and an edge between any two trees that differ by a single flip. The folklore happy edge conjecture suggests that any edge shared by both the initial and the target tree (a happy edge) is never flipped in a shortest flip sequence. This conjecture was computationally verified for up to 10 points in convex position. In this paper, we disprove this conjecture both on general point sets and on point sets in convex position by providing two counterexamples on $16$ and $30$ points, respectively. The counterexample for point sets in general position also holds for compatible flips, while the happy edge property holds for compatible flips in convex point sets as shown by Aichholzer, Dorfer, and Vogtenhuber [GD 2025].

Authors: Oswin Aichholzer, Joseph Dorfer, Peter Kramer, Christian Rieck, Birgit Vogtenhuber

We study the problem of reconfiguring plane, straight-line spanning trees on point sets in the plane. A flip is a reconfiguration step in which one edge of a tree is replaced with another, yielding again a plane spanning tree. For a given point set $P$, the flip graph has a vertex for each plane spanning tree on $P$, and an edge between any two trees that differ by a single flip. The folklore happy edge conjecture suggests that any edge shared by both the initial and the target tree (a happy edge) is never flipped in a shortest flip sequence. This conjecture was computationally verified for up to 10 points in convex position. In this paper, we disprove this conjecture both on general point sets and on point sets in convex position by providing two counterexamples on $16$ and $30$ points, respectively. The counterexample for point sets in general position also holds for compatible flips, while the happy edge property holds for compatible flips in convex point sets as shown by Aichholzer, Dorfer, and Vogtenhuber [GD 2025].

Polyominoes Folding into Four and Five Distinct Boxes

from arXiv: Computational Geometry

Authors: Erik D. Demaine, Toshihiro Shirakawa, Conner Silverstein, Reed Silverstein, Michael Tardibuono, Ryuhei Uehara

We present new results on polyomino nets that fold into four or five distinct boxes. With the help of AI, we first found examples with areas 532 and 2128, respectively. We analyze these examples and reduce their areas to 178 and 358, respectively. We also generalize the construction of the five-box net and show that infinitely many polyomino nets fold into five distinct boxes.

Authors: Erik D. Demaine, Toshihiro Shirakawa, Conner Silverstein, Reed Silverstein, Michael Tardibuono, Ryuhei Uehara

We present new results on polyomino nets that fold into four or five distinct boxes. With the help of AI, we first found examples with areas 532 and 2128, respectively. We analyze these examples and reduce their areas to 178 and 358, respectively. We also generalize the construction of the five-box net and show that infinitely many polyomino nets fold into five distinct boxes.

Breaking the quantum barriers for Graph Collision

from arXiv: Data Structures and Algorithms

Authors: Arjan Cornelissen, Amin Shiraz Gilani, Shiva Kintali, François Le Gall, Subhasree Patro, Xingyu Zhou

In the graph collision problem, the task is to decide whether a known graph on $n$ vertices contains an edge with both endpoints marked, given query access to the input bits specifying the marked vertices. We first refute the conjectured $Ω(n^{2/3})$ query lower bound with a bounded error quantum algorithm using $\widetilde O(n^{92/139})$ queries. We then refine this approach to obtain a $\widetilde O(n^{11/17})$ query algorithm. We also obtain a $\widetilde O((nq)^{11/17})$ query bound for all (partial) Boolean output functions with positive certificate complexity at most 2 and alphabet size $q$, contrasting a known $Ω(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $Ω(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.

Authors: Arjan Cornelissen, Amin Shiraz Gilani, Shiva Kintali, François Le Gall, Subhasree Patro, Xingyu Zhou

In the graph collision problem, the task is to decide whether a known graph on $n$ vertices contains an edge with both endpoints marked, given query access to the input bits specifying the marked vertices. We first refute the conjectured $Ω(n^{2/3})$ query lower bound with a bounded error quantum algorithm using $\widetilde O(n^{92/139})$ queries. We then refine this approach to obtain a $\widetilde O(n^{11/17})$ query algorithm. We also obtain a $\widetilde O((nq)^{11/17})$ query bound for all (partial) Boolean output functions with positive certificate complexity at most 2 and alphabet size $q$, contrasting a known $Ω(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $Ω(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.

Agnostic Learning by Imitating

from arXiv: Data Structures and Algorithms

Authors: Sílvia Casacuberta, Parikshit Gopalan, Varun Kanade, Omer Reingold, Konstantinos Stavropoulos, Pranay Tankala

Given a class of machine learning models and sample access to a data distribution with binary labels, which computational task is harder: training a discriminative model that distinguishes positive and negative labels as well as the best member of the class, or training a generative model whose outputs are indistinguishable from real data by all members of the class? We prove that these two fundamental tasks in computational learning theory are not just closely related, but actually equivalent to each other in complexity. One direction of this equivalence was previously known. Indeed, the powerful regularity lemma in complexity theory gives an efficient oracle reduction from the indistinguishability task (a.k.a. multiaccuracy) to the distinguishing task (a.k.a. agnostic learning) in the style of boosting algorithms. Our main contribution is an efficient oracle reduction in the other direction. Taking this idea further, we show that a multiaccuracy oracle is a remarkably powerful primitive, achieving not just agnostic learning but also a wide range of seemingly more demanding learning objectives via similarly efficient and direct constructions. Our results crucially use the power of making adaptive queries to the oracle, and stand in stark contrast to prior work, which showed that post-processing a single multiaccurate predictor is insufficient for even weak forms of learning. On a technical level, we draw a connection between our (offline) problem and a version of no-regret learning in the online setting, which allows us to use follow-the-leader type algorithms to modify the label distribution and aggregate the predictions of the multiaccuracy oracle.

Authors: Sílvia Casacuberta, Parikshit Gopalan, Varun Kanade, Omer Reingold, Konstantinos Stavropoulos, Pranay Tankala

Given a class of machine learning models and sample access to a data distribution with binary labels, which computational task is harder: training a discriminative model that distinguishes positive and negative labels as well as the best member of the class, or training a generative model whose outputs are indistinguishable from real data by all members of the class? We prove that these two fundamental tasks in computational learning theory are not just closely related, but actually equivalent to each other in complexity. One direction of this equivalence was previously known. Indeed, the powerful regularity lemma in complexity theory gives an efficient oracle reduction from the indistinguishability task (a.k.a. multiaccuracy) to the distinguishing task (a.k.a. agnostic learning) in the style of boosting algorithms. Our main contribution is an efficient oracle reduction in the other direction. Taking this idea further, we show that a multiaccuracy oracle is a remarkably powerful primitive, achieving not just agnostic learning but also a wide range of seemingly more demanding learning objectives via similarly efficient and direct constructions. Our results crucially use the power of making adaptive queries to the oracle, and stand in stark contrast to prior work, which showed that post-processing a single multiaccurate predictor is insufficient for even weak forms of learning. On a technical level, we draw a connection between our (offline) problem and a version of no-regret learning in the online setting, which allows us to use follow-the-leader type algorithms to modify the label distribution and aggregate the predictions of the multiaccuracy oracle.

Finding Hypercliques Faster Than Brute Force

from arXiv: Data Structures and Algorithms

Authors: Josh Alman, Baitian Li, Neha Pant, Kevin Pratt, Ryan Williams

We show that for any fixed $k$, there exists $\varepsilon_k > 0$ such that, given a $(k-1)$-uniform hypergraph $H$ on $n$ vertices, one can count the number of $k$-hypercliques in $H$ in $O(n^{k-\varepsilon_k})$ time. This refutes the hyperclique conjecture of Lincoln, Vassilevska Williams, and Williams (SODA 2018). Our result follows from the uniform construction of a family of arithmetic circuits of size $O(n^{k-\varepsilon_k})$ for the $k$-hyperclique tensor, a natural higher-order generalization of the matrix multiplication tensor. We obtain two corollaries. First, for every $k$, there exists $\varepsilon > 0$ such that Max-$k$SAT can be solved in $2^{(1-\varepsilon)n} \cdot \mathrm{poly}(m,n)$ time. Second, for every $k$, there exists $\varepsilon > 0$ such that arithmetic progressions of length $k$ in subsets of $\{0,1,\ldots,n\}$ can be detected in $n^{2-\varepsilon}$ time. Both corollaries refute other prominent conjectures in the literature.

Authors: Josh Alman, Baitian Li, Neha Pant, Kevin Pratt, Ryan Williams

We show that for any fixed $k$, there exists $\varepsilon_k > 0$ such that, given a $(k-1)$-uniform hypergraph $H$ on $n$ vertices, one can count the number of $k$-hypercliques in $H$ in $O(n^{k-\varepsilon_k})$ time. This refutes the hyperclique conjecture of Lincoln, Vassilevska Williams, and Williams (SODA 2018). Our result follows from the uniform construction of a family of arithmetic circuits of size $O(n^{k-\varepsilon_k})$ for the $k$-hyperclique tensor, a natural higher-order generalization of the matrix multiplication tensor. We obtain two corollaries. First, for every $k$, there exists $\varepsilon > 0$ such that Max-$k$SAT can be solved in $2^{(1-\varepsilon)n} \cdot \mathrm{poly}(m,n)$ time. Second, for every $k$, there exists $\varepsilon > 0$ such that arithmetic progressions of length $k$ in subsets of $\{0,1,\ldots,n\}$ can be detected in $n^{2-\varepsilon}$ time. Both corollaries refute other prominent conjectures in the literature.

Differentially Oblivious Priority Queues with Sublogarithmic Cost

from arXiv: Data Structures and Algorithms

Authors: Zimai Guo, Cheng Hong, Zhechen Li

In this paper, we study exact online integer priority queues under differential obliviousness in the balls-and-bins model. Given a public upper bound $T$ on the number of operations and assuming that the priorities are bounded by a polynomial in $T$, we construct an $(\eps,δ)$-differentially oblivious priority queue supporting insertion and minimum extraction, which requires $\polylog(T)$ private memory. For constant $\varepsilon>0$ and $δ=\exp(-Θ(\log^2 T))$, our construction incurs $O(\log T/\log\log T)$ expected amortized total work per operation, which breaks the $Ω(\log T)$ barrier for fully oblivious priority queues~[Jacob et al., SODA'2019]. Further, we prove a matching lower bound for $(\eps,δ)$-differentially oblivious priority queues with $\varepsilon=O(1)$ and $δ=o(1)$, which indicates that our construction is asymptotically optimal in terms of total work.

Authors: Zimai Guo, Cheng Hong, Zhechen Li

In this paper, we study exact online integer priority queues under differential obliviousness in the balls-and-bins model. Given a public upper bound $T$ on the number of operations and assuming that the priorities are bounded by a polynomial in $T$, we construct an $(\eps,δ)$-differentially oblivious priority queue supporting insertion and minimum extraction, which requires $\polylog(T)$ private memory. For constant $\varepsilon>0$ and $δ=\exp(-Θ(\log^2 T))$, our construction incurs $O(\log T/\log\log T)$ expected amortized total work per operation, which breaks the $Ω(\log T)$ barrier for fully oblivious priority queues~[Jacob et al., SODA'2019]. Further, we prove a matching lower bound for $(\eps,δ)$-differentially oblivious priority queues with $\varepsilon=O(1)$ and $δ=o(1)$, which indicates that our construction is asymptotically optimal in terms of total work.

Abelian embeddings make useful promises: approximation of detrivialised equations

from arXiv: Data Structures and Algorithms

Authors: Tamio-Vesa Nakajima, Zephyr Verwimp, Stanislav Živný

We study the following family of search problems: given a system of equations $x+y+z\equiv 0\pmod p$ promised to admit a solution that is nonconstant on every equation, find a $c$-colouring of the variables that makes no equation monochromatic. For $p\in\{2,3\}$, the problem recovers known tractable Promise Constraint Satisfaction Problems (PCSPs). For $p > 3$, we obtain a complete complexity classification: the problem is polynomial-time solvable for $c \geq 4$, yet even its decision version is NP-hard for $c = 3$, and it is not well-defined for $c = 2$. Our main conceptual contribution is a general tractability framework for PCSPs based on Abelian embeddings, a notion introduced by Bhangale, Khot, and Minzer [STOC 2022] in the study of quantitative approximation of satisfiable CSPs. We show that suitable Abelian embeddings yield polynomial-time PCSP reductions realised by singleton algorithms in the sense of Zhuk [FOCS 2026]. On the hardness side, the promise relation is far removed from the reconfigurable regime covered by the conditional hardness framework of Banakh, Barsukov, and Nakajima [LICS 2026]; our proof instead relies on a new additive-combinatorial rigidity theorem classifying special $3$-colourings of $\mathbb{Z}_p$.

Authors: Tamio-Vesa Nakajima, Zephyr Verwimp, Stanislav Živný

We study the following family of search problems: given a system of equations $x+y+z\equiv 0\pmod p$ promised to admit a solution that is nonconstant on every equation, find a $c$-colouring of the variables that makes no equation monochromatic. For $p\in\{2,3\}$, the problem recovers known tractable Promise Constraint Satisfaction Problems (PCSPs). For $p > 3$, we obtain a complete complexity classification: the problem is polynomial-time solvable for $c \geq 4$, yet even its decision version is NP-hard for $c = 3$, and it is not well-defined for $c = 2$. Our main conceptual contribution is a general tractability framework for PCSPs based on Abelian embeddings, a notion introduced by Bhangale, Khot, and Minzer [STOC 2022] in the study of quantitative approximation of satisfiable CSPs. We show that suitable Abelian embeddings yield polynomial-time PCSP reductions realised by singleton algorithms in the sense of Zhuk [FOCS 2026]. On the hardness side, the promise relation is far removed from the reconfigurable regime covered by the conditional hardness framework of Banakh, Barsukov, and Nakajima [LICS 2026]; our proof instead relies on a new additive-combinatorial rigidity theorem classifying special $3$-colourings of $\mathbb{Z}_p$.

Entropy-Weighted Soundness for Multi-Answer Birthday Graphs

from arXiv: Data Structures and Algorithms

Authors: Alexandru Popa

The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ ρ^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.

Authors: Alexandru Popa

The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ ρ^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.

Binary Huffman Coding with Unequal Letter Costs Is NP-Complete

from arXiv: Data Structures and Algorithms

Authors: L. Sunil Chandran, Rishikesh Gajjala

Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.

Authors: L. Sunil Chandran, Rishikesh Gajjala

Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.

Space-Optimal Algorithms for Spanning Forests in Dynamic Streams at Every Confidence Level: With and Without Deletion Constraints

from arXiv: Data Structures and Algorithms

Authors: Zemin Chao, Jianhe Cen, Hengzhao Ma, Hongzhi Wang

We characterize the optimal space for one-pass spanning-forest computation in dynamic streams of $v$-vertex simple graphs with failure probability at most $δ$. Nelson and Yu proved an $Ω(v\log^2v\log(v/δ))$-bit lower bound, establishing the AGM algorithm's optimality for each fixed $0<ε<1$ and $2^{-v^{1-ε}}<δ\le1/4$. We resolve the remaining high-confidence regime: for every $0<δ\le1/4$, matching lower bounds and an algorithm give \[ Θ\!\left(\min\left\{v^2,\;v\log(v/δ)\log^2\!\left(2+\frac{v}{\log(v/δ)}\right)\right\}\right) \] bits, improving space by up to a factor of $Θ((\log v/\log\log v)^2)$ in this regime. We also study streams with at most $Z$ previously inserted edges deleted and not yet reinserted at report time, where the integer $Z\ge0$ is given in advance. Total deletions, restorations, and intermediate absence counts are unrestricted. For a single report and every $0<δ\le1/4$, matching lower bounds and a one-pass randomized algorithm yield optimal space $Θ(\mathcal L(v,Z,\log(1/δ)))$ bits, where $x_Z(t)=\min\{Z+1,v,\sqrt{vt}\}$ and \[ \mathcal L(v,Z,t)=v\log v\min\{\log^2v,\log^2(Z+2)\} +vt\log\!\left(1+\frac{x_Z(t)}t\right)\log\!\left(1+\frac v{x_Z(t)}\right). \] At zero error, a lower bound for all zero-error algorithms and a matching deterministic algorithm give optimal space \[ Θ\!\left(\min\{v^2,v(Z+1)\log(1+v/(Z+1))\}\right) \] bits. Together, these results characterize the memory required across all confidence levels and report-time edge-loss bounds.

Authors: Zemin Chao, Jianhe Cen, Hengzhao Ma, Hongzhi Wang

We characterize the optimal space for one-pass spanning-forest computation in dynamic streams of $v$-vertex simple graphs with failure probability at most $δ$. Nelson and Yu proved an $Ω(v\log^2v\log(v/δ))$-bit lower bound, establishing the AGM algorithm's optimality for each fixed $0<ε<1$ and $2^{-v^{1-ε}}<δ\le1/4$. We resolve the remaining high-confidence regime: for every $0<δ\le1/4$, matching lower bounds and an algorithm give \[ Θ\!\left(\min\left\{v^2,\;v\log(v/δ)\log^2\!\left(2+\frac{v}{\log(v/δ)}\right)\right\}\right) \] bits, improving space by up to a factor of $Θ((\log v/\log\log v)^2)$ in this regime. We also study streams with at most $Z$ previously inserted edges deleted and not yet reinserted at report time, where the integer $Z\ge0$ is given in advance. Total deletions, restorations, and intermediate absence counts are unrestricted. For a single report and every $0<δ\le1/4$, matching lower bounds and a one-pass randomized algorithm yield optimal space $Θ(\mathcal L(v,Z,\log(1/δ)))$ bits, where $x_Z(t)=\min\{Z+1,v,\sqrt{vt}\}$ and \[ \mathcal L(v,Z,t)=v\log v\min\{\log^2v,\log^2(Z+2)\} +vt\log\!\left(1+\frac{x_Z(t)}t\right)\log\!\left(1+\frac v{x_Z(t)}\right). \] At zero error, a lower bound for all zero-error algorithms and a matching deterministic algorithm give optimal space \[ Θ\!\left(\min\{v^2,v(Z+1)\log(1+v/(Z+1))\}\right) \] bits. Together, these results characterize the memory required across all confidence levels and report-time edge-loss bounds.

State Evolution and Universality of Iterated Linear Programs for the Binary Perceptron

from arXiv: Data Structures and Algorithms

Authors: Jianan Zhang, Shuangping Li

The binary perceptron is a random constraint satisfaction problem that asks for a sign vector satisfying a system of random linear inequalities. Its solution space has a strikingly extreme geometry: at every positive constraint density, all but a vanishing fraction of the solutions are isolated. Yet polynomial-time algorithms find solutions at small enough constraint density, which raises the question of how efficient algorithms navigate such a landscape. Linear programming is a natural algorithm to study here, but its analysis has so far been limited to a single round. Li, Schramm, and Zhou (STOC 2025) analyzed one linear program followed by iterative rounding. In this work, we overcome this limitation and give a precise analysis of iterated linear programming for the binary perceptron. (1) For Gaussian matrices and any fixed number of rounds, we characterize the iterated linear programs exactly through a deterministic state-evolution recursion. (2) Adding a final rounding step, we obtain a randomized polynomial-time algorithm for the binary perceptron. Numerical evaluations of the recursion provide evidence that it succeeds at constraint density $α=0.60$ with zero margin, well beyond the range $α\le0.1$ guaranteed by the single-LP analysis and $α\approx0.327$ (Jo and Lee, 2026), the best guarantee so far. (3) We establish universality, showing that the same characterization holds for matrices with i.i.d. entries drawn from any centered, unit-variance subgaussian law. The key idea of our proof is that, although conditioning on earlier rounds makes the matrix non-Gaussian, its conditional law can be computed explicitly and remains close to a Gaussian law, so the convex Gaussian min-max theorem applies in every round. For universality, we compare Gaussian and subgaussian matrices through the Lindeberg principle, using a regularized version of each linear program.

Authors: Jianan Zhang, Shuangping Li

The binary perceptron is a random constraint satisfaction problem that asks for a sign vector satisfying a system of random linear inequalities. Its solution space has a strikingly extreme geometry: at every positive constraint density, all but a vanishing fraction of the solutions are isolated. Yet polynomial-time algorithms find solutions at small enough constraint density, which raises the question of how efficient algorithms navigate such a landscape. Linear programming is a natural algorithm to study here, but its analysis has so far been limited to a single round. Li, Schramm, and Zhou (STOC 2025) analyzed one linear program followed by iterative rounding. In this work, we overcome this limitation and give a precise analysis of iterated linear programming for the binary perceptron. (1) For Gaussian matrices and any fixed number of rounds, we characterize the iterated linear programs exactly through a deterministic state-evolution recursion. (2) Adding a final rounding step, we obtain a randomized polynomial-time algorithm for the binary perceptron. Numerical evaluations of the recursion provide evidence that it succeeds at constraint density $α=0.60$ with zero margin, well beyond the range $α\le0.1$ guaranteed by the single-LP analysis and $α\approx0.327$ (Jo and Lee, 2026), the best guarantee so far. (3) We establish universality, showing that the same characterization holds for matrices with i.i.d. entries drawn from any centered, unit-variance subgaussian law. The key idea of our proof is that, although conditioning on earlier rounds makes the matrix non-Gaussian, its conditional law can be computed explicitly and remains close to a Gaussian law, so the convex Gaussian min-max theorem applies in every round. For universality, we compare Gaussian and subgaussian matrices through the Lindeberg principle, using a regularized version of each linear program.

Semidirect sums of tensors and the matrix multiplication exponent

from arXiv: Data Structures and Algorithms

Authors: Baitian Li

This note is an exposition of the recent preprints from OpenAI that establish the bound $ω\leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a generally applicable method for analyzing tensors. We also discuss further consequences of this method, as well as the relation between the OpenAI proof and previous work.

Authors: Baitian Li

This note is an exposition of the recent preprints from OpenAI that establish the bound $ω\leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a generally applicable method for analyzing tensors. We also discuss further consequences of this method, as well as the relation between the OpenAI proof and previous work.

On the optimality of truncation among binary linear sketches

from arXiv: Data Structures and Algorithms

Authors: Robinson D. H. Cung, Adway Girish, Emre Telatar

We study statistical inference and function evaluation problems in which the input binary vector is available only through a binary linear sketch, i.e., a fixed number of parity measurements. We show that under a certain monotonicity condition, simply truncating the input by retaining a subset of its coordinates is optimal among all linear sketches of the same size. In particular, this applies to distributed testing of correlations of binary vectors when they are of opposite signs and evaluating unate Boolean functions on uniformly random inputs. Additionally, given that truncation is optimal, we immediately have lower bounds for arbitrary binary linear sketches. To illustrate this application, we consider the one-way communication complexity of a distributional graph connectivity problem on $n$ vertices and show that truncation, and hence all binary linear sketches, requires $Θ(n \log n)$ bits to achieve constant error, while a nonlinear protocol only requires $O(\log n)$ bits.

Authors: Robinson D. H. Cung, Adway Girish, Emre Telatar

We study statistical inference and function evaluation problems in which the input binary vector is available only through a binary linear sketch, i.e., a fixed number of parity measurements. We show that under a certain monotonicity condition, simply truncating the input by retaining a subset of its coordinates is optimal among all linear sketches of the same size. In particular, this applies to distributed testing of correlations of binary vectors when they are of opposite signs and evaluating unate Boolean functions on uniformly random inputs. Additionally, given that truncation is optimal, we immediately have lower bounds for arbitrary binary linear sketches. To illustrate this application, we consider the one-way communication complexity of a distributional graph connectivity problem on $n$ vertices and show that truncation, and hence all binary linear sketches, requires $Θ(n \log n)$ bits to achieve constant error, while a nonlinear protocol only requires $O(\log n)$ bits.

A King for Almost Everyone

from arXiv: Data Structures and Algorithms

Authors: Ziad Ismaili Alaoui, Oded Lachish, Nikhil S. Mande

A tournament is an orientation of a complete graph. A vertex $v$ controls a vertex $u\neq v$ if there is a directed path of length at most two from $v$ to $u$, and a $δ$-king is one that controls at least $δ(n-1)$ vertices. In the query complexity model, each query reveals the orientation of an edge. Shen, Sheng and Wu [SICOMP'03] established an $Ω(n^{4/3})$ lower bound on the deterministic query complexity of finding a $1$-king, and a non-matching upper bound of $O(n^{3/2})$. Neither bound has been improved in over two decades, despite substantial progress on variants of the problem. For $δ$-kings with $δ< 1$, Biswas, Jayapaul, Raman and Satti [Discrete Applied Mathematics'22] showed that $O(n^{4/3})$ queries suffice to find a $\frac12$-king. Lachish, Reidl and Trehan [FSTTCS'22] improved this guarantee to $\frac12+\frac{2}{17}$ using $\widetilde{O}(n^{4/3})$ queries, and asked whether a larger fraction could be achieved within the same query bound. We answer this question affirmatively for every fixed fraction below one: for every fixed $\varepsilon\in(0,1)$, we give a deterministic algorithm that finds a $(1-\varepsilon)$-king using $O(n^{4/3})$ queries. The design of our algorithm is informed by a structural analysis combining a sparse regularity lemma due to Allen, Kohayakawa, Mota and Parente [Electronic Journal of Combinatorics'14] with a new sampling argument based on a reachability theorem of Afek, Gafni and Linial [arXiv'19].

Authors: Ziad Ismaili Alaoui, Oded Lachish, Nikhil S. Mande

A tournament is an orientation of a complete graph. A vertex $v$ controls a vertex $u\neq v$ if there is a directed path of length at most two from $v$ to $u$, and a $δ$-king is one that controls at least $δ(n-1)$ vertices. In the query complexity model, each query reveals the orientation of an edge. Shen, Sheng and Wu [SICOMP'03] established an $Ω(n^{4/3})$ lower bound on the deterministic query complexity of finding a $1$-king, and a non-matching upper bound of $O(n^{3/2})$. Neither bound has been improved in over two decades, despite substantial progress on variants of the problem. For $δ$-kings with $δ< 1$, Biswas, Jayapaul, Raman and Satti [Discrete Applied Mathematics'22] showed that $O(n^{4/3})$ queries suffice to find a $\frac12$-king. Lachish, Reidl and Trehan [FSTTCS'22] improved this guarantee to $\frac12+\frac{2}{17}$ using $\widetilde{O}(n^{4/3})$ queries, and asked whether a larger fraction could be achieved within the same query bound. We answer this question affirmatively for every fixed fraction below one: for every fixed $\varepsilon\in(0,1)$, we give a deterministic algorithm that finds a $(1-\varepsilon)$-king using $O(n^{4/3})$ queries. The design of our algorithm is informed by a structural analysis combining a sparse regularity lemma due to Allen, Kohayakawa, Mota and Parente [Electronic Journal of Combinatorics'14] with a new sampling argument based on a reachability theorem of Afek, Gafni and Linial [arXiv'19].

Faster Planar Diameter using Pseudoline Envelopes

from arXiv: Data Structures and Algorithms

Authors: Da Wei Zheng, Daniel Prigan

We give an improved algorithm for weighted directed planar diameter that runs in $\widetilde{O}(n^{3/2})$ time. This improves upon the $\widetilde{O}(n^{5/3})$ time algorithm for weighted directed planar diameter of Gawrychowski, Kaplan, Mozes, Sharir, and Weimann [SIAM J. Comput. 2021] and the recent $\widetilde{O}(n^{8/5})$ time algorithm for unweighted undirected planar diameter of Fredslund-Hansen, Mozes, and Weimann [SOSA 2027]. The main new result is an improved data structure for computing the farthest vertex when given feasible planar distances to the vertices on the outer face. The critical insight is the fact that normalized distances to the vertices of a face behave like pseudolines, for which we can leverage range searching and upper envelope data structures. Using the farthest vertex data structure, we are able to compute all vertex eccentricities and diameter in $\widetilde{O}(n^{3/2})$.

Authors: Da Wei Zheng, Daniel Prigan

We give an improved algorithm for weighted directed planar diameter that runs in $\widetilde{O}(n^{3/2})$ time. This improves upon the $\widetilde{O}(n^{5/3})$ time algorithm for weighted directed planar diameter of Gawrychowski, Kaplan, Mozes, Sharir, and Weimann [SIAM J. Comput. 2021] and the recent $\widetilde{O}(n^{8/5})$ time algorithm for unweighted undirected planar diameter of Fredslund-Hansen, Mozes, and Weimann [SOSA 2027]. The main new result is an improved data structure for computing the farthest vertex when given feasible planar distances to the vertices on the outer face. The critical insight is the fact that normalized distances to the vertices of a face behave like pseudolines, for which we can leverage range searching and upper envelope data structures. Using the farthest vertex data structure, we are able to compute all vertex eccentricities and diameter in $\widetilde{O}(n^{3/2})$.

Geometric Bin Packing Beyond Convexity: Approximation Schemes without Augmentation

from arXiv: Data Structures and Algorithms

Authors: Lucas de Oliveira Silva, Elisa Dell'Arriva, Flávio Keidi Miyazawa

We study the geometric bin packing problem for simple polygonal regions, i.e., non-degenerate, connected, rational regions, in the minimum number of unit-square bins, allowing only translations. We even consider packing general non-convex objects, possibly with holes, where we can then pack smaller objects. Previous schemes cover squares (MOR'06), circles and other $L_p$-balls (Algorithmica'16), sufficiently skewed rectangles (Algorithmica'23), and convex fat objects with a constant number of distinct shapes under resource augmentation (arXiv'24, WAOA'23). The result for circles and other $L_p$-balls needs bin augmentation to obtain rational packing realizations. For restricted classes of $L$-shaped polygons, Gerlach et al. (SoCG'26) give constant worst-case guarantees, while for general non-convex objects, bounding-box reductions give just approximation factors that depend on fatness. Against this background, we present a general framework that provides asymptotic polynomial-time approximation schemes (APTAS) for bin packing of any fixed class of simple polygonal regions satisfying just two assumptions. First, covering any object by the grid cells intersecting its interior in any sufficiently fine square grid, relative to the object's bounding box, increases its area by at most a small fraction. Second, the class can be partitioned into a constant number of sequences such that each object in a sequence can be translated to fit inside every preceding object. In particular, certain classes of star-shaped polygons and more general polygonal regions, including some with simple holes, satisfy both assumptions. In addition, our framework does not require a bound on the objects' boundary complexity. We emphasize that our results require no bin augmentation whatsoever and yield explicit, exact rational packings of polynomial encoding length for all input objects.

Authors: Lucas de Oliveira Silva, Elisa Dell'Arriva, Flávio Keidi Miyazawa

We study the geometric bin packing problem for simple polygonal regions, i.e., non-degenerate, connected, rational regions, in the minimum number of unit-square bins, allowing only translations. We even consider packing general non-convex objects, possibly with holes, where we can then pack smaller objects. Previous schemes cover squares (MOR'06), circles and other $L_p$-balls (Algorithmica'16), sufficiently skewed rectangles (Algorithmica'23), and convex fat objects with a constant number of distinct shapes under resource augmentation (arXiv'24, WAOA'23). The result for circles and other $L_p$-balls needs bin augmentation to obtain rational packing realizations. For restricted classes of $L$-shaped polygons, Gerlach et al. (SoCG'26) give constant worst-case guarantees, while for general non-convex objects, bounding-box reductions give just approximation factors that depend on fatness. Against this background, we present a general framework that provides asymptotic polynomial-time approximation schemes (APTAS) for bin packing of any fixed class of simple polygonal regions satisfying just two assumptions. First, covering any object by the grid cells intersecting its interior in any sufficiently fine square grid, relative to the object's bounding box, increases its area by at most a small fraction. Second, the class can be partitioned into a constant number of sequences such that each object in a sequence can be translated to fit inside every preceding object. In particular, certain classes of star-shaped polygons and more general polygonal regions, including some with simple holes, satisfy both assumptions. In addition, our framework does not require a bound on the objects' boundary complexity. We emphasize that our results require no bin augmentation whatsoever and yield explicit, exact rational packings of polynomial encoding length for all input objects.

Roundtrip All-Pairs Shortest Paths in Directed Graphs

from arXiv: Data Structures and Algorithms

Authors: Shiri Chechik, Gur Lifshitz

We study approximation algorithms for All-Pairs Shortest Paths (APSP) in directed graphs under the roundtrip metric. In directed graphs, even finite-factor approximation of ordinary one-way distances must distinguish reachable from unreachable vertex pairs, and therefore computes transitive closure, a problem equivalent to Boolean matrix multiplication (BMM) under standard reductions. The roundtrip distance, defined as $\text{dist}_G(u \leftrightarrows v) = \text{dist}_G(u \to v) + \text{dist}_G(v \to u)$, offers a natural and robust relaxation. It measures the cost of traveling from $u$ to $v$ and back, satisfies the triangle inequality, and is inherently symmetric. While this metric has become central to the study of directed roundtrip spanners and compact routing, the fundamental problem of all-pairs roundtrip distance approximation has remained largely unexplored. In this work, we present a comprehensive study of roundtrip APSP and distance oracles, presenting several new lower and upper bounds. On the algorithmic side, our main result is a combinatorial algorithm that computes a stretch-$3$ approximation to all-pairs roundtrip distances in $\widetilde{O}(n^2 + \min\{m\sqrt{n},\, m^{1/3}n^{5/3}\})$ time. For edge weights in $[1,W]$, we also obtain a $(3+\varepsilon)$-approximation in $\widetilde O((n^{2.037}+nm^{2/3})\log(nW)/\varepsilon)$ time and a $(4+\varepsilon)$-approximation in $\widetilde O(n^{2.032}\log(nW)/\varepsilon)$ time using fast matrix multiplication. On the hardness side, we show that any stretch below $3$ would imply a breakthrough for BMM. We also obtain additional tradeoffs between stretch and running time.

Authors: Shiri Chechik, Gur Lifshitz

We study approximation algorithms for All-Pairs Shortest Paths (APSP) in directed graphs under the roundtrip metric. In directed graphs, even finite-factor approximation of ordinary one-way distances must distinguish reachable from unreachable vertex pairs, and therefore computes transitive closure, a problem equivalent to Boolean matrix multiplication (BMM) under standard reductions. The roundtrip distance, defined as $\text{dist}_G(u \leftrightarrows v) = \text{dist}_G(u \to v) + \text{dist}_G(v \to u)$, offers a natural and robust relaxation. It measures the cost of traveling from $u$ to $v$ and back, satisfies the triangle inequality, and is inherently symmetric. While this metric has become central to the study of directed roundtrip spanners and compact routing, the fundamental problem of all-pairs roundtrip distance approximation has remained largely unexplored. In this work, we present a comprehensive study of roundtrip APSP and distance oracles, presenting several new lower and upper bounds. On the algorithmic side, our main result is a combinatorial algorithm that computes a stretch-$3$ approximation to all-pairs roundtrip distances in $\widetilde{O}(n^2 + \min\{m\sqrt{n},\, m^{1/3}n^{5/3}\})$ time. For edge weights in $[1,W]$, we also obtain a $(3+\varepsilon)$-approximation in $\widetilde O((n^{2.037}+nm^{2/3})\log(nW)/\varepsilon)$ time and a $(4+\varepsilon)$-approximation in $\widetilde O(n^{2.032}\log(nW)/\varepsilon)$ time using fast matrix multiplication. On the hardness side, we show that any stretch below $3$ would imply a breakthrough for BMM. We also obtain additional tradeoffs between stretch and running time.

Gaussian Partitioning for Reliable Learning of Multi-Index Models

from arXiv: Data Structures and Algorithms

Authors: Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane, Mingchen Ma

We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+ε$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/ε))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/ε)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{Ω(1/ε^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.

Authors: Ilias Diakonikolas, Giannis Iakovidis, Daniel M. Kane, Mingchen Ma

We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+ε$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/ε))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/ε)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{Ω(1/ε^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.

Catalan Hashing: Efficient Linear Probing for Variable-Length Items

from arXiv: Data Structures and Algorithms

Authors: Alex Conway, Hanna Komlós, William Kuszmaul

A defining feature of the linear-probing hash table is its spatial locality: at load factor $1 - \varepsilon$, the hash table can be implemented so that each operation scans a single contiguous region of expected size $O(1/\varepsilon)$ (this is known as the \emph{scan cost}). This guarantee, however, relies on all items having unit size. In this paper, we ask whether it is possible to achieve a similar guarantee, but in a hash table that stores items of arbitrary lengths $|x| \ge 1$. The main result of the paper is a new hash table called \emph{Catalan hashing}, which supports amortized expected scan cost $O(|x|/\varepsilon)$ for positive queries, $O(1/\varepsilon)$ for negative queries, and $O(|x|\varepsilon^{-1}\log^{1+δ}|x|)$ for insertions/deletions, for any constant $δ> 0$. As a corollary, we obtain the same bounds in the cache-oblivious model, with a scan cost of $S$ translating to $1 + O(S/B)$ I/Os. Along the way, we introduce several high-level techniques that may be of independent interest; this includes a novel way of using erasure codes in order to perform write optimization within a hash table, and a connection to scheduling that allows us to place elements in a way which avoids clustering at high load factors.

Authors: Alex Conway, Hanna Komlós, William Kuszmaul

A defining feature of the linear-probing hash table is its spatial locality: at load factor $1 - \varepsilon$, the hash table can be implemented so that each operation scans a single contiguous region of expected size $O(1/\varepsilon)$ (this is known as the \emph{scan cost}). This guarantee, however, relies on all items having unit size. In this paper, we ask whether it is possible to achieve a similar guarantee, but in a hash table that stores items of arbitrary lengths $|x| \ge 1$. The main result of the paper is a new hash table called \emph{Catalan hashing}, which supports amortized expected scan cost $O(|x|/\varepsilon)$ for positive queries, $O(1/\varepsilon)$ for negative queries, and $O(|x|\varepsilon^{-1}\log^{1+δ}|x|)$ for insertions/deletions, for any constant $δ> 0$. As a corollary, we obtain the same bounds in the cache-oblivious model, with a scan cost of $S$ translating to $1 + O(S/B)$ I/Os. Along the way, we introduce several high-level techniques that may be of independent interest; this includes a novel way of using erasure codes in order to perform write optimization within a hash table, and a connection to scheduling that allows us to place elements in a way which avoids clustering at high load factors.

Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations

from arXiv: Data Structures and Algorithms

Authors: Yuhang Chen, Yuda Feng, Weijiang Hu, Shi Li, Ruilong Zhang

We study two fairness objectives for allocating $m$ indivisible items among $n$ agents with subadditive valuations: the max-min objective, also known as the Santa Claus problem, which maximizes the smallest value of an agent, and the weighted Nash social welfare, which maximizes the weighted geometric mean of the agents' values. For the max-min allocation problem, we give a randomized polynomial-time $O(\sqrt n)$-approximation algorithm for subadditive valuations with demand queries, improving the $2n-1$ bound of Khot and Ponnuswami. For additive valuations and for submodular valuations with value queries, we give simpler $O(\sqrt n)$-approximation algorithms. For additive valuations, this improves the $O(\sqrt{n\log n})$ bound of Saha and Srinivasan. For these two classes, $O(m^\varepsilon)$-approximation algorithms are also known for every constant $\varepsilon>0$. Our ratios depend only on $n$ and match, up to constants, the $Ω(\sqrt n)$ integrality gap of the configuration LP. For the weighted Nash social welfare problem with subadditive valuations and demand queries, we give an $O(Γ(w))$-approximation algorithm, where $Γ(w)=\exp\bigl((\ln n-H(w))/2\bigr)$ and $H(w)$ is the entropy of the weight vector $w$. The factor $Γ(w)$ equals $1$ for equal weights and is at most $\sqrt n$ for all weights. We show that this ratio is tight for algorithms analyzed against the configuration LP: for every weight vector, the configuration LP has an integrality gap of $Θ(Γ(w))$, even for XOS valuations, and this gap is $Ω(\sqrt n)$ when one agent has almost all the weight. For submodular valuations, the configuration LP has a constant integrality gap for all weights.

Authors: Yuhang Chen, Yuda Feng, Weijiang Hu, Shi Li, Ruilong Zhang

We study two fairness objectives for allocating $m$ indivisible items among $n$ agents with subadditive valuations: the max-min objective, also known as the Santa Claus problem, which maximizes the smallest value of an agent, and the weighted Nash social welfare, which maximizes the weighted geometric mean of the agents' values. For the max-min allocation problem, we give a randomized polynomial-time $O(\sqrt n)$-approximation algorithm for subadditive valuations with demand queries, improving the $2n-1$ bound of Khot and Ponnuswami. For additive valuations and for submodular valuations with value queries, we give simpler $O(\sqrt n)$-approximation algorithms. For additive valuations, this improves the $O(\sqrt{n\log n})$ bound of Saha and Srinivasan. For these two classes, $O(m^\varepsilon)$-approximation algorithms are also known for every constant $\varepsilon>0$. Our ratios depend only on $n$ and match, up to constants, the $Ω(\sqrt n)$ integrality gap of the configuration LP. For the weighted Nash social welfare problem with subadditive valuations and demand queries, we give an $O(Γ(w))$-approximation algorithm, where $Γ(w)=\exp\bigl((\ln n-H(w))/2\bigr)$ and $H(w)$ is the entropy of the weight vector $w$. The factor $Γ(w)$ equals $1$ for equal weights and is at most $\sqrt n$ for all weights. We show that this ratio is tight for algorithms analyzed against the configuration LP: for every weight vector, the configuration LP has an integrality gap of $Θ(Γ(w))$, even for XOS valuations, and this gap is $Ω(\sqrt n)$ when one agent has almost all the weight. For submodular valuations, the configuration LP has a constant integrality gap for all weights.

Beyond the Bochner Method: Optimal Spectral Gaps for Block Dynamics of Spin Systems like the $O(N)$ Model

from arXiv: Data Structures and Algorithms

Authors: Jonathan Leake, Anqi Li

Let $μ$ be a probability distribution on a spin system with $n$ sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix $A$ with zero diagonal and $λ_{\max}(A)<1$ encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance $(I-A)^{-1}$ attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the $O(N)$ model, with $N\geq 2$, arbitrary external fields, and couplings of either sign, has spectral gap at least $(1-\|J_{\mathrm{abs}}\|_{\mathrm{op}}/N)/n$, where $J$ is the interaction matrix and $J_{\mathrm{abs}}=(|J_{ij}|)$. We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over $\mathbb F_q$, we obtain a one-sided local spectral expansion bound of $2\sqrt q/(q+1)$. This bound is independent of the number and sizes of each group, and is nontrivial for every prime power $q$. Finally, for random forests $F$ of any finite graph with arbitrary positive edge weights, we prove that \[ \Pr(e,f\in F)\le \frac{4}{3} \Pr(e\in F)\Pr(f\in F) \] for distinct edges $e,f$, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.

Authors: Jonathan Leake, Anqi Li

Let $μ$ be a probability distribution on a spin system with $n$ sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix $A$ with zero diagonal and $λ_{\max}(A)<1$ encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance $(I-A)^{-1}$ attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the $O(N)$ model, with $N\geq 2$, arbitrary external fields, and couplings of either sign, has spectral gap at least $(1-\|J_{\mathrm{abs}}\|_{\mathrm{op}}/N)/n$, where $J$ is the interaction matrix and $J_{\mathrm{abs}}=(|J_{ij}|)$. We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over $\mathbb F_q$, we obtain a one-sided local spectral expansion bound of $2\sqrt q/(q+1)$. This bound is independent of the number and sizes of each group, and is nontrivial for every prime power $q$. Finally, for random forests $F$ of any finite graph with arbitrary positive edge weights, we prove that \[ \Pr(e,f\in F)\le \frac{4}{3} \Pr(e\in F)\Pr(f\in F) \] for distinct edges $e,f$, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.

Efficient Equilibrium Computation in Colonel Blotto Games

from arXiv: Data Structures and Algorithms

Authors: Martin Bichler, Abheek Ghosh

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.

Authors: Martin Bichler, Abheek Ghosh

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.

Logarithmic Embeddings into Probabilistic Spanning Trees for Distances and Capacities

from arXiv: Data Structures and Algorithms

Authors: Mohammad Hajiaghayi

We prove that every connected $n$-vertex graph admits a probabilistic embedding into its spanning trees with expected distance stretch $O(\log n)$. This settles the logarithmic stretch conjecture of Alon, Karp, Peleg, and West, whose spanning-tree game appeared at the 1991 DIMACS Workshop on On-Line Algorithms and in SIAM Journal on Computing in 1995. A weighted form of the theorem also yields a distribution over spanning trees with capacities exactly equal to their fundamental graph cuts. Every tree dominates every graph cut, every cut is approximated within $O(\log n)$ in expectation, and the same capacity distribution defines an $O(\log n)$-competitive fractional oblivious routing. The distance and capacity distributions may differ. Diamond graphs give matching logarithmic lower bounds for stretch, congestion, and cut distortion. For rational input, both distributions are constructed deterministically in polynomial time and use at most $m$ trees, where $m$ is the number of graph edges. We prove an $Ω(m/ρ)$ support lower bound for congestion $ρ$; in particular, $(1-o(1))m$ trees can be necessary for near-unit congestion, even when the optimum tends to one. Thus linear dependence on $m$ is unavoidable in this regime. The weighted theorem combines a protected-geodesic petal decomposition with a pathwise entropy bound for locally selected centers. A local averaging inequality derandomizes the radii, and an adaptive rational packing procedure constructs the distributions.

Authors: Mohammad Hajiaghayi

We prove that every connected $n$-vertex graph admits a probabilistic embedding into its spanning trees with expected distance stretch $O(\log n)$. This settles the logarithmic stretch conjecture of Alon, Karp, Peleg, and West, whose spanning-tree game appeared at the 1991 DIMACS Workshop on On-Line Algorithms and in SIAM Journal on Computing in 1995. A weighted form of the theorem also yields a distribution over spanning trees with capacities exactly equal to their fundamental graph cuts. Every tree dominates every graph cut, every cut is approximated within $O(\log n)$ in expectation, and the same capacity distribution defines an $O(\log n)$-competitive fractional oblivious routing. The distance and capacity distributions may differ. Diamond graphs give matching logarithmic lower bounds for stretch, congestion, and cut distortion. For rational input, both distributions are constructed deterministically in polynomial time and use at most $m$ trees, where $m$ is the number of graph edges. We prove an $Ω(m/ρ)$ support lower bound for congestion $ρ$; in particular, $(1-o(1))m$ trees can be necessary for near-unit congestion, even when the optimum tends to one. Thus linear dependence on $m$ is unavoidable in this regime. The weighted theorem combines a protected-geodesic petal decomposition with a pathwise entropy bound for locally selected centers. A local averaging inequality derandomizes the radii, and an adaptive rational packing procedure constructs the distributions.

Path Decompositions of Sparse Graphs via Gaussian Sweeps and Faster Exponential Algorithms

from arXiv: Data Structures and Algorithms

Authors: Serge Gaspers

Using the bisection bound of Monien and Preis, Fomin and Høie proved in 2006 that every graph of maximum degree 3 on $n$ vertices has pathwidth at most $n/6+o(n)$. We prove that the pathwidth, bisection width and cutwidth of these graphs are at most $c_3 n+o(n)$, where $c_3=\frac{3}{4π}\arccos\frac56\approx0.1398$. For bisection width this improves the bound $n/6+o(n)$ of Monien and Preis (2006) and extends to all such graphs the bound that Lichev and Mitsche (2023) obtained for random cubic graphs. For maximum degree 4, we prove that bisection width and cutwidth are at most $n/3+o(n)$, improving the bound $2n/5+o(n)$ of Monien and Preis for 4-regular graphs, and that pathwidth is at most $0.2458n$, improving the bound $n/3$ of Fomin, Gaspers, Saurabh and Stepanov (2009). We show that every graph with $m$ edges has pathwidth at most $7m/48+o(n)$, improving the bound $13m/75$ of Scott and Sorkin (2007), and we give further bounds of the form $\sum_{d\ge3}a_d n_d+o(n)$ with $a_d<1$, where $n_d$ is the number of vertices of degree $d$. The path decompositions, and linear orderings of the stated cutwidth, can be computed in polynomial time. Instead of deriving pathwidth bounds from bisection width, we give each vertex a Gaussian label, a weighted sum of independent standard normal variables over a ball around it, and sweep a threshold through the labels; the bags are frontiers, the vertices above the threshold with a neighbor below it. We bound the probability that a vertex lies in a frontier by comparison with an explicit Gaussian model of a vertex and its neighbors. Faster exact algorithms follow: for feedback vertex set and the traveling salesperson problem on graphs of bounded maximum or average degree, and on general graphs for minimum weight dominating set (by branching until the instance is sparse), Max 2-CSP and counting subgraphs with $k$ edges (by the bound in $m$).

Authors: Serge Gaspers

Using the bisection bound of Monien and Preis, Fomin and Høie proved in 2006 that every graph of maximum degree 3 on $n$ vertices has pathwidth at most $n/6+o(n)$. We prove that the pathwidth, bisection width and cutwidth of these graphs are at most $c_3 n+o(n)$, where $c_3=\frac{3}{4π}\arccos\frac56\approx0.1398$. For bisection width this improves the bound $n/6+o(n)$ of Monien and Preis (2006) and extends to all such graphs the bound that Lichev and Mitsche (2023) obtained for random cubic graphs. For maximum degree 4, we prove that bisection width and cutwidth are at most $n/3+o(n)$, improving the bound $2n/5+o(n)$ of Monien and Preis for 4-regular graphs, and that pathwidth is at most $0.2458n$, improving the bound $n/3$ of Fomin, Gaspers, Saurabh and Stepanov (2009). We show that every graph with $m$ edges has pathwidth at most $7m/48+o(n)$, improving the bound $13m/75$ of Scott and Sorkin (2007), and we give further bounds of the form $\sum_{d\ge3}a_d n_d+o(n)$ with $a_d<1$, where $n_d$ is the number of vertices of degree $d$. The path decompositions, and linear orderings of the stated cutwidth, can be computed in polynomial time. Instead of deriving pathwidth bounds from bisection width, we give each vertex a Gaussian label, a weighted sum of independent standard normal variables over a ball around it, and sweep a threshold through the labels; the bags are frontiers, the vertices above the threshold with a neighbor below it. We bound the probability that a vertex lies in a frontier by comparison with an explicit Gaussian model of a vertex and its neighbors. Faster exact algorithms follow: for feedback vertex set and the traveling salesperson problem on graphs of bounded maximum or average degree, and on general graphs for minimum weight dominating set (by branching until the instance is sparse), Max 2-CSP and counting subgraphs with $k$ edges (by the bound in $m$).

Recognizing Label-Connected Graphs

from arXiv: Data Structures and Algorithms

Authors: Ralf Klasing, George B. Mertzios, Hendrik Molter, Nils Morawietz, Paul G. Spirakis

In classical gossiping, each participant initially knows a distinct piece of information. Whenever two participants call each other, they exchange everything they know. We study Gossiping: given an undirected multigraph whose vertices represent participants and whose edges represent available calls, decide whether the calls can be ordered so that every participant learns all information, using each edge at most once. Equivalently, can one time label per edge make the graph temporally connected by strict temporal paths? This recognition problem is NP-complete; we study its parameterized complexity. After linear-time preprocessing, we give an exact $O(3^{n/3}n^3)$-time algorithm and a deterministic $2^{O(tw)}n$-time algorithm, where $n$ is the number of vertices and $tw$ is the treewidth of the underlying simple graph. We introduce a Tree Contraction Rule that extends the treewidth algorithm to single-exponential algorithms for parameters that may remain small on dense graphs, such as tree-independence number and clique-width, without requiring a decomposition as input. For the excess $δ=m-(2n-4)$ over the minimum number of calls, where $m$ is the number of edges, we give a randomized $2^{O(δ)}n^3$-time algorithm with one-sided error. Its main subroutine solves Rooted Independent Cycle: finding a cycle through four prescribed vertices whose edges are independent in a represented linear matroid. We solve this problem deterministically in $2^{O(r)}m$ arithmetic operations over the finite field representing the matroid, where $r$ is its rank. Gossiping is polynomial-time solvable at maximum degree four, counting parallel edges, but NP-complete on planar simple graphs of maximum degree five. The treewidth algorithm yields a $2^{O(\sqrt n)}$-time algorithm on planar graphs.

Authors: Ralf Klasing, George B. Mertzios, Hendrik Molter, Nils Morawietz, Paul G. Spirakis

In classical gossiping, each participant initially knows a distinct piece of information. Whenever two participants call each other, they exchange everything they know. We study Gossiping: given an undirected multigraph whose vertices represent participants and whose edges represent available calls, decide whether the calls can be ordered so that every participant learns all information, using each edge at most once. Equivalently, can one time label per edge make the graph temporally connected by strict temporal paths? This recognition problem is NP-complete; we study its parameterized complexity. After linear-time preprocessing, we give an exact $O(3^{n/3}n^3)$-time algorithm and a deterministic $2^{O(tw)}n$-time algorithm, where $n$ is the number of vertices and $tw$ is the treewidth of the underlying simple graph. We introduce a Tree Contraction Rule that extends the treewidth algorithm to single-exponential algorithms for parameters that may remain small on dense graphs, such as tree-independence number and clique-width, without requiring a decomposition as input. For the excess $δ=m-(2n-4)$ over the minimum number of calls, where $m$ is the number of edges, we give a randomized $2^{O(δ)}n^3$-time algorithm with one-sided error. Its main subroutine solves Rooted Independent Cycle: finding a cycle through four prescribed vertices whose edges are independent in a represented linear matroid. We solve this problem deterministically in $2^{O(r)}m$ arithmetic operations over the finite field representing the matroid, where $r$ is its rank. Gossiping is polynomial-time solvable at maximum degree four, counting parallel edges, but NP-complete on planar simple graphs of maximum degree five. The treewidth algorithm yields a $2^{O(\sqrt n)}$-time algorithm on planar graphs.

Quantum Query Lower Bounds from Classical Moment Matching

from arXiv: Data Structures and Algorithms

Authors: Chandrima Kayal, Frédéric Magniez, Sayantan Sen, Dániel Szabó

We prove a quantum analogue of the classical moment matching method, a popular framework used to prove lower bounds in property testing. For two random variables $X_0$ and $X_1$ whose moments match up to the first $r-1$ moments, the classical moment matching method says that no classical algorithm can reliably distinguish between input instances of length $N$ constructed from $X_0$ and $X_1$, respectively, unless it performs at least $Ω(N^{1-1/r})$ queries. While this method has been widely explored since its inception in the seminal work of Raskhodnikova, Ron, Shpilka, and Smith (STOC 2007) to prove lower bounds for symmetric properties, it has remained an open question whether a quantum analogue of it could be proved. We prove two variants of this quantum moment matching method, which are incomparable. Our first quantum query lower bound scales as $Ω(N^{1/2-1/(2r)})$, while our second moment matching theorem yields a sharper lower bound of $Ω(N^{1/2-1/(2(2^r-1))})$, albeit for smaller parameter ranges. Our results combine a new continuous hybrid argument with the polynomial method. The continuous hybrid argument compares two hard input distributions generated from i.i.d. samples by continuously varying both distributions, keeping them identically distributed at every intermediate stage. It reduces the comparison to the effect of changing the distribution of a single sample. The polynomial method then cancels the terms determined by the matched moments. We apply these moment matching theorems to prove new optimal lower bounds on $k$-distinctness, $k$-star-freeness, planarity and outerplanarity testing, as well as to improve prior lower bounds on image size testing and $K$-junta testing by logarithmic factors. A key advantage of our framework is its simplicity: it reduces quantum lower bound proofs to the construction of suitable moment-matching distributions.

Authors: Chandrima Kayal, Frédéric Magniez, Sayantan Sen, Dániel Szabó

We prove a quantum analogue of the classical moment matching method, a popular framework used to prove lower bounds in property testing. For two random variables $X_0$ and $X_1$ whose moments match up to the first $r-1$ moments, the classical moment matching method says that no classical algorithm can reliably distinguish between input instances of length $N$ constructed from $X_0$ and $X_1$, respectively, unless it performs at least $Ω(N^{1-1/r})$ queries. While this method has been widely explored since its inception in the seminal work of Raskhodnikova, Ron, Shpilka, and Smith (STOC 2007) to prove lower bounds for symmetric properties, it has remained an open question whether a quantum analogue of it could be proved. We prove two variants of this quantum moment matching method, which are incomparable. Our first quantum query lower bound scales as $Ω(N^{1/2-1/(2r)})$, while our second moment matching theorem yields a sharper lower bound of $Ω(N^{1/2-1/(2(2^r-1))})$, albeit for smaller parameter ranges. Our results combine a new continuous hybrid argument with the polynomial method. The continuous hybrid argument compares two hard input distributions generated from i.i.d. samples by continuously varying both distributions, keeping them identically distributed at every intermediate stage. It reduces the comparison to the effect of changing the distribution of a single sample. The polynomial method then cancels the terms determined by the matched moments. We apply these moment matching theorems to prove new optimal lower bounds on $k$-distinctness, $k$-star-freeness, planarity and outerplanarity testing, as well as to improve prior lower bounds on image size testing and $K$-junta testing by logarithmic factors. A key advantage of our framework is its simplicity: it reduces quantum lower bound proofs to the construction of suitable moment-matching distributions.

Beyond Multi-Index Models: Learning Intersections of Degree-2 PTFs

from arXiv: Data Structures and Algorithms

Authors: Ilias Diakonikolas, Daniel M. Kane, Mingchen Ma, Lisheng Ren, Trung Tran

We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\log^3(1/ε))}$ time and samples, improving over the $d^{O(k^2/ε^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{Ω(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\mathrm{poly}_k(1/ε)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/ε$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.

Authors: Ilias Diakonikolas, Daniel M. Kane, Mingchen Ma, Lisheng Ren, Trung Tran

We give an algorithm for PAC learning intersections of $k$ degree-2 polynomial threshold functions over $R^d$ under a Gaussian distribution. Our algorithm achieves classification error $ε$ using $d^{O(k)}2^{O_k(\log^3(1/ε))}$ time and samples, improving over the $d^{O(k^2/ε^2)}$ running time of the polynomial regression approach. To our knowledge, our result is the first to obtain a dimension exponent independent of the accuracy parameter, even for $k=2$. We complement our algorithmic result with an SQ lower bound showing that the complexity dependence of $d^{Ω(k)}$ is unavoidable. Lifting the input to polynomial features turns the target into an intersection of $k$ halfspaces. This suggests using the first two labeled moments for subspace recovery. In the lifted space, however, these moments can reveal nothing. Our starting point is a Gaussian comparison theorem showing that, for sufficiently regular defining quadratics, the labeled moments needed for subspace recovery are approximately preserved. To handle arbitrary defining quadratics, we establish a new structural result showing that every direction with significant resampling influence can be detected through degree-$O(k)$ moments. These results yield an initial $d^{O(k)}2^{\mathrm{poly}_k(1/ε)}$-time learner using adaptive restrictions. To improve the exponential dependence on $1/ε$, we develop a two-stage dimension-reduction approach that avoids the costly adaptive partitioning. After identifying a subspace containing all directions with large influence, we use generalized regularity and Gaussian comparison to recover the remaining relevant quadratic features through a new derivative-based moment operator. The recovered features support an intersection close to the target. Combining them with robust hinge-loss minimization gives our quasipolynomial-time learner.

Semi-Streaming Reachability and Shortest Paths in $n^{1/3+o(1)}$ Passes

from arXiv: Data Structures and Algorithms

Authors: Gary Hoppenworth, George Z. Li, Sepideh Mahabadi, Jakub Tarnawski

In the semi-streaming model, the edges of an $n$-vertex graph arrive in a stream, and an algorithm makes multiple passes over that stream while using only $\widetilde{O}(n)$ space. The number of passes needed to compute single-source reachability in directed graphs is a long-standing open question in streaming algorithms, going back to Henzinger, Raghavan, and Rajagopalan (1998). The best known semi-streaming algorithm uses $n^{1/2+o(1)}$ passes due to Jambulapati, Liu, and Sidford (2019), while the best lower bound is $Ω(\log n/\log\log n)$ passes due to Guruswami and Onak (2016). We give streaming algorithms for computing single-source reachability and shortest paths on $n$-vertex directed graphs using $\widetilde{O}(n)$ space and $n^{1/3+o(1)}$ passes. Our algorithms are based on implicit hopsets, which are large hopsets that can be represented implicitly in $\widetilde{O}(n)$ space. We introduce this notion and give new constructions for implicit hopsets with small hopbound. Along the way, we obtain new work-depth tradeoffs for parallel shortest paths in directed graphs, improving the recent result of Ashvinkumar, Bernstein, Probst Gutenberg, and Saranurak (2026).

Authors: Gary Hoppenworth, George Z. Li, Sepideh Mahabadi, Jakub Tarnawski

In the semi-streaming model, the edges of an $n$-vertex graph arrive in a stream, and an algorithm makes multiple passes over that stream while using only $\widetilde{O}(n)$ space. The number of passes needed to compute single-source reachability in directed graphs is a long-standing open question in streaming algorithms, going back to Henzinger, Raghavan, and Rajagopalan (1998). The best known semi-streaming algorithm uses $n^{1/2+o(1)}$ passes due to Jambulapati, Liu, and Sidford (2019), while the best lower bound is $Ω(\log n/\log\log n)$ passes due to Guruswami and Onak (2016). We give streaming algorithms for computing single-source reachability and shortest paths on $n$-vertex directed graphs using $\widetilde{O}(n)$ space and $n^{1/3+o(1)}$ passes. Our algorithms are based on implicit hopsets, which are large hopsets that can be represented implicitly in $\widetilde{O}(n)$ space. We introduce this notion and give new constructions for implicit hopsets with small hopbound. Along the way, we obtain new work-depth tradeoffs for parallel shortest paths in directed graphs, improving the recent result of Ashvinkumar, Bernstein, Probst Gutenberg, and Saranurak (2026).

A Low-Diameter Nash-Williams--Tutte Theorem

from arXiv: Data Structures and Algorithms

Authors: Pin-Hsian Lee, Peng-Ting Lin, Meng-Tsung Tsai, Ting-An Wu

The Nash-Williams--Tutte theorem characterizes the existence of $k$ edge-disjoint spanning trees by a condition on every vertex partition. We prove that, for simple graphs, the same condition guarantees $k$ edge-disjoint spanning trees, each of diameter $O(n/δ)$ and with $O((n/δ)\log(δ+1))$ internal vertices, where $n$ is the number of vertices and $δ$ is the minimum degree. Both bounds hold simultaneously for every tree, with absolute implicit constants independent of $k$, and each is asymptotically optimal in general. Our proof is constructive and yields a randomized algorithm with expected running time $\tilde{O}(n^5)$. We derive applications to short edge-disjoint routes and weak linkages, small Steiner trees, and survivable networks with short paths. The joint bounds also yield low-diameter spanning Eulerian subgraphs and short tours for postman and general routing problems.

Authors: Pin-Hsian Lee, Peng-Ting Lin, Meng-Tsung Tsai, Ting-An Wu

The Nash-Williams--Tutte theorem characterizes the existence of $k$ edge-disjoint spanning trees by a condition on every vertex partition. We prove that, for simple graphs, the same condition guarantees $k$ edge-disjoint spanning trees, each of diameter $O(n/δ)$ and with $O((n/δ)\log(δ+1))$ internal vertices, where $n$ is the number of vertices and $δ$ is the minimum degree. Both bounds hold simultaneously for every tree, with absolute implicit constants independent of $k$, and each is asymptotically optimal in general. Our proof is constructive and yields a randomized algorithm with expected running time $\tilde{O}(n^5)$. We derive applications to short edge-disjoint routes and weak linkages, small Steiner trees, and survivable networks with short paths. The joint bounds also yield low-diameter spanning Eulerian subgraphs and short tours for postman and general routing problems.

The Power of Interaction in Locally Edge-Private Subgraph Counting

from arXiv: Data Structures and Algorithms

Authors: Xian Chen, Pan Peng

We study (non-induced) subgraph counting under interactive local edge differential privacy (LEDP). Given a fixed pattern graph $H$ with $k$ vertices, $\ell\ge1$ edges, and $q$ connected components, our goal is to approximately count its copies in an arbitrary $n$-vertex undirected graph while preserving edge privacy. We give two complementary accuracy guarantees. First, for every $H$, we give Linear Noise Cancellation (LNC), a two-round $\varepsilon$-LEDP algorithm with additive error and root mean square error both $O(n^{k-3/2}/\varepsilon^\ell)$ on every $n$-vertex graph. For fixed sufficiently small $\varepsilon$, this improves the optimal noninteractive additive error for triangles by a factor of $\sqrt n$ and gives the same improvement in worst-case root mean square error for all cliques on at least three vertices. To our knowledge, this is the first separation between interactive and noninteractive LEDP subgraph counting, showing that a single additional round can improve worst-case accuracy polynomially. Second, we determine how accuracy improves when the graph has maximum degree at most a public bound $d$. Our Predict and Correct (PC) framework gives an $(\ell+2)$-round $\varepsilon$-LEDP algorithm with additive error $O(n^{q-1/2}d^{k-q-1}\log^{3\ell-2}n/\varepsilon^\ell)$. For connected patterns, its polynomial factor is $\sqrt n\,d^{k-2}$, giving sublinear error on constant-degree graphs at fixed privacy. We prove matching lower bounds in $n$ and $d$, up to logarithmic factors at fixed privacy, even under arbitrary interaction and approximate privacy with sufficiently small $δ$. At $d=Θ(n)$, the lower bound also establishes the optimality of LNC's dependence on $n$.

Authors: Xian Chen, Pan Peng

We study (non-induced) subgraph counting under interactive local edge differential privacy (LEDP). Given a fixed pattern graph $H$ with $k$ vertices, $\ell\ge1$ edges, and $q$ connected components, our goal is to approximately count its copies in an arbitrary $n$-vertex undirected graph while preserving edge privacy. We give two complementary accuracy guarantees. First, for every $H$, we give Linear Noise Cancellation (LNC), a two-round $\varepsilon$-LEDP algorithm with additive error and root mean square error both $O(n^{k-3/2}/\varepsilon^\ell)$ on every $n$-vertex graph. For fixed sufficiently small $\varepsilon$, this improves the optimal noninteractive additive error for triangles by a factor of $\sqrt n$ and gives the same improvement in worst-case root mean square error for all cliques on at least three vertices. To our knowledge, this is the first separation between interactive and noninteractive LEDP subgraph counting, showing that a single additional round can improve worst-case accuracy polynomially. Second, we determine how accuracy improves when the graph has maximum degree at most a public bound $d$. Our Predict and Correct (PC) framework gives an $(\ell+2)$-round $\varepsilon$-LEDP algorithm with additive error $O(n^{q-1/2}d^{k-q-1}\log^{3\ell-2}n/\varepsilon^\ell)$. For connected patterns, its polynomial factor is $\sqrt n\,d^{k-2}$, giving sublinear error on constant-degree graphs at fixed privacy. We prove matching lower bounds in $n$ and $d$, up to logarithmic factors at fixed privacy, even under arbitrary interaction and approximate privacy with sufficiently small $δ$. At $d=Θ(n)$, the lower bound also establishes the optimality of LNC's dependence on $n$.

Matchings under One-Sided Preferences with Soft Quotas

from arXiv: Data Structures and Algorithms

Authors: Santhini K A, Raghu Raman Ravi, Meghana Nasre

Assigning applicants to posts in the presence of applicant preferences and quotas associated with posts is extensively investigated. For a post, a lower quota guarantees, and an upper quota limits the number of applicants assigned to it. Typically, quotas are assumed to be fixed, which need not be the case in practice. We address this by introducing a soft quota setting, in which every post is associated with two values - lower target and upper target - which together denote a range for the intended number of applicants in any assignment. Unlike the fixed quota setting, we allow the number of applicants assigned to a post to fall outside the range. This leads to assignments with deviation. Here, we study the problem of computing an assignment that has two orthogonal optimization objectives - minimizing the deviation (maximum or total) with respect to soft quotas and ensuring optimality with respect to preferences of applicants (rank-maximality or fairness). The order in which these objectives are considered, the different possibilities to optimize deviation combined with the well-studied notions of optimality with respect to preferences open up a range of optimization problems of practical importance. We present efficient algorithms based on flow networks to solve these optimization problems. We also consider a generalization of the soft quota setting, where every deviation is associated with a cost. We show extensions of our algorithms for solving the problems in this setting.

Authors: Santhini K A, Raghu Raman Ravi, Meghana Nasre

Assigning applicants to posts in the presence of applicant preferences and quotas associated with posts is extensively investigated. For a post, a lower quota guarantees, and an upper quota limits the number of applicants assigned to it. Typically, quotas are assumed to be fixed, which need not be the case in practice. We address this by introducing a soft quota setting, in which every post is associated with two values - lower target and upper target - which together denote a range for the intended number of applicants in any assignment. Unlike the fixed quota setting, we allow the number of applicants assigned to a post to fall outside the range. This leads to assignments with deviation. Here, we study the problem of computing an assignment that has two orthogonal optimization objectives - minimizing the deviation (maximum or total) with respect to soft quotas and ensuring optimality with respect to preferences of applicants (rank-maximality or fairness). The order in which these objectives are considered, the different possibilities to optimize deviation combined with the well-studied notions of optimality with respect to preferences open up a range of optimization problems of practical importance. We present efficient algorithms based on flow networks to solve these optimization problems. We also consider a generalization of the soft quota setting, where every deviation is associated with a cost. We show extensions of our algorithms for solving the problems in this setting.

Nearly-Linear Query Algorithm for Submodular Maximization subject to a Matroid Constraint

from arXiv: Data Structures and Algorithms

Authors: Tatsuya Terao

We consider monotone submodular maximization subject to a matroid constraint. The seminal continuous greedy algorithm of Calinescu, Chekuri, Pál, and Vondrák~[SICOMP'11] achieves the tight approximation of $(1-1/e)$. Since then, a line of works by Badanidiyuru and Vondrák~[SODA'14], Buchbinder, Feldman, and Schwartz~[SODA'15 \& MOR'17], Kobayashi and Terao~[ICALP'24], Buchbinder and Feldman~[FOCS'24 \& SICOMP'25], and Buchbinder and Feldman~[STOC'25] has developed efficient $(1-1/e-\varepsilon)$-approximation algorithms for this problem. We present a combinatorial randomized $(1-1/e-\varepsilon)$-approximation algorithm that requires $\tilde{O}_{\varepsilon}(n)$ independence-oracle and value-oracle queries, where $n$ is the number of elements in the matroid. This is the first $(1-1/e-\varepsilon)$-approximation algorithm with both independence-oracle and value-oracle query complexities nearly-linear in $n$ and polynomial in $1/\varepsilon$. To achieve this, we develop a new local search algorithm, which we call \emph{protected local search}, based on the potential function of Buchbinder and Feldman~[FOCS'24 \& SICOMP'25].

Authors: Tatsuya Terao

We consider monotone submodular maximization subject to a matroid constraint. The seminal continuous greedy algorithm of Calinescu, Chekuri, Pál, and Vondrák~[SICOMP'11] achieves the tight approximation of $(1-1/e)$. Since then, a line of works by Badanidiyuru and Vondrák~[SODA'14], Buchbinder, Feldman, and Schwartz~[SODA'15 \& MOR'17], Kobayashi and Terao~[ICALP'24], Buchbinder and Feldman~[FOCS'24 \& SICOMP'25], and Buchbinder and Feldman~[STOC'25] has developed efficient $(1-1/e-\varepsilon)$-approximation algorithms for this problem. We present a combinatorial randomized $(1-1/e-\varepsilon)$-approximation algorithm that requires $\tilde{O}_{\varepsilon}(n)$ independence-oracle and value-oracle queries, where $n$ is the number of elements in the matroid. This is the first $(1-1/e-\varepsilon)$-approximation algorithm with both independence-oracle and value-oracle query complexities nearly-linear in $n$ and polynomial in $1/\varepsilon$. To achieve this, we develop a new local search algorithm, which we call \emph{protected local search}, based on the potential function of Buchbinder and Feldman~[FOCS'24 \& SICOMP'25].

Metric Facility Location under Batched Random Order

from arXiv: Data Structures and Algorithms

Authors: Fabrizio Grandoni, Stefano Leonardi, Matteo Russo

We study online metric facility location in the batched random-order model. An adversary partitions $n$ clients into $B$ nonempty batches, which arrive in uniformly random order. Each batch is revealed in full, and its clients must be connected to open facilities before the next batch arrives. Openings and assignments are irrevocable, and the objective is to minimize the total opening and connection cost. For both uniform and nonuniform opening costs, we prove that the tight competitive ratio is \[ Θ\!\left(\min\left\{\log(B+1), \sqrt{\log(n-B+2)}\right\}\right). \] Thus, the ratio is constant both when there is a single batch and when every batch is a singleton, but intermediate numbers of batches can require a $Θ(\sqrt{\log n})$ ratio. Our algorithms build on Meyerson's algorithm (2001) and do not need to know $n$ nor $B$. A key algorithmic novelty is that we process the clients in a batch one by one in uniform random order (hence neglecting the global properties of the batch). We present a novel analysis of this algorithm that hinges on the random order of the batches, as well as of the clients within each batch. We also provide a matching competitive ratio lower bound already for uniform costs, and even when the entire metric is known in advance.

Authors: Fabrizio Grandoni, Stefano Leonardi, Matteo Russo

We study online metric facility location in the batched random-order model. An adversary partitions $n$ clients into $B$ nonempty batches, which arrive in uniformly random order. Each batch is revealed in full, and its clients must be connected to open facilities before the next batch arrives. Openings and assignments are irrevocable, and the objective is to minimize the total opening and connection cost. For both uniform and nonuniform opening costs, we prove that the tight competitive ratio is \[ Θ\!\left(\min\left\{\log(B+1), \sqrt{\log(n-B+2)}\right\}\right). \] Thus, the ratio is constant both when there is a single batch and when every batch is a singleton, but intermediate numbers of batches can require a $Θ(\sqrt{\log n})$ ratio. Our algorithms build on Meyerson's algorithm (2001) and do not need to know $n$ nor $B$. A key algorithmic novelty is that we process the clients in a batch one by one in uniform random order (hence neglecting the global properties of the batch). We present a novel analysis of this algorithm that hinges on the random order of the batches, as well as of the clients within each batch. We also provide a matching competitive ratio lower bound already for uniform costs, and even when the entire metric is known in advance.

Sorting in PRAM, Parallel Comparison Model and Parallel External Memory Model

from arXiv: Data Structures and Algorithms

Authors: Gopalan Sajith, Manas Kumar Panda, Ranajit Senko

We study the problem of sorting on the Parallel Random Access Machine (PRAM), Parallel Comparison Model (PCM) and Parallel External Memory Model (PEM). On PRAM, we present a Concurrent Read Exclusive Write algorithm that runs in $O(\log n)$ time with $n$ processors; our algorithm is very different from Cole's Merge Sort. On PCM, we present the first optimal algorithm for sorting after AKS; our algorithm runs in $O(\log n/\logα)$ comparison rounds with $nα$ processors, for $1\leqα\leq n$. On PEM, we present a Concurrent Read Exclusive Write algorithm that sorts $N$ elements in $O\left(\log\frac{N}{B}\right)$ I/O rounds with $p\geq \frac{N}{B}$ processors.

Authors: Gopalan Sajith, Manas Kumar Panda, Ranajit Senko

We study the problem of sorting on the Parallel Random Access Machine (PRAM), Parallel Comparison Model (PCM) and Parallel External Memory Model (PEM). On PRAM, we present a Concurrent Read Exclusive Write algorithm that runs in $O(\log n)$ time with $n$ processors; our algorithm is very different from Cole's Merge Sort. On PCM, we present the first optimal algorithm for sorting after AKS; our algorithm runs in $O(\log n/\logα)$ comparison rounds with $nα$ processors, for $1\leqα\leq n$. On PEM, we present a Concurrent Read Exclusive Write algorithm that sorts $N$ elements in $O\left(\log\frac{N}{B}\right)$ I/O rounds with $p\geq \frac{N}{B}$ processors.

Near-Linear Lower Bounds for Testing Directed Acyclicity

from arXiv: Data Structures and Algorithms

Authors: Artur Czumaj, Pan Peng, Yuyang Wang

We study testing acyclicity of directed graphs in the bounded-degree incidence-lists model, where a tester can query the outneighbors of a vertex but not its inneighbors. Despite significant effort, no tester with $o(n)$ queries is known, while the best established lower bounds are $\widetildeΩ(n^{5/9})$ for one-sided testers (Chen, Randolph, Servedio, and Sun, 2022) and $Ω(n^{1/3})$ for two-sided testers (Bender and Ron, 2002). We show that a query exponent arbitrarily close to one is necessary: for every fixed $η\in(0,1)$, there exists a constant $\varepsilon_η>0$ such that, for every constant $d\ge2$, every randomized adaptive two-sided $\varepsilon_η$-tester for acyclicity of simple $n$-vertex directed graphs whose indegrees and outdegrees are at most $d$ makes $Ω(n^{1-η})$ queries. Our proof overcomes two barriers. First, in previous constructions, the acyclic part of the graph is easy to recognize, which helps an algorithm stay inside the cyclic part. We hide a high-girth cyclic core inside a hierarchy of acyclic masking regions, in which each region imitates the levels below it and can be told apart from them only through rare events, and an iterated information-theoretic argument shows that every exploration observes only small, low-diameter forest components of the core. Second, a two-sided tester may exploit the statistics of collisions without ever seeing a cycle, and at a near-linear budget collisions are unavoidable. We compare complete transcripts with those of an acyclic input built on a fixed core by counting injective embeddings of the observed forest: in high-girth cores, all connected collision patterns of bounded order contribute identically, which makes the two transcript distributions statistically indistinguishable.

Authors: Artur Czumaj, Pan Peng, Yuyang Wang

We study testing acyclicity of directed graphs in the bounded-degree incidence-lists model, where a tester can query the outneighbors of a vertex but not its inneighbors. Despite significant effort, no tester with $o(n)$ queries is known, while the best established lower bounds are $\widetildeΩ(n^{5/9})$ for one-sided testers (Chen, Randolph, Servedio, and Sun, 2022) and $Ω(n^{1/3})$ for two-sided testers (Bender and Ron, 2002). We show that a query exponent arbitrarily close to one is necessary: for every fixed $η\in(0,1)$, there exists a constant $\varepsilon_η>0$ such that, for every constant $d\ge2$, every randomized adaptive two-sided $\varepsilon_η$-tester for acyclicity of simple $n$-vertex directed graphs whose indegrees and outdegrees are at most $d$ makes $Ω(n^{1-η})$ queries. Our proof overcomes two barriers. First, in previous constructions, the acyclic part of the graph is easy to recognize, which helps an algorithm stay inside the cyclic part. We hide a high-girth cyclic core inside a hierarchy of acyclic masking regions, in which each region imitates the levels below it and can be told apart from them only through rare events, and an iterated information-theoretic argument shows that every exploration observes only small, low-diameter forest components of the core. Second, a two-sided tester may exploit the statistics of collisions without ever seeing a cycle, and at a near-linear budget collisions are unavoidable. We compare complete transcripts with those of an acyclic input built on a fixed core by counting injective embeddings of the observed forest: in high-girth cores, all connected collision patterns of bounded order contribute identically, which makes the two transcript distributions statistically indistinguishable.

The Power of Bidegeneracy for Hypergraph Data Structures

from arXiv: Data Structures and Algorithms

Authors: Marco Bressan, Stefano Clemente, Giacomo Fumagalli, Daniel Paul-Pena, C. Seshadhri

A hypergraph $H = (V,E)$ describes a set system over a vertex universe $V$. Hypergraphs are increasingly popular in network science, as they represent interactions between subsets of vertices. A natural challenge is to design a data structure for $H$ to quickly answer basic questions about it, e.g. how many neighbors a vertex $v$ has or whether a hyperedge contains a vertex subset $S$. Such questions are trivial for graphs, for which we simply inspect the adjacency list. The gold standard, from a data structures perspective, would be near-linear time preprocessing and query answering in (poly)logarithmic time. Since this is unlikely for arbitrary $H$, we ask for well-motivated hypergraph properties that can make such a goal possible. We introduce bidegeneracy, a measure of hypergraph sparsity that strictly generalizes many other measures, including bounded degree, bounded rank, and bounded $\infty$-degeneracy. We show that bounded bidegeneracy yields fast hypergraph data structures. For a large set of fundamental queries, including their listing and sampling versions, we obtain near-linear preprocessing and (poly)logarithmic query time. Moreover, direct reductions from assumptions like the Orthogonal Vectors Conjecture suggest that, without assumptions on the input, these structures require almost-quadratic preprocessing or almost-linear query time. We use bidegeneracy to compute a compressed representation of $H$ and perform careful reductions to stabbing and searching problems in $\mathbb{R}^d$. Our framework significantly extends the well-studied "partial matching" problem of finding database entries through wildcard search. It also uncovers an intriguing connection between hypergraph sparsity, compression, and the complexity of hypergraph primitives. We also find remarkably low bidegeneracy in hypergraph datasets from the literature, suggesting practical consequences.

Authors: Marco Bressan, Stefano Clemente, Giacomo Fumagalli, Daniel Paul-Pena, C. Seshadhri

A hypergraph $H = (V,E)$ describes a set system over a vertex universe $V$. Hypergraphs are increasingly popular in network science, as they represent interactions between subsets of vertices. A natural challenge is to design a data structure for $H$ to quickly answer basic questions about it, e.g. how many neighbors a vertex $v$ has or whether a hyperedge contains a vertex subset $S$. Such questions are trivial for graphs, for which we simply inspect the adjacency list. The gold standard, from a data structures perspective, would be near-linear time preprocessing and query answering in (poly)logarithmic time. Since this is unlikely for arbitrary $H$, we ask for well-motivated hypergraph properties that can make such a goal possible. We introduce bidegeneracy, a measure of hypergraph sparsity that strictly generalizes many other measures, including bounded degree, bounded rank, and bounded $\infty$-degeneracy. We show that bounded bidegeneracy yields fast hypergraph data structures. For a large set of fundamental queries, including their listing and sampling versions, we obtain near-linear preprocessing and (poly)logarithmic query time. Moreover, direct reductions from assumptions like the Orthogonal Vectors Conjecture suggest that, without assumptions on the input, these structures require almost-quadratic preprocessing or almost-linear query time. We use bidegeneracy to compute a compressed representation of $H$ and perform careful reductions to stabbing and searching problems in $\mathbb{R}^d$. Our framework significantly extends the well-studied "partial matching" problem of finding database entries through wildcard search. It also uncovers an intriguing connection between hypergraph sparsity, compression, and the complexity of hypergraph primitives. We also find remarkably low bidegeneracy in hypergraph datasets from the literature, suggesting practical consequences.

Local Distributed Computation under Crash Failures

from arXiv: Data Structures and Algorithms

Authors: Diana Ghinea, Fabian Kuhn, Chen-Da Liu-Zhang, Zahra Parsaeian

We study deterministic algorithms for locally checkable graph problems in the synchronous LOCAL model with crash failures, where a crashing node may send messages to only some neighbors. We introduce consistent neighborhoods: partial radius-$R$ views that agree locally across surviving edges and are complete without crashes. For $R=T+\ell+1$, these views allow local simulation of $T$-round fault-free algorithms that remain correct on subgraphs, satisfying constraints wherever the radius-$\ell$ checking neighborhood contains no crash. On graphs with at most $n$ nodes and maximum degree at most $Δ\geq3$, we construct these views in $O(\min\{n,(Δ-1)^R,RΔf\})$ rounds when a crash bound $f\geq1$ is globally known; the first two bounds require no knowledge of $f$. For $Δ=2$ or girth greater than $2R$, including forests, $O(R)$ rounds suffice. Our upper bounds are essentially tight in suitable parameter regimes. One crash can force $Ω(n)$ rounds even for $R=3$ on diameter-two graphs. For each fixed $Δ\geq3$ and sufficiently large $R$, we prove a $(Δ-1)^{R-O(\sqrt R)}$ lower bound even with crashes restricted to round beginnings. We also prove an $Ω(ΔRf)$ lower bound when $R\geq4\lceil\log_{Δ-1}f\rceil+4$ and $n\geqΔRf$. For locally checkable labeling (LCL) problems, we preserve $O(\log^* n)$ complexity under any number of crashes whenever this bound holds without faults on all graphs with a fixed degree bound. In contrast, we construct an LCL on graphs of maximum degree at most five with fault-free complexity $Θ(\log n)$ and crash-tolerant complexity $Θ(n)$. Finally, any fault-free sinkless-orientation algorithm can be transformed to tolerate any number of crashes with only one additional round, guaranteeing correctness even at surviving nodes adjacent to crashes.

Authors: Diana Ghinea, Fabian Kuhn, Chen-Da Liu-Zhang, Zahra Parsaeian

We study deterministic algorithms for locally checkable graph problems in the synchronous LOCAL model with crash failures, where a crashing node may send messages to only some neighbors. We introduce consistent neighborhoods: partial radius-$R$ views that agree locally across surviving edges and are complete without crashes. For $R=T+\ell+1$, these views allow local simulation of $T$-round fault-free algorithms that remain correct on subgraphs, satisfying constraints wherever the radius-$\ell$ checking neighborhood contains no crash. On graphs with at most $n$ nodes and maximum degree at most $Δ\geq3$, we construct these views in $O(\min\{n,(Δ-1)^R,RΔf\})$ rounds when a crash bound $f\geq1$ is globally known; the first two bounds require no knowledge of $f$. For $Δ=2$ or girth greater than $2R$, including forests, $O(R)$ rounds suffice. Our upper bounds are essentially tight in suitable parameter regimes. One crash can force $Ω(n)$ rounds even for $R=3$ on diameter-two graphs. For each fixed $Δ\geq3$ and sufficiently large $R$, we prove a $(Δ-1)^{R-O(\sqrt R)}$ lower bound even with crashes restricted to round beginnings. We also prove an $Ω(ΔRf)$ lower bound when $R\geq4\lceil\log_{Δ-1}f\rceil+4$ and $n\geqΔRf$. For locally checkable labeling (LCL) problems, we preserve $O(\log^* n)$ complexity under any number of crashes whenever this bound holds without faults on all graphs with a fixed degree bound. In contrast, we construct an LCL on graphs of maximum degree at most five with fault-free complexity $Θ(\log n)$ and crash-tolerant complexity $Θ(n)$. Finally, any fault-free sinkless-orientation algorithm can be transformed to tolerate any number of crashes with only one additional round, guaranteeing correctness even at surviving nodes adjacent to crashes.

NP-Hardness and a PTAS for Maximum Interval Multicover, with Applications to Dense Subgraphs

from arXiv: Data Structures and Algorithms

Authors: Morteza Alimi, Tobias Mömke, Alexandru Popa, Michael Ruderer

Given intervals and points on a line, each point with a demand, and a budget $K$, Maximum Interval Multicover asks for at most $K$ intervals that satisfy as many points as possible, a point being satisfied if it lies in at least as many chosen intervals as its demand. One application is committee elections in which each candidate is approved by consecutive voters and voters have individual demands. Xu, Ran and Zhang introduced the problem and gave a $1/2$-approximation; the objective is not submodular when demands exceed one. Li, Tu, Zhang and Cao gave a polynomial-time approximation scheme (PTAS) for the minimization counterpart, unweighted partial interval multicover, and asked for its complexity. We prove that both problems are NP-hard, even for intervals of equal length; as their optima are integers bounded by the input size, neither has a fully polynomial-time approximation scheme unless $\mathrm{P}=\mathrm{NP}$. We then give a PTAS for Maximum Interval Multicover, also for arbitrary nonnegative nondecreasing rewards for the coverage of each point. Building on a decomposition from Nonner's PTAS for Densest $k$-Subgraph on interval graphs, our scheme cuts the line into pieces at a small loss and solves every piece exactly. The key dynamic program handles intervals crossing a common point by choosing an interleaving of their left and right endpoints rather than sweeping along the line; it needs only two counters to track these intervals. The same algorithm yields PTASes on convex bipartite graphs for Bipartite Dense Subgraph, improving on the ratio $2/3$ of Backer and Keil for the subclass of bipartite permutation graphs, and on interval graphs for Maximum $k$-Vertex Cover and weighted clique counts.

Authors: Morteza Alimi, Tobias Mömke, Alexandru Popa, Michael Ruderer

Given intervals and points on a line, each point with a demand, and a budget $K$, Maximum Interval Multicover asks for at most $K$ intervals that satisfy as many points as possible, a point being satisfied if it lies in at least as many chosen intervals as its demand. One application is committee elections in which each candidate is approved by consecutive voters and voters have individual demands. Xu, Ran and Zhang introduced the problem and gave a $1/2$-approximation; the objective is not submodular when demands exceed one. Li, Tu, Zhang and Cao gave a polynomial-time approximation scheme (PTAS) for the minimization counterpart, unweighted partial interval multicover, and asked for its complexity. We prove that both problems are NP-hard, even for intervals of equal length; as their optima are integers bounded by the input size, neither has a fully polynomial-time approximation scheme unless $\mathrm{P}=\mathrm{NP}$. We then give a PTAS for Maximum Interval Multicover, also for arbitrary nonnegative nondecreasing rewards for the coverage of each point. Building on a decomposition from Nonner's PTAS for Densest $k$-Subgraph on interval graphs, our scheme cuts the line into pieces at a small loss and solves every piece exactly. The key dynamic program handles intervals crossing a common point by choosing an interleaving of their left and right endpoints rather than sweeping along the line; it needs only two counters to track these intervals. The same algorithm yields PTASes on convex bipartite graphs for Bipartite Dense Subgraph, improving on the ratio $2/3$ of Backer and Keil for the subclass of bipartite permutation graphs, and on interval graphs for Maximum $k$-Vertex Cover and weighted clique counts.

Best-of-Many-Worlds MMS Allocation for Chores with Sublinear Disposals

from arXiv: Data Structures and Algorithms

Authors: Zehan Lin, Xiaowei Wu, Shengwei Zhou

We study randomized allocations (lotteries) of indivisible chores under additive costs and introduce the notion of MMS1, which requires each agent's bundle to satisfy the maximin share (MMS) bound after the removal of at most one chore. Our main result establishes that every instance admits an ex-ante proportional lottery in whose support every allocation is MMS1 and all but $o(n)$ agents satisfy MMS. Consequently, every realized allocation can be made MMS by disposing of at most $o(n)$ chores. The proposed framework further achieves ex-ante proportionality alongside ex-post $4/3$-MMS, $o(n)$-AMMS, or $1$-out-of-$d$ MMS for $d=(1-o(1))n$. These guarantees substantially improve upon state-of-the-art bounds for MMS allocation for chores with disposal or subsidy, best-of-both-worlds fairness, approximate AMMS, and ordinal MMS approximations.

Authors: Zehan Lin, Xiaowei Wu, Shengwei Zhou

We study randomized allocations (lotteries) of indivisible chores under additive costs and introduce the notion of MMS1, which requires each agent's bundle to satisfy the maximin share (MMS) bound after the removal of at most one chore. Our main result establishes that every instance admits an ex-ante proportional lottery in whose support every allocation is MMS1 and all but $o(n)$ agents satisfy MMS. Consequently, every realized allocation can be made MMS by disposing of at most $o(n)$ chores. The proposed framework further achieves ex-ante proportionality alongside ex-post $4/3$-MMS, $o(n)$-AMMS, or $1$-out-of-$d$ MMS for $d=(1-o(1))n$. These guarantees substantially improve upon state-of-the-art bounds for MMS allocation for chores with disposal or subsidy, best-of-both-worlds fairness, approximate AMMS, and ordinal MMS approximations.

Faster Regular Expression Matching via Two-Dimensional Tabulation

from arXiv: Data Structures and Algorithms

Authors: Philip Bille, Inge Li Gørtz

Regular expression matching asks whether a string of length $n$ belongs to the language described by a regular expression of length $m$. We give an algorithm that solves this problem in linear space and $O(nm \log \log n/\log^2 n + n + m)$ time. This improves the leading term of the previous best worst-case constant-alphabet time bound by a factor of $\sqrt{\log n}$. The key new technique is a two-dimensional tabulation specialized to the regular expression.

Authors: Philip Bille, Inge Li Gørtz

Regular expression matching asks whether a string of length $n$ belongs to the language described by a regular expression of length $m$. We give an algorithm that solves this problem in linear space and $O(nm \log \log n/\log^2 n + n + m)$ time. This improves the leading term of the previous best worst-case constant-alphabet time bound by a factor of $\sqrt{\log n}$. The key new technique is a two-dimensional tabulation specialized to the regular expression.

Real-to-Integer Reduction for All-Pairs Shortest Paths

from arXiv: Data Structures and Algorithms

Authors: Nick Fischer, Adam Polak, Jonas Schmidt

Efficient graph algorithms often use techniques that are specific to integer-valued inputs and do not work for real-valued versions of the problems. Is it a limitation of current techniques, or are the real-valued variants inherently harder? One way to study this question is through real-to-integer reductions. We give (randomized, combinatorial) subcubic real-to-integer reductions for two central graph problems in fine-grained complexity: All-Pairs Shortest Paths and Exact Triangle. That is, we show that if the integer variants of these problems can be solved in truly subcubic time, then the real variants can also be solved in truly subcubic time. Our work extends and combines techniques from three previous works: (1) our real-to-integer reduction for the 3SUM problem (Fischer, Polak, Schmidt, FOCS 2026), which itself combines the Frank--Tardos theorem from optimization with Freiman's theorem from additive combinatorics; (2) universe reduction for APSP (Fischer, STOC 2026), which we generalize to reals and use to reduce the doubling of the set of edge weights; and (3) the ``Fredman's trick meets dominance product'' technique (Chan, Vassilevska Williams, Xu, STOC 2023), which is needed to adapt the prior real-to-integer framework from equality constraints to inequalities. As a corollary, our reduction combined with Yuval's fast Integer APSP algorithm in the real RAM model with multiplication (IPL 1976) shows that the Real APSP Hypothesis is false in this model, which also follows from the very recent breakthrough by Alman and Vassilevska Williams using a very different approach.

Authors: Nick Fischer, Adam Polak, Jonas Schmidt

Efficient graph algorithms often use techniques that are specific to integer-valued inputs and do not work for real-valued versions of the problems. Is it a limitation of current techniques, or are the real-valued variants inherently harder? One way to study this question is through real-to-integer reductions. We give (randomized, combinatorial) subcubic real-to-integer reductions for two central graph problems in fine-grained complexity: All-Pairs Shortest Paths and Exact Triangle. That is, we show that if the integer variants of these problems can be solved in truly subcubic time, then the real variants can also be solved in truly subcubic time. Our work extends and combines techniques from three previous works: (1) our real-to-integer reduction for the 3SUM problem (Fischer, Polak, Schmidt, FOCS 2026), which itself combines the Frank--Tardos theorem from optimization with Freiman's theorem from additive combinatorics; (2) universe reduction for APSP (Fischer, STOC 2026), which we generalize to reals and use to reduce the doubling of the set of edge weights; and (3) the ``Fredman's trick meets dominance product'' technique (Chan, Vassilevska Williams, Xu, STOC 2023), which is needed to adapt the prior real-to-integer framework from equality constraints to inequalities. As a corollary, our reduction combined with Yuval's fast Integer APSP algorithm in the real RAM model with multiplication (IPL 1976) shows that the Real APSP Hypothesis is false in this model, which also follows from the very recent breakthrough by Alman and Vassilevska Williams using a very different approach.

Near-linear-Time Sampling of the Ferromagnetic Ising Model at the Tree Uniqueness Threshold

from arXiv: Data Structures and Algorithms

Authors: Tianshun Miao

We give a near-linear-time sampler for the zero-field ferromagnetic Ising model throughout the tree-uniqueness regime, including the threshold itself. The method also gives sampling algorithms for the XY and integer Potts models. On an $n$-vertex graph with $m$ edges and maximum degree at most $Δ$, we obtain the following expected-work bounds, with total variation error at most $\varepsilon$. 1.Near-linear Ising sampling. For $Δ\ge3$, the work is $O(n+m\log^3(n/\varepsilon))$ throughout the tree-uniqueness regime, including the threshold $β=Δ/(Δ-2)$. 2.Almost-linear XY sampling. For $Δ\ge8$, we sample the XY model in $\widetilde O(m+n^{1+o(1)})$ expected work at inverse-polynomial accuracy up to and including the robust phase-transition threshold on the $Δ$-regular tree. 3.Near-linear Potts sampling. For integer $q\ge3$, we sample the $q$-state Potts model in $\widetilde O(n+q^{O(1)}m)$ expected work whenever $β-1\le 0.1(β_{\mathrm{od}}(q,Δ)-1)$, where $β_{\mathrm{od}}$ is the ordered/disordered coexistence parameter. We express the even-subgraph representation of critical Ising as a polymer model that admits near-linear-time sampling, thereby bypassing the critical slowdown of Glauber and Swendsen--Wang dynamics. Building on the low-degree approximation approach of Miao and Yin, we establish general spectral conditions for fast polymer sampling. We verify these conditions through explorations that also implement the sampler efficiently, and apply the same framework to flow representations of the XY and Potts models. We also establish polynomial-time counting and sampling for the ferromagnetic XY model at every fixed temperature.

Authors: Tianshun Miao

We give a near-linear-time sampler for the zero-field ferromagnetic Ising model throughout the tree-uniqueness regime, including the threshold itself. The method also gives sampling algorithms for the XY and integer Potts models. On an $n$-vertex graph with $m$ edges and maximum degree at most $Δ$, we obtain the following expected-work bounds, with total variation error at most $\varepsilon$. 1.Near-linear Ising sampling. For $Δ\ge3$, the work is $O(n+m\log^3(n/\varepsilon))$ throughout the tree-uniqueness regime, including the threshold $β=Δ/(Δ-2)$. 2.Almost-linear XY sampling. For $Δ\ge8$, we sample the XY model in $\widetilde O(m+n^{1+o(1)})$ expected work at inverse-polynomial accuracy up to and including the robust phase-transition threshold on the $Δ$-regular tree. 3.Near-linear Potts sampling. For integer $q\ge3$, we sample the $q$-state Potts model in $\widetilde O(n+q^{O(1)}m)$ expected work whenever $β-1\le 0.1(β_{\mathrm{od}}(q,Δ)-1)$, where $β_{\mathrm{od}}$ is the ordered/disordered coexistence parameter. We express the even-subgraph representation of critical Ising as a polymer model that admits near-linear-time sampling, thereby bypassing the critical slowdown of Glauber and Swendsen--Wang dynamics. Building on the low-degree approximation approach of Miao and Yin, we establish general spectral conditions for fast polymer sampling. We verify these conditions through explorations that also implement the sampler efficiently, and apply the same framework to flow representations of the XY and Potts models. We also establish polynomial-time counting and sampling for the ferromagnetic XY model at every fixed temperature.

Optimal Approximation for Constrained Correlation Clustering via Repulsion

from arXiv: Data Structures and Algorithms

Authors: Yixin Cao, Ying Xu

The best-known approximation factor for constrained correlation clustering stood at three for nearly two decades, until a recent improvement to below $16/7$. We give a randomized polynomial-time $(2+\varepsilon)$-approximation using \emph{Repulsion}, a recursive operation that separates a metric ball from its complement. Repulsion naturally respects hard constraints and admits a short potential-based analysis; as a warm-up, radius-$2/3$ Repulsion rounds the standard metric LP into a feasible partition of cost at most three times its objective. The $(2+\varepsilon)$-approximation takes a two-stage approach, first removing must-links with arbitrarily small loss in the approximation ratio. The second stage combines lossless, constraint-aware preclustering with the bounded sub-cluster LP framework of Cao, Cohen-Addad, Lee, Li, Newman, and Vogl (STOC 2024). Repeated positive conditioning and sampling produce candidate sets, which radius-one Repulsion partitions into feasible clusters at controlled cost. We also give an approximation-preserving reduction from vertex cover to cluster deletion, a special case with only cannot-links. Apart from establishing a matching factor-two hardness threshold under the Unique Games Conjecture, it motivates our two-stage approach and Repulsion.

Authors: Yixin Cao, Ying Xu

The best-known approximation factor for constrained correlation clustering stood at three for nearly two decades, until a recent improvement to below $16/7$. We give a randomized polynomial-time $(2+\varepsilon)$-approximation using \emph{Repulsion}, a recursive operation that separates a metric ball from its complement. Repulsion naturally respects hard constraints and admits a short potential-based analysis; as a warm-up, radius-$2/3$ Repulsion rounds the standard metric LP into a feasible partition of cost at most three times its objective. The $(2+\varepsilon)$-approximation takes a two-stage approach, first removing must-links with arbitrarily small loss in the approximation ratio. The second stage combines lossless, constraint-aware preclustering with the bounded sub-cluster LP framework of Cao, Cohen-Addad, Lee, Li, Newman, and Vogl (STOC 2024). Repeated positive conditioning and sampling produce candidate sets, which radius-one Repulsion partitions into feasible clusters at controlled cost. We also give an approximation-preserving reduction from vertex cover to cluster deletion, a special case with only cannot-links. Apart from establishing a matching factor-two hardness threshold under the Unique Games Conjecture, it motivates our two-stage approach and Repulsion.

Testing for a Large Induced Expander: Sublinear Algorithms and Query Lower Bounds

from arXiv: Data Structures and Algorithms

Authors: Pan Peng, Ranran Shen

Testing expansion asks to distinguish graphs of large conductance from graphs that are far from having large conductance. We study a relaxation in which only part of the graph is required to expand: does an $n$-vertex graph of bounded degree contain an induced expander on a constant fraction of its vertices, when the rest of the graph, including the edges leaving the expander, is arbitrary? Let $P_{α,φ}$ denote the property of containing an induced subgraph on at least $αn$ vertices with inner conductance at least $φ$. For every constant $κ\in(0,1)$, we give a tester that accepts graphs in $P_{α,φ}$ and rejects graphs outside $P_{κα,μφ^2}$, where $μ=Θ_{α,κ}(1/\log n)$, using $O_{α,κ}(\sqrt n\,φ^{-2}\log n)$ queries. For fixed $α\in(0,1]$, $κ\in(0,1)$, and $0<φ\le9/20$, we further show that distinguishing these two classes requires $Ω_{α,κ,φ}(n^{1/2+c_{κ,φ}μ})$ queries for every sufficiently large $n$ and every $0<μ\le1$, where $c_{κ,φ}>0$. The bound holds for adaptive algorithms, uniformly in $μ$. Hence, for testers with $\sqrt n\,\mathrm{polylog}(n)$ queries, the logarithmic loss in conductance is necessary up to a $\log\log n$ factor. The same tester also permits distance-based rejection at a weaker conductance while retaining the original core size. Our tester compares the endpoint distributions of random walks from different vertices with each other, rather than with the uniform distribution. Its analysis uses the trace chain, which observes the walk only at its visits to the unknown expanding core. Our lower bound builds on the noisy-parity construction of [CKK+18], using vector-valued parity labels in $\mathbb F_2^r$ to create many hidden classes, each too small to contain a qualifying induced expander.

Authors: Pan Peng, Ranran Shen

Testing expansion asks to distinguish graphs of large conductance from graphs that are far from having large conductance. We study a relaxation in which only part of the graph is required to expand: does an $n$-vertex graph of bounded degree contain an induced expander on a constant fraction of its vertices, when the rest of the graph, including the edges leaving the expander, is arbitrary? Let $P_{α,φ}$ denote the property of containing an induced subgraph on at least $αn$ vertices with inner conductance at least $φ$. For every constant $κ\in(0,1)$, we give a tester that accepts graphs in $P_{α,φ}$ and rejects graphs outside $P_{κα,μφ^2}$, where $μ=Θ_{α,κ}(1/\log n)$, using $O_{α,κ}(\sqrt n\,φ^{-2}\log n)$ queries. For fixed $α\in(0,1]$, $κ\in(0,1)$, and $0<φ\le9/20$, we further show that distinguishing these two classes requires $Ω_{α,κ,φ}(n^{1/2+c_{κ,φ}μ})$ queries for every sufficiently large $n$ and every $0<μ\le1$, where $c_{κ,φ}>0$. The bound holds for adaptive algorithms, uniformly in $μ$. Hence, for testers with $\sqrt n\,\mathrm{polylog}(n)$ queries, the logarithmic loss in conductance is necessary up to a $\log\log n$ factor. The same tester also permits distance-based rejection at a weaker conductance while retaining the original core size. Our tester compares the endpoint distributions of random walks from different vertices with each other, rather than with the uniform distribution. Its analysis uses the trace chain, which observes the walk only at its visits to the unknown expanding core. Our lower bound builds on the noisy-parity construction of [CKK+18], using vector-valued parity labels in $\mathbb F_2^r$ to create many hidden classes, each too small to contain a qualifying induced expander.

Optimal (Randomized) Compact Routing Schemes

from arXiv: Data Structures and Algorithms

Authors: Shiri Chechik, Avi Kadria, Gur Lifshitz, Liam Roditty

Compact routing schemes route messages along short paths while every vertex stores little information. We study the trade-off between the stretch and the storage of compact routing schemes in undirected graphs, in four models: labeled and name-independent schemes, with average or worst-case local storage. We improve the best known stretch in all four models. Our main result is a labeled routing scheme with expected stretch $2k-1$ and $\widetilde O(n^{1/k})$ average local storage, in which only the routing is randomized. Under the Erdős girth conjecture, no scheme with this storage has stretch below $2k-1$, so our scheme is optimal, up to polylogarithmic factors in the storage. With deterministic routing, we obtain stretch $\approx2.47k$, the best known deterministic stretch with average storage. Worst-case local storage is the classical setting of compact routing. The leading coefficient $4$ in the stretch $4k-5$ of Thorup and Zwick~[SPAA'01] was improved only by Chechik~[PODC'13], to $\approx3.68$ in unweighted graphs. In unweighted graphs, we obtain expected stretch $(\frac83k+O(1),O(k))$, and deterministic stretch $\approx3.05k$. Name-independent routing is harder, since the source knows only the name of the destination. The best known stretch with $\widetilde O(n^{1/k})$ local storage in unweighted graphs was $96k$, by Abraham, Gavoille, and Malkhi~[DISC'04]. We improve it by more than a factor of $8$. With average storage, we obtain expected stretch $4k-5$, which matches the labeled scheme of Thorup and Zwick. With worst-case storage in unweighted graphs, we obtain expected stretch $(8k+O(1),O(k))$. All our schemes for unweighted graphs extend to weighted graphs with a small loss in the stretch, and still improve the previous unweighted bounds.

Authors: Shiri Chechik, Avi Kadria, Gur Lifshitz, Liam Roditty

Compact routing schemes route messages along short paths while every vertex stores little information. We study the trade-off between the stretch and the storage of compact routing schemes in undirected graphs, in four models: labeled and name-independent schemes, with average or worst-case local storage. We improve the best known stretch in all four models. Our main result is a labeled routing scheme with expected stretch $2k-1$ and $\widetilde O(n^{1/k})$ average local storage, in which only the routing is randomized. Under the Erdős girth conjecture, no scheme with this storage has stretch below $2k-1$, so our scheme is optimal, up to polylogarithmic factors in the storage. With deterministic routing, we obtain stretch $\approx2.47k$, the best known deterministic stretch with average storage. Worst-case local storage is the classical setting of compact routing. The leading coefficient $4$ in the stretch $4k-5$ of Thorup and Zwick~[SPAA'01] was improved only by Chechik~[PODC'13], to $\approx3.68$ in unweighted graphs. In unweighted graphs, we obtain expected stretch $(\frac83k+O(1),O(k))$, and deterministic stretch $\approx3.05k$. Name-independent routing is harder, since the source knows only the name of the destination. The best known stretch with $\widetilde O(n^{1/k})$ local storage in unweighted graphs was $96k$, by Abraham, Gavoille, and Malkhi~[DISC'04]. We improve it by more than a factor of $8$. With average storage, we obtain expected stretch $4k-5$, which matches the labeled scheme of Thorup and Zwick. With worst-case storage in unweighted graphs, we obtain expected stretch $(8k+O(1),O(k))$. All our schemes for unweighted graphs extend to weighted graphs with a small loss in the stretch, and still improve the previous unweighted bounds.

Approximation Algorithms for Multi-Criteria Network Design Problems

from arXiv: Data Structures and Algorithms

Authors: Mehrshad Taziki, Ali Vakilian

Min-max multi-criteria optimization asks for a single feasible solution minimizing the maximum of its costs under $\ell$ nonnegative cost functions, a formulation that captures both fairness across groups and robustness across scenarios. It has proved hard to approximate even for basic problems: for robust shortest paths, polylogarithmic approximations are known only in quasi-polynomial time. For network design, the best known guarantee is an $\ell$-approximation, which is tight for the natural LP relaxation. We show that exponentially better and nearly optimal guarantees are achievable for two fundamental network design problems. For multi-criteria connectivity augmentation (MC-CAP), which asks to increase the edge connectivity of a given undirected graph by one, we give a randomized polynomial-time $O(\log \ell)$-approximation algorithm. Applying our augmentation procedure iteratively and analyzing its randomized choices jointly across iterations yields an $O(\log k \log \ell)$-approximation for the multi-criteria $k$-edge-connected spanning subgraph (MC-$k$-ECSS) problem, for every $k \geq 2$. We complement these algorithms with a nearly matching lower bound: for every fixed $\varepsilon > 0$, MC-CAP admits no polynomial-time $O(\log^{1-\varepsilon} \ell)$-approximation, even when the underlying graph is a star, unless $\mathsf{NP}\subseteq\mathsf{DTIME}\bigl(n^{\log^{O(1)}n}\bigr)$. The same hardness extends to the MC-$k$-ECSS problem.

Authors: Mehrshad Taziki, Ali Vakilian

Min-max multi-criteria optimization asks for a single feasible solution minimizing the maximum of its costs under $\ell$ nonnegative cost functions, a formulation that captures both fairness across groups and robustness across scenarios. It has proved hard to approximate even for basic problems: for robust shortest paths, polylogarithmic approximations are known only in quasi-polynomial time. For network design, the best known guarantee is an $\ell$-approximation, which is tight for the natural LP relaxation. We show that exponentially better and nearly optimal guarantees are achievable for two fundamental network design problems. For multi-criteria connectivity augmentation (MC-CAP), which asks to increase the edge connectivity of a given undirected graph by one, we give a randomized polynomial-time $O(\log \ell)$-approximation algorithm. Applying our augmentation procedure iteratively and analyzing its randomized choices jointly across iterations yields an $O(\log k \log \ell)$-approximation for the multi-criteria $k$-edge-connected spanning subgraph (MC-$k$-ECSS) problem, for every $k \geq 2$. We complement these algorithms with a nearly matching lower bound: for every fixed $\varepsilon > 0$, MC-CAP admits no polynomial-time $O(\log^{1-\varepsilon} \ell)$-approximation, even when the underlying graph is a star, unless $\mathsf{NP}\subseteq\mathsf{DTIME}\bigl(n^{\log^{O(1)}n}\bigr)$. The same hardness extends to the MC-$k$-ECSS problem.

Smallest Multiplicative Stretch for an $α$-APASP Below $\widetilde O \left(n^ω\right)$ Runtime

from arXiv: Data Structures and Algorithms

Authors: Lakshay Saggi, Ariel Sapir

Dor, Halperin and Zwick [FOCS'96; SICOMP'00] showed that $(2-\varepsilon)$-APASP is at least as hard as BMM for every fixed $0<\varepsilon<1$. Zwick [FOCS'98; JACM'02] gave a $2$-APASP algorithm in $\widetilde O(n^{ω+o(1)}\log W_{\max})$ runtime, where $W_{\max}$ is the maximum edge weight. Bringmann, K{ü}nnemann and W{ę}grzycki [STOC'19] removed this logarithmic factor. Using OpenAI's recent bound $ω\leq\frac{9}{4}$ gives $\widetilde O\left(n^{\frac{9}{4}+o(1)}\right)$ time. We present two weighted undirected $2$-APASP algorithms. The first uses $(\min,\max)$-products. Earlier rectangular FMM bounds of Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'24] and Alman, Duan, Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'25] give $\widetilde O(n^{2.36999465})$ runtime, below the previous $ω\leq2.371177$ of Dupont, Eisenberger, Kozlovskii, Mehrabian, Ruiz, See, Zhou, Alman, Vassilevska Williams and Balog [arXiv'26]. Current rectangular FMM bounds give $\widetilde O(n^{2.34549151})$, independent of $W_{\max}$. For integer weights, our second algorithm has $\widetilde O(n^{ω-\frac{(ω-2)^2}{ω-1}+o(1)}\log^{O(1)}W_{\max})$ runtime, with exponent below $ω$ whenever $ω>2$. The rectangular bounds of Uzna{ń}ski [arXiv'26] or of Amanov, van den Berg and Zuiddam [arXiv'26] give $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.

Authors: Lakshay Saggi, Ariel Sapir

Dor, Halperin and Zwick [FOCS'96; SICOMP'00] showed that $(2-\varepsilon)$-APASP is at least as hard as BMM for every fixed $0<\varepsilon<1$. Zwick [FOCS'98; JACM'02] gave a $2$-APASP algorithm in $\widetilde O(n^{ω+o(1)}\log W_{\max})$ runtime, where $W_{\max}$ is the maximum edge weight. Bringmann, K{ü}nnemann and W{ę}grzycki [STOC'19] removed this logarithmic factor. Using OpenAI's recent bound $ω\leq\frac{9}{4}$ gives $\widetilde O\left(n^{\frac{9}{4}+o(1)}\right)$ time. We present two weighted undirected $2$-APASP algorithms. The first uses $(\min,\max)$-products. Earlier rectangular FMM bounds of Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'24] and Alman, Duan, Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'25] give $\widetilde O(n^{2.36999465})$ runtime, below the previous $ω\leq2.371177$ of Dupont, Eisenberger, Kozlovskii, Mehrabian, Ruiz, See, Zhou, Alman, Vassilevska Williams and Balog [arXiv'26]. Current rectangular FMM bounds give $\widetilde O(n^{2.34549151})$, independent of $W_{\max}$. For integer weights, our second algorithm has $\widetilde O(n^{ω-\frac{(ω-2)^2}{ω-1}+o(1)}\log^{O(1)}W_{\max})$ runtime, with exponent below $ω$ whenever $ω>2$. The rectangular bounds of Uzna{ń}ski [arXiv'26] or of Amanov, van den Berg and Zuiddam [arXiv'26] give $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.

Tree Covers for Directed Graphs: Distance Oracles, Routing Schemes and Spanners

from arXiv: Data Structures and Algorithms

Authors: Shiri Chechik, Gur Lifshitz

In a seminal work [STOC'01; JACM'05], Thorup and Zwick showed that every weighted undirected graph admits a tree cover where each vertex belongs to $O(kn^{1/k})$ trees in expectation, and every pair has a path of stretch at most $2k-1$ in some tree. In this paper, we extend this structural property to directed graphs with nonnegative edge weights. Let $\mathrm{dist}_G(u \leftrightarrows v) = \mathrm{dist}_G(u \to v) + \mathrm{dist}_G(v \to u)$ denote the roundtrip distance. For every integer $k$, we construct a collection $\mathcal{T}$ of tree-pairs, each consisting of an in-tree and an out-tree rooted at the same vertex. Each vertex belongs to $O(kn^{1/k})$ tree-pairs in expectation, and for every ordered pair $(u, v)$ some tree-pair $T_{in}\cup T_{out} \in \mathcal{T}$ satisfies $\mathrm{dist}_{T_{in}\cup T_{out}}(u \to v) \le \mathrm{dist}_G(u \to v) + (k-1)\mathrm{dist}_G(u \leftrightarrows v).$ We call this a \emph{$k$-strong (roundtrip) tree cover}. While prior techniques partition distances into geometric scales and build separate structures for each scale, our construction handles all distances simultaneously within a single, unified hierarchy. This yields the first directed tree cover for general $k$ matching the classic Thorup-Zwick stretch-size trade-off. Our cover immediately implies improved constructions of roundtrip spanners, path-reporting distance oracles, and compact routing schemes, as well as their source-wise and prioritized variants. We also provide faster algorithms for these structures: for general $k \ge 4$, all the above objects can be constructed in $\widetilde{O}(\min\{mn, n^{2.5}\})$ expected time, breaking the $mn$ barrier in dense directed graphs.

Authors: Shiri Chechik, Gur Lifshitz

In a seminal work [STOC'01; JACM'05], Thorup and Zwick showed that every weighted undirected graph admits a tree cover where each vertex belongs to $O(kn^{1/k})$ trees in expectation, and every pair has a path of stretch at most $2k-1$ in some tree. In this paper, we extend this structural property to directed graphs with nonnegative edge weights. Let $\mathrm{dist}_G(u \leftrightarrows v) = \mathrm{dist}_G(u \to v) + \mathrm{dist}_G(v \to u)$ denote the roundtrip distance. For every integer $k$, we construct a collection $\mathcal{T}$ of tree-pairs, each consisting of an in-tree and an out-tree rooted at the same vertex. Each vertex belongs to $O(kn^{1/k})$ tree-pairs in expectation, and for every ordered pair $(u, v)$ some tree-pair $T_{in}\cup T_{out} \in \mathcal{T}$ satisfies $\mathrm{dist}_{T_{in}\cup T_{out}}(u \to v) \le \mathrm{dist}_G(u \to v) + (k-1)\mathrm{dist}_G(u \leftrightarrows v).$ We call this a \emph{$k$-strong (roundtrip) tree cover}. While prior techniques partition distances into geometric scales and build separate structures for each scale, our construction handles all distances simultaneously within a single, unified hierarchy. This yields the first directed tree cover for general $k$ matching the classic Thorup-Zwick stretch-size trade-off. Our cover immediately implies improved constructions of roundtrip spanners, path-reporting distance oracles, and compact routing schemes, as well as their source-wise and prioritized variants. We also provide faster algorithms for these structures: for general $k \ge 4$, all the above objects can be constructed in $\widetilde{O}(\min\{mn, n^{2.5}\})$ expected time, breaking the $mn$ barrier in dense directed graphs.

Finding Integer Roots of Polynomials in Nearly Optimal Time

from arXiv: Data Structures and Algorithms

Authors: Yansong Feng, Itamar Nir

Let $f\in\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\widetilde O(nb+\min(n,b)^3b^2)$, which is $\widetilde O(nb)$ whenever $b\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\widetilde O(nb)$ bit operations.

Authors: Yansong Feng, Itamar Nir

Let $f\in\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\widetilde O(nb+\min(n,b)^3b^2)$, which is $\widetilde O(nb)$ whenever $b\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\widetilde O(nb)$ bit operations.

Pure Differentially Private Query Release via Laplace Input Smoothing

from arXiv: Data Structures and Algorithms

Authors: Zongrui Zou

We study the offline release of a public family of $k$ statistical queries on a database of $N$ records from a public universe of size $d$, under pure $\varepsilon$-differential privacy. We give an efficient algorithm that, with probability at least $1-β$, answers all queries simultaneously to error $$O\left(\min\left\{1,\sqrt{\frac{\log(d/β)\log k}{N\varepsilon}}\right\}\right).$$ The output is a synthetic histogram on the data universe, and its running time depends polynomially on $N,d$ and $k$. This resolves a conjecture of Nikolov and Ullman (2021) with an efficient, polynomial-time construction. Fitzsimons (2026) recently established the same square-root error scale via an information-theoretic construction that relies on a computationally intractable Lipschitz envelope over the exponential space of possible outputs. In contrast, we introduce an input-side smoothing approach: we run a selection-only multiplicative-weights process after a single sparse Laplace perturbation of the input histogram. Departing from standard first-order stability analyses, we investigate the stability of stability by establishing a uniform lower bound on the second-order log-derivatives of the output transcript across $T$ rounds of multiplicative weight updates. We then develop an analytic smoothing principle showing that this curvature bound allows a sparse, coordinate-wise perturbation to guarantee pure differential privacy globally, attaining optimal utility through a conceptually simple, polynomial-time pure-DP mechanism.

Authors: Zongrui Zou

We study the offline release of a public family of $k$ statistical queries on a database of $N$ records from a public universe of size $d$, under pure $\varepsilon$-differential privacy. We give an efficient algorithm that, with probability at least $1-β$, answers all queries simultaneously to error $$O\left(\min\left\{1,\sqrt{\frac{\log(d/β)\log k}{N\varepsilon}}\right\}\right).$$ The output is a synthetic histogram on the data universe, and its running time depends polynomially on $N,d$ and $k$. This resolves a conjecture of Nikolov and Ullman (2021) with an efficient, polynomial-time construction. Fitzsimons (2026) recently established the same square-root error scale via an information-theoretic construction that relies on a computationally intractable Lipschitz envelope over the exponential space of possible outputs. In contrast, we introduce an input-side smoothing approach: we run a selection-only multiplicative-weights process after a single sparse Laplace perturbation of the input histogram. Departing from standard first-order stability analyses, we investigate the stability of stability by establishing a uniform lower bound on the second-order log-derivatives of the output transcript across $T$ rounds of multiplicative weight updates. We then develop an analytic smoothing principle showing that this curvature bound allows a sparse, coordinate-wise perturbation to guarantee pure differential privacy globally, attaining optimal utility through a conceptually simple, polynomial-time pure-DP mechanism.

Glauber Dynamics for Sampling Colorings: Rapid Mixing above $(1+1/\mathrm{e})Δ$ via Hidden Hard-Core Models

from arXiv: Data Structures and Algorithms

Authors: Xiongxin Yang

We show that for every $γ>0$ and every sufficiently large $Δ$, the Glauber dynamics for proper $q$-colorings of any $n$-vertex graph of maximum degree $Δ$ has spectral gap $Ω_γ(1/n)$, and hence mixing time $O_γ(n^2\log q)$, whenever $q\ge(1+1/\mathrm{e}+γ)Δ$. Our proof establishes a Bochner-type inequality by exploiting the spectral gaps of hard-core models hidden in the color classes. The constant $1+1/\mathrm{e}\approx1.368$ comes from the uniqueness threshold $\approx\mathrm{e}/Δ$ of the hard-core model on the infinite $Δ$-regular tree.

Authors: Xiongxin Yang

We show that for every $γ>0$ and every sufficiently large $Δ$, the Glauber dynamics for proper $q$-colorings of any $n$-vertex graph of maximum degree $Δ$ has spectral gap $Ω_γ(1/n)$, and hence mixing time $O_γ(n^2\log q)$, whenever $q\ge(1+1/\mathrm{e}+γ)Δ$. Our proof establishes a Bochner-type inequality by exploiting the spectral gaps of hard-core models hidden in the color classes. The constant $1+1/\mathrm{e}\approx1.368$ comes from the uniqueness threshold $\approx\mathrm{e}/Δ$ of the hard-core model on the infinite $Δ$-regular tree.

First Near-Additive All-Pairs Approximate Shortest Paths in Decremental Weighted Graphs

from arXiv: Data Structures and Algorithms

Authors: Eden Koveshi, Liam Roditty, Ariel Sapir

We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.

Authors: Eden Koveshi, Liam Roditty, Ariel Sapir

We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.

Distributionally Robust Search: The Power of Information and the Limits of Moments

from arXiv: Data Structures and Algorithms

Authors: Nicole Megow, Cliff Stein, Jose Verschae

We study binary search for a target drawn from a distribution on $n$ ordered elements. The cost is the expected number of comparison queries. In our distributionally robust framework, the algorithm knows only an uncertainty set containing the distribution and minimizes its worst-case approximation ratio against an optimal strategy that knows the distribution. Collections of uncertainty sets capture different forms of information: quantiles, moments, general linear measurements, or approximate CDFs. This framework quantifies how much and what form of information is needed for near-optimal search. Our main result separates moments from general linear measurements. Although $n-1$ moments determine the distribution exactly, for every constant $δ\in(0,1)$, the first $n^{1-δ}$ exact moments can be as uninformative for search as no information: the worst-case approximation ratio remains $Ω(\log n)$. In contrast, for every constant $\varepsilon\in(0,1)$, $O(n^\varepsilon\log^4 n)$ suitably chosen linear measurements suffice for an $O(1/\varepsilon)$-approximation. The limitation lies in the measurements chosen, rather than linearity itself. Using LP duality and polynomial bounds of Rakhmanov and Bernstein, we construct distributions with identical moments and mass concentrated at different positions. For the upper bound, we combine dyadic sketches and Johnson--Lindenstrauss embeddings with a truncated median-search algorithm that converts uniform CDF approximation into search guarantees. Measuring information in bits, we prove a lower bound of $Ω(1/\varepsilon)$ on the approximation ratio for summaries of any form using at most $\frac12 n^\varepsilon\log n$ bits, for every sufficiently small constant $\varepsilon>0$. Quantiles match this within constant factors. For Kolmogorov balls of radius $n^{-\varepsilon}$ the optimal approximation ratio is $Θ(1/\varepsilon)$.

Authors: Nicole Megow, Cliff Stein, Jose Verschae

We study binary search for a target drawn from a distribution on $n$ ordered elements. The cost is the expected number of comparison queries. In our distributionally robust framework, the algorithm knows only an uncertainty set containing the distribution and minimizes its worst-case approximation ratio against an optimal strategy that knows the distribution. Collections of uncertainty sets capture different forms of information: quantiles, moments, general linear measurements, or approximate CDFs. This framework quantifies how much and what form of information is needed for near-optimal search. Our main result separates moments from general linear measurements. Although $n-1$ moments determine the distribution exactly, for every constant $δ\in(0,1)$, the first $n^{1-δ}$ exact moments can be as uninformative for search as no information: the worst-case approximation ratio remains $Ω(\log n)$. In contrast, for every constant $\varepsilon\in(0,1)$, $O(n^\varepsilon\log^4 n)$ suitably chosen linear measurements suffice for an $O(1/\varepsilon)$-approximation. The limitation lies in the measurements chosen, rather than linearity itself. Using LP duality and polynomial bounds of Rakhmanov and Bernstein, we construct distributions with identical moments and mass concentrated at different positions. For the upper bound, we combine dyadic sketches and Johnson--Lindenstrauss embeddings with a truncated median-search algorithm that converts uniform CDF approximation into search guarantees. Measuring information in bits, we prove a lower bound of $Ω(1/\varepsilon)$ on the approximation ratio for summaries of any form using at most $\frac12 n^\varepsilon\log n$ bits, for every sufficiently small constant $\varepsilon>0$. Quantiles match this within constant factors. For Kolmogorov balls of radius $n^{-\varepsilon}$ the optimal approximation ratio is $Θ(1/\varepsilon)$.

Dynamic Steiner Tree under Edge Updates

from arXiv: Data Structures and Algorithms

Authors: Jeff Giliberti, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Xinchen Yang

The Steiner tree problem is the canonical generalization of minimum spanning tree (MST). When connectivity is required only among a fixed subset of vertices (terminals), the goal is to maintain a minimum-cost tree connecting these terminals as the graph undergoes edge updates. While dynamic MST has been studied since the 1980s~[Frederickson, STOC'83, SICOMP'85], and the first sublinear-in-$n$ dynamic MST algorithm dates back almost thirty-five years to Eppstein, Galil, Italiano, and Nissenzweig~[FOCS'92, JACM'97], achieving sublinear-in-$n$ update time for dynamic Steiner tree has long remained open, as a single edge update can invalidate many routes in the underlying terminal metric closure. We present the first fully dynamic algorithm for Steiner tree that achieves a constant-factor approximation with sublinear-in-$n$ update time. For any fixed $0<ζ<1/2$, our deterministic algorithm maintains an $O_ζ(1)$-approximate Steiner tree for undirected graphs with polynomially bounded edge weights in $\widetilde O(n^{1/2+ζ})$ amortized update time, answering cost queries in $O(1)$ time, and reporting an explicit tree in $\widetilde O(n^{1+ζ})$ time. Our approach combines periodic terminal-metric rebuilds on a dynamic spanner with a top-tree dynamic program that localizes edge deletions and reconnects surviving components through a dynamic distance sparsifier. Along the way, we give a simpler randomized insertion-only algorithm with a better, explicit approximation ratio.

Authors: Jeff Giliberti, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Xinchen Yang

The Steiner tree problem is the canonical generalization of minimum spanning tree (MST). When connectivity is required only among a fixed subset of vertices (terminals), the goal is to maintain a minimum-cost tree connecting these terminals as the graph undergoes edge updates. While dynamic MST has been studied since the 1980s~[Frederickson, STOC'83, SICOMP'85], and the first sublinear-in-$n$ dynamic MST algorithm dates back almost thirty-five years to Eppstein, Galil, Italiano, and Nissenzweig~[FOCS'92, JACM'97], achieving sublinear-in-$n$ update time for dynamic Steiner tree has long remained open, as a single edge update can invalidate many routes in the underlying terminal metric closure. We present the first fully dynamic algorithm for Steiner tree that achieves a constant-factor approximation with sublinear-in-$n$ update time. For any fixed $0<ζ<1/2$, our deterministic algorithm maintains an $O_ζ(1)$-approximate Steiner tree for undirected graphs with polynomially bounded edge weights in $\widetilde O(n^{1/2+ζ})$ amortized update time, answering cost queries in $O(1)$ time, and reporting an explicit tree in $\widetilde O(n^{1+ζ})$ time. Our approach combines periodic terminal-metric rebuilds on a dynamic spanner with a top-tree dynamic program that localizes edge deletions and reconnects surviving components through a dynamic distance sparsifier. Along the way, we give a simpler randomized insertion-only algorithm with a better, explicit approximation ratio.

XOR-Functions and Beyond in the Random Query Model

from arXiv: Data Structures and Algorithms

Authors: Sumegha Garg, Janani Sundaresan, Santhoshini Velusamy

We study time--space tradeoffs for computing Boolean functions $f:\{0,1\}^n \to \{0,1\}$ in the random query model. On input $x$, a branching program receives independent samples $(i,x(i))$, where $i$ is uniform in $[n]$. We want to evaluate $f(x)$ while minimizing the width $2^S$ and the length $T$ of the branching program. There is a simple upper bound of $T = \tilde{O}(n)$, $S = O(n)$: coupon-collector shows that $\tilde{O}(n)$ samples suffice, and they can be stored in $O(n)$ space. Raz and Zhan (ITCS 2020) introduced this model and studied the zero-error setting. In the bounded-error setting, Dinur (SODA 2024) proved $TS = \tildeΩ(n \cdot inf[f])$, where $inf[f]$ is the total influence of $f$. For many functions with low influence such as Equality and Tribes, this lower bound can leave a substantial gap from the best known upper bounds. Our main result is a stronger lower bound of $T(S+\log T) = Ω(n \cdot sens[g])$ for any XOR-function $g(x,y)= f(x \oplus y)$, where $sens[g]$ is the sensitivity of $g$. Through simple corollaries, we get tight nearly-quadratic time--space lower bounds for Tribes and Equality, despite their low influence. Our proof introduces a new notion of information complexity tailored to the random query model along with a corresponding direct-sum argument which may be of independent interest. Dinur also proved a tight bound of $TS = Ω(n^2)$ for Majority, whose influence is only $Θ(\sqrt{n})$, and asked whether an alternative characterization of Boolean functions could give a unified proof of tight results for a broader class of functions, including Majority. We answer this open question by proving $TS = \tildeΩ(n \cdot dgn[f])$, where $dgn[f]$ is the degeneracy of the sensitivity graph of $f$ (for Majority, this is $Θ(n)$). We also prove that this bound is tight for all symmetric functions up to logarithmic factors.

Authors: Sumegha Garg, Janani Sundaresan, Santhoshini Velusamy

We study time--space tradeoffs for computing Boolean functions $f:\{0,1\}^n \to \{0,1\}$ in the random query model. On input $x$, a branching program receives independent samples $(i,x(i))$, where $i$ is uniform in $[n]$. We want to evaluate $f(x)$ while minimizing the width $2^S$ and the length $T$ of the branching program. There is a simple upper bound of $T = \tilde{O}(n)$, $S = O(n)$: coupon-collector shows that $\tilde{O}(n)$ samples suffice, and they can be stored in $O(n)$ space. Raz and Zhan (ITCS 2020) introduced this model and studied the zero-error setting. In the bounded-error setting, Dinur (SODA 2024) proved $TS = \tildeΩ(n \cdot inf[f])$, where $inf[f]$ is the total influence of $f$. For many functions with low influence such as Equality and Tribes, this lower bound can leave a substantial gap from the best known upper bounds. Our main result is a stronger lower bound of $T(S+\log T) = Ω(n \cdot sens[g])$ for any XOR-function $g(x,y)= f(x \oplus y)$, where $sens[g]$ is the sensitivity of $g$. Through simple corollaries, we get tight nearly-quadratic time--space lower bounds for Tribes and Equality, despite their low influence. Our proof introduces a new notion of information complexity tailored to the random query model along with a corresponding direct-sum argument which may be of independent interest. Dinur also proved a tight bound of $TS = Ω(n^2)$ for Majority, whose influence is only $Θ(\sqrt{n})$, and asked whether an alternative characterization of Boolean functions could give a unified proof of tight results for a broader class of functions, including Majority. We answer this open question by proving $TS = \tildeΩ(n \cdot dgn[f])$, where $dgn[f]$ is the degeneracy of the sensitivity graph of $f$ (for Majority, this is $Θ(n)$). We also prove that this bound is tight for all symmetric functions up to logarithmic factors.

Gaussian Difference Dynamics: Breaking the $2^{n/2}$ Barrier for the Shortest Vector Problem and Discrete Gaussian Sampling

from arXiv: Data Structures and Algorithms

Authors: Yiming Gao, Xuyuan Han, Honggang Hu

We give classical randomized algorithms in a word RAM model that break the $2^{n/2}$ barrier for both discrete Gaussian sampling (DGS) and the Shortest Vector Problem (SVP). For every fixed $α>1$, shifted DGS on arbitrary lattices at width $σ\geαη_{1/2}(L)$ takes $$ (3/2)^{n/2+o(n)}\text{ time} \qquad\text{and}\qquad (4/3)^{n/2+o(n)}\text{ space}. $$ These bounds match the Becker--Ducas--Gama--Laarhoven (BDGL) heuristic sieving exponents. At widths at least $3η_{1/2}(L)/2$, the same total cost suffices for an entire batch of $\lfloor(4/3)^{n/2}\rfloor$ samples jointly close to independent target Gaussians. For SVP on arbitrary lattices, our algorithm returns a shortest nonzero vector with high probability in $$ 2^{0.4130n+o(n)}\text{ time} \qquad\text{and}\qquad 2^{0.3138n+o(n)}\text{ space}. $$ As a corollary, we obtain centered DGS at every width with the same exponents, answering a question of Stephens-Davidowitz (SODA 2016). We introduce Gaussian difference dynamics (GDD), a framework for tracking expected counts and pairwise clustering of vectors through repeated differencing. A closed recursion for the first two counting moments controls these quantities relative to a Gaussian reference, without requiring independence within intermediate lists. Accurate moment bounds let us extract a Gaussian sample by weighted selection, while the weaker bounds needed to establish a shortest vector representation allow us to retain more differences. We perform these transitions on thin spherical shells, where the Gaussian rejection weights can be renormalized at subexponential cost. For batch sampling, we combine this geometry with the double rejection technique of Doulgerakis and Laarhoven to control the joint distribution of outputs while reusing vectors.

Authors: Yiming Gao, Xuyuan Han, Honggang Hu

We give classical randomized algorithms in a word RAM model that break the $2^{n/2}$ barrier for both discrete Gaussian sampling (DGS) and the Shortest Vector Problem (SVP). For every fixed $α>1$, shifted DGS on arbitrary lattices at width $σ\geαη_{1/2}(L)$ takes $$ (3/2)^{n/2+o(n)}\text{ time} \qquad\text{and}\qquad (4/3)^{n/2+o(n)}\text{ space}. $$ These bounds match the Becker--Ducas--Gama--Laarhoven (BDGL) heuristic sieving exponents. At widths at least $3η_{1/2}(L)/2$, the same total cost suffices for an entire batch of $\lfloor(4/3)^{n/2}\rfloor$ samples jointly close to independent target Gaussians. For SVP on arbitrary lattices, our algorithm returns a shortest nonzero vector with high probability in $$ 2^{0.4130n+o(n)}\text{ time} \qquad\text{and}\qquad 2^{0.3138n+o(n)}\text{ space}. $$ As a corollary, we obtain centered DGS at every width with the same exponents, answering a question of Stephens-Davidowitz (SODA 2016). We introduce Gaussian difference dynamics (GDD), a framework for tracking expected counts and pairwise clustering of vectors through repeated differencing. A closed recursion for the first two counting moments controls these quantities relative to a Gaussian reference, without requiring independence within intermediate lists. Accurate moment bounds let us extract a Gaussian sample by weighted selection, while the weaker bounds needed to establish a shortest vector representation allow us to retain more differences. We perform these transitions on thin spherical shells, where the Gaussian rejection weights can be renormalized at subexponential cost. For batch sampling, we combine this geometry with the double rejection technique of Doulgerakis and Laarhoven to control the joint distribution of outputs while reusing vectors.

Metric TSP in Sublinear Time: Breaking Two Approximation

from arXiv: Data Structures and Algorithms

Authors: Amir Azarmehr, Soheil Behnezhad

The seminal result of Czumaj and Sohler [STOC'04] showed that a 2-approximation of the value of metric TSP can be obtained with sublinear queries. Whether sublinear algorithms can break the 2-approximation barrier has remained a major open problem. See problem 71 on sublinear.info. This paper resolves this open problem. We show there are fixed $\varepsilon, δ> 0$ such that a $(2-δ)$-approximation of metric TSP value can be obtained with $O(n^{2-\varepsilon})$ queries. Here $n$ is the number of vertices, and the input size is $Θ(n^2)$.

Authors: Amir Azarmehr, Soheil Behnezhad

The seminal result of Czumaj and Sohler [STOC'04] showed that a 2-approximation of the value of metric TSP can be obtained with sublinear queries. Whether sublinear algorithms can break the 2-approximation barrier has remained a major open problem. See problem 71 on sublinear.info. This paper resolves this open problem. We show there are fixed $\varepsilon, δ> 0$ such that a $(2-δ)$-approximation of metric TSP value can be obtained with $O(n^{2-\varepsilon})$ queries. Here $n$ is the number of vertices, and the input size is $Θ(n^2)$.

Testing Implies Distance Estimation with Polynomial Loss

from arXiv: Data Structures and Algorithms

Authors: Gaia Carenini, Cameron Seth

A tester decides whether a graph has a property or is far from having it. A distance estimator asks for more: it must approximately determine how many edges need to be changed. A property is said to be testable if there is a tester with a number of queries bounded independently of the graph's order for each conceivable distance. Similarly, if a property admits an estimator with a number of queries bounded independently of the graph's order, it is said to be estimable. Fischer and Newman proved that, in the dense graph model, every testable property is also estimable. We show that the passage from testing to estimation requires only a polynomial increase in query complexity. This resolves a long-standing open problem in property testing, previously tackled by Fiat and Ron, by Hoppen, Kohayakawa, Lang, Lefmann, and Stagni, and by Gishboliner, Kushnir, and Shapira.

Authors: Gaia Carenini, Cameron Seth

A tester decides whether a graph has a property or is far from having it. A distance estimator asks for more: it must approximately determine how many edges need to be changed. A property is said to be testable if there is a tester with a number of queries bounded independently of the graph's order for each conceivable distance. Similarly, if a property admits an estimator with a number of queries bounded independently of the graph's order, it is said to be estimable. Fischer and Newman proved that, in the dense graph model, every testable property is also estimable. We show that the passage from testing to estimation requires only a polynomial increase in query complexity. This resolves a long-standing open problem in property testing, previously tackled by Fiat and Ron, by Hoppen, Kohayakawa, Lang, Lefmann, and Stagni, and by Gishboliner, Kushnir, and Shapira.

Online Metric Matching with Unknown i.i.d. Arrivals

from arXiv: Data Structures and Algorithms

Authors: Tsubasa Harada

We study online metric matching with $n$ servers and $n$ requests drawn independently from an unknown distribution. Each request must be matched immediately and irrevocably to an unused server, and the algorithm receives no samples before the process starts. We give a randomized algorithm that is $O((\log\log\log n)^2)$-competitive on arbitrary metric spaces and $O(1)$-competitive on tree metrics. These guarantees match the corresponding bounds for known i.i.d. request distributions in [GGPW19] up to constant factors. More generally, our competitive ratio is $O(α(n))$ whenever the Fair-Bias algorithm introduced in that work has competitive ratio $α(n)$ for known i.i.d. arrivals on the same class of metrics. Consequently, proving that Fair-Bias is constant-competitive on general metrics would also yield a constant-competitive algorithm for unknown i.i.d. arrivals.

Authors: Tsubasa Harada

We study online metric matching with $n$ servers and $n$ requests drawn independently from an unknown distribution. Each request must be matched immediately and irrevocably to an unused server, and the algorithm receives no samples before the process starts. We give a randomized algorithm that is $O((\log\log\log n)^2)$-competitive on arbitrary metric spaces and $O(1)$-competitive on tree metrics. These guarantees match the corresponding bounds for known i.i.d. request distributions in [GGPW19] up to constant factors. More generally, our competitive ratio is $O(α(n))$ whenever the Fair-Bias algorithm introduced in that work has competitive ratio $α(n)$ for known i.i.d. arrivals on the same class of metrics. Consequently, proving that Fair-Bias is constant-competitive on general metrics would also yield a constant-competitive algorithm for unknown i.i.d. arrivals.

Incremental Open Addressing Has No Time-Space Tradeoff

from arXiv: Data Structures and Algorithms

Authors: Martín Farach-Colton, Andrew Krapivin, William Kuszmaul

In this paper, we show that there is no time-space tradeoff in incremental open addressing: there is an incremental open-addressed hash table that supports $O(1)$ worst-case expected time insertions, positive queries, and negative queries, at any load factor (even load factor $1$). As an intermediate result, we also revisit the problem of constructing an incremental open-addressed hash table that does not perform reordering (does not move elements around over time). In this setting, we show that it is possible to get $O(1)$ amortized expected positive query time together with $O(1)$ worst-case expected insertion time, a combination that was not previously known to be possible.

Authors: Martín Farach-Colton, Andrew Krapivin, William Kuszmaul

In this paper, we show that there is no time-space tradeoff in incremental open addressing: there is an incremental open-addressed hash table that supports $O(1)$ worst-case expected time insertions, positive queries, and negative queries, at any load factor (even load factor $1$). As an intermediate result, we also revisit the problem of constructing an incremental open-addressed hash table that does not perform reordering (does not move elements around over time). In this setting, we show that it is possible to get $O(1)$ amortized expected positive query time together with $O(1)$ worst-case expected insertion time, a combination that was not previously known to be possible.

A Constant-Factor Approximation for Capacitated Facility Location with Outliers

from arXiv: Data Structures and Algorithms

Authors: Rajni Dabas

We give a deterministic polynomial-time $(60/11+\eps)$-approximation for metric capacitated facility location with outliers, with arbitrary opening costs and nonuniform hard capacities. The algorithm respects both capacities and the outlier budget. Its guarantee, approximately $5.455$, is close to the best known factor $5+\eps$ for capacitated facility location without outliers. The main ingredient is a local-search comparison for a common rejection penalty that retains the signed difference between the algorithm's and the comparator's rejection counts. Ordinary approximation guarantees for the penalty objective do not provide this comparison. We extend the exchange-forest accounting of Bansal, Garg, and Gupta \cite{BGG18} so that a nonnegative assignment remainder pays for rejection and cancels in the final bound. We then merge two penalty assignments whose served counts bracket the outlier quota. Grouping alternating paths by their terminal facility preserves capacities and shares opening costs. These ingredients give a $(20/3+\eps)$-approximation; starting each penalty run from the preceding solution improves the guarantee to $60/11+\eps$.

Authors: Rajni Dabas

We give a deterministic polynomial-time $(60/11+\eps)$-approximation for metric capacitated facility location with outliers, with arbitrary opening costs and nonuniform hard capacities. The algorithm respects both capacities and the outlier budget. Its guarantee, approximately $5.455$, is close to the best known factor $5+\eps$ for capacitated facility location without outliers. The main ingredient is a local-search comparison for a common rejection penalty that retains the signed difference between the algorithm's and the comparator's rejection counts. Ordinary approximation guarantees for the penalty objective do not provide this comparison. We extend the exchange-forest accounting of Bansal, Garg, and Gupta \cite{BGG18} so that a nonnegative assignment remainder pays for rejection and cancels in the final bound. We then merge two penalty assignments whose served counts bracket the outlier quota. Grouping alternating paths by their terminal facility preserves capacities and shares opening costs. These ingredients give a $(20/3+\eps)$-approximation; starting each penalty run from the preceding solution improves the guarantee to $60/11+\eps$.

The Hyperclique Hypothesis is False

from arXiv: Data Structures and Algorithms

Authors: Nick Fischer

The Hyperclique Hypothesis postulates that, for all constants $k > h \geq 3$, detecting a $k$-clique in an $h$-uniform hypergraph with $n$ vertices requires time $n^{k-o(1)}$. This hypothesis is false: there is an algebraic algorithm that detects (and even counts) $k$-cliques in $h$-uniform hypergraphs in time $O(n^{(1-ε_h)k})$. Via known reductions, this also refutes the recently proposed $k$-AP Hypothesis. As a consequence, many problems with conditional lower bounds based on these two hypotheses may admit faster algorithms after all. This result is almost entirely due to AI.

Authors: Nick Fischer

The Hyperclique Hypothesis postulates that, for all constants $k > h \geq 3$, detecting a $k$-clique in an $h$-uniform hypergraph with $n$ vertices requires time $n^{k-o(1)}$. This hypothesis is false: there is an algebraic algorithm that detects (and even counts) $k$-cliques in $h$-uniform hypergraphs in time $O(n^{(1-ε_h)k})$. Via known reductions, this also refutes the recently proposed $k$-AP Hypothesis. As a consequence, many problems with conditional lower bounds based on these two hypotheses may admit faster algorithms after all. This result is almost entirely due to AI.

A Family of Well-Behaved Multiflow Instances

from arXiv: Data Structures and Algorithms

Authors: Joseph Poremba, F. Bruce Shepherd

We introduce a family of cut-sufficient instances for undirected multiflow that generalizes single-source planar and fully-planar instances. We also show the family has an additive $6 d_{\max}$ error for converting a fractional flow into an unsplittable one.

Authors: Joseph Poremba, F. Bruce Shepherd

We introduce a family of cut-sufficient instances for undirected multiflow that generalizes single-source planar and fully-planar instances. We also show the family has an additive $6 d_{\max}$ error for converting a fractional flow into an unsplittable one.

A Fully Polynomial Approximation Algorithm for the Sherrington-Kirkpatrick Model

from arXiv: Data Structures and Algorithms

Authors: Dylan Herman, Shouvanik Chakrabarti, Junhyung Lyle Kim, Guneykan Ozgul

Montanari's incremental approximate message passing (IAMP) algorithm [FOCS 2019; SIAM J. Comput., 2025] and the zero-temperature formulation of El Alaoui, Montanari and Sellke (AMS) [Ann. Probab., 2021] find a $(1-\varepsilon)$ approximation to the maximum Sherrington-Kirkpatrick (SK) energy in $C(\varepsilon)N^2$ operations for fixed accuracy, with probability tending to one as $N\to\infty$, under their respective full-support and access-model assumptions. However, it was unknown whether this approximation could be achieved with work polynomial in both $N$ and $1/\varepsilon$, together with quantitative guarantees at finite dimension. Under the same assumptions as AMS, we introduce grouped IAMP with exact Gaussian query outputs (GEG-IAMP), which is the first SK approximation algorithm with total work polynomial in $N$ and $1/\varepsilon$ and a finite-size guarantee above a dimension threshold polynomial in $1/\varepsilon$ and $\log(1/α)$. More precisely, for every $α\in(0,1/2)$, it returns a $(1-\varepsilon)$ approximation with probability at least $1-α$ whenever $N\ge \mathsf{K}\varepsilon^{-2}\log^2(e/\varepsilon)\log(\mathsf{K}/α)$, using $O(N^2/\varepsilon)$ work and $O(N^2)$ storage. GEG-IAMP replaces the Onsager correction by Reeves's coupling construction [arXiv:2508.10782] and groups Euler-Maruyama steps into $O(1/\varepsilon)$ matrix-vector products. We also prove a relative guarantee for AMS IAMP with $O(N^2\varepsilon^{-3}\log^2(e/\varepsilon))$ work but a sufficient dimension exponential in inverse accuracy. By returning a fixed configuration below its dimension threshold, GEG-IAMP also provides an asymptotic fully polynomial randomized approximation scheme (AFPRAS), with an additive error independent of dimension.

Authors: Dylan Herman, Shouvanik Chakrabarti, Junhyung Lyle Kim, Guneykan Ozgul

Montanari's incremental approximate message passing (IAMP) algorithm [FOCS 2019; SIAM J. Comput., 2025] and the zero-temperature formulation of El Alaoui, Montanari and Sellke (AMS) [Ann. Probab., 2021] find a $(1-\varepsilon)$ approximation to the maximum Sherrington-Kirkpatrick (SK) energy in $C(\varepsilon)N^2$ operations for fixed accuracy, with probability tending to one as $N\to\infty$, under their respective full-support and access-model assumptions. However, it was unknown whether this approximation could be achieved with work polynomial in both $N$ and $1/\varepsilon$, together with quantitative guarantees at finite dimension. Under the same assumptions as AMS, we introduce grouped IAMP with exact Gaussian query outputs (GEG-IAMP), which is the first SK approximation algorithm with total work polynomial in $N$ and $1/\varepsilon$ and a finite-size guarantee above a dimension threshold polynomial in $1/\varepsilon$ and $\log(1/α)$. More precisely, for every $α\in(0,1/2)$, it returns a $(1-\varepsilon)$ approximation with probability at least $1-α$ whenever $N\ge \mathsf{K}\varepsilon^{-2}\log^2(e/\varepsilon)\log(\mathsf{K}/α)$, using $O(N^2/\varepsilon)$ work and $O(N^2)$ storage. GEG-IAMP replaces the Onsager correction by Reeves's coupling construction [arXiv:2508.10782] and groups Euler-Maruyama steps into $O(1/\varepsilon)$ matrix-vector products. We also prove a relative guarantee for AMS IAMP with $O(N^2\varepsilon^{-3}\log^2(e/\varepsilon))$ work but a sufficient dimension exponential in inverse accuracy. By returning a fixed configuration below its dimension threshold, GEG-IAMP also provides an asymptotic fully polynomial randomized approximation scheme (AFPRAS), with an additive error independent of dimension.

Tuesday, November 03

TR26-296 | Exponential Quantum Advantage in Testing Fourier Dimensionality | Kenny Chen

from ECCC Papers
A boolean function $f$ has Fourier dimension $k$ if its strings with nonzero Fourier coefficients span a subspace of dimension $k$. We first consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is an $O(k/\sqrt{\epsilon})$-query quantum property tester for this problem, which we show to be almost optimal. Combined with Gopalan \etal's classical lower bound of $\Omega(2^{k/2})$, this demonstrates an exponential quantum advantage for this task. We complement this result with a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester, giving a quadratic improvement over the previous best tester, and essentially settling the classical query complexity. We then introduce and study the tolerant version of this problem. That is, determining whether a function is $\alpha\epsilon$-close to having Fourier dimension at most $k$, or is $\epsilon$-far from being so. Perhaps surprisingly, we show that this tolerant version of the problem is essentially just as hard as the standard version of the problem. That is, we show this problem has a $\tilde{O}(k/\epsilon)$ quantum tester, and a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester.
A boolean function $f$ has Fourier dimension $k$ if its strings with nonzero Fourier coefficients span a subspace of dimension $k$. We first consider the property testing task of determining whether a function has Fourier dimension at most $k$, or is $\epsilon$-far from being so. We show that there is an $O(k/\sqrt{\epsilon})$-query quantum property tester for this problem, which we show to be almost optimal. Combined with Gopalan \etal's classical lower bound of $\Omega(2^{k/2})$, this demonstrates an exponential quantum advantage for this task. We complement this result with a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester, giving a quadratic improvement over the previous best tester, and essentially settling the classical query complexity. We then introduce and study the tolerant version of this problem. That is, determining whether a function is $\alpha\epsilon$-close to having Fourier dimension at most $k$, or is $\epsilon$-far from being so. Perhaps surprisingly, we show that this tolerant version of the problem is essentially just as hard as the standard version of the problem. That is, we show this problem has a $\tilde{O}(k/\epsilon)$ quantum tester, and a $\tilde{O}(2^{k/2}/\epsilon)$ classical tester.

TR26-295 | Fast Algorithms for Low Rank Tensor Decomposition | Shubhangi Saraf, Devansh Shringi

from ECCC Papers
We give fast algorithms for low-rank tensor decomposition. Over $\mathbb R$ and $\mathbb C$ we give a deterministic algorithm that takes as input an order-$d$, width-$n$, rank-$k$ tensor, gets measurement access to the tensor, runs in time $k^{O(k^2)}\cdot \mathrm{poly}(n,d)$, and finds a minimal decomposition as a sum of rank-one tensors. This algorithm also immediately gives an algorithm for tensor rank computation with similar time complexity. This improves upon the previous best runtime of $k^{O(k^{16})}\cdot \mathrm{poly}( n,d)$ by the work of Bhargava and Shringi. In particular, we get the first subexponential time algorithms for tensor decomposition of order-$d$ tensors of rank as large as $\sqrt n/\log n$. One interesting consequence is that the run time of our algorithm for decomposing order-$d$ tensors approaches the fastest algorithms we know for decomposing order-3 tensors ($2^{O(k^2)}\cdot \mathrm{poly}(n)$). Moreover, known NP-hardness results for tensor rank computation suggest that an exponential dependence on $k$ is necessary. Our approach is via the equivalent problem of proper learning for the class of set multilinear depth-3 circuits of top fan-in $k$. The main technical tools involve iteratively applying appropriate pseudorandom projections combined with solving multivariate polynomial systems.
We give fast algorithms for low-rank tensor decomposition. Over $\mathbb R$ and $\mathbb C$ we give a deterministic algorithm that takes as input an order-$d$, width-$n$, rank-$k$ tensor, gets measurement access to the tensor, runs in time $k^{O(k^2)}\cdot \mathrm{poly}(n,d)$, and finds a minimal decomposition as a sum of rank-one tensors. This algorithm also immediately gives an algorithm for tensor rank computation with similar time complexity. This improves upon the previous best runtime of $k^{O(k^{16})}\cdot \mathrm{poly}( n,d)$ by the work of Bhargava and Shringi. In particular, we get the first subexponential time algorithms for tensor decomposition of order-$d$ tensors of rank as large as $\sqrt n/\log n$. One interesting consequence is that the run time of our algorithm for decomposing order-$d$ tensors approaches the fastest algorithms we know for decomposing order-3 tensors ($2^{O(k^2)}\cdot \mathrm{poly}(n)$). Moreover, known NP-hardness results for tensor rank computation suggest that an exponential dependence on $k$ is necessary. Our approach is via the equivalent problem of proper learning for the class of set multilinear depth-3 circuits of top fan-in $k$. The main technical tools involve iteratively applying appropriate pseudorandom projections combined with solving multivariate polynomial systems.

Limits to Simulated Growth

from Ben Recht
Jay Forrester, Systems Dynamics, and the allure of the simulated society.

Hi there, argmin readers! Today’s post is a live blog of Class 17 of my graduate seminar “Forecasting: A Critical Retrospective.” In this class, we’re going to discuss simulation as a means of prediction. The syllabus and list of past posts are here.

In a surprisingly dark survey paper from 1971, “Counterintuitive Behavior of Social Systems,” electrical engineer turned social scientist Jay Forrester made a case for informing public policy with computer simulations. Forrester argued that policymakers made all decisions with models, but his computer models were more explicit and concrete than people’s mental models. Computer models laid all of the assumptions on the table and let stakeholders adjust the parameters until they conformed to reality. They also let people create a variety of prefactual worlds, demonstrating plausibility and illustrating the consequences of various decision-making paths. Most importantly for Forrester, the computer models captured the surprising and unintuitive nature of feedback effects, showing pathways that mental models might not have predicted.

Forrester’s survey gives three examples, each with more ambitious aims than the last. He describes his time as a management consultant and how computer simulations revealed that decisions meant to fix things led to worse outcomes. Good intentions could lead to a “downward spiral,” and this would often convince managers to double down on their initial wrong decision. Forrester argues that a simulation would have predicted these troubles and informed management of a better path.

He then turns these ideas toward urban planning. He claims that his simulations can explain and fix the “depressed nature of central cities.” Indeed, his simulations argue that four well-intentioned liberal programs are causing urban blight, not fixing it.

“One program was creation of jobs by busing the unemployed to suburban jobs or through governmental jobs as employer of last resort. Second was a training program to increase skills of the lowest-income group. Third was financial aid to depressed cities from federal subsidies. Fourth was construction of low-cost housing.”

Forrester goes on to explain the vicious cycle here, where a surplus of cheap housing draws in more new residents than the job market can support. This leads to overcrowding and a less attractive urban environment. This, in turn, repels new affluent residents. Forrester says his simulations “both confirm and explain much of what has been happening over the last several decades in cities.”

A dispassionate observer might say that his simulation matched his politics. But Forrester saw modeling power, not confirmation bias. He went on to extend his models to a global scale.

Forrester goes on to describe work he had done that predicted the end of the world by 2071. Current trends could not support population growth, and pollution would run amok, and we’d run out of natural resources. The only solution was to impose policies that swiftly dampened population growth and austerely conserved resources. Applying Forrester’s simulator, a team led by Dennis Meadows assembled the dire predictions into the 1972 book The Limits to Growth, which sold 12 million copies worldwide.

Forrester’s report is filled with feedback diagrams and simulation traces. The ability to predict so many different possibilities under different scenarios is intoxicating. Historian Kevin Baker argues that this sort of interactive experience was core to the allure of the simulation model. By adding knobs and switches, people could play out different worlds, making the simulation feel more trustworthy and real. This would eventually lead to enthralling games with bizarre hidden politics, notably the world bestseller SimCity. As Baker puts it, “The capacity of simulation models to captivate and enthrall is taken for granted as a natural property of simulation. But these abilities were often deliberately engineered into software for historically-specific purposes.” Forrester loved his simulators because they quantified and externalized mental models. He underemphasized how they created new mental models, letting policymakers play video games in imaginary worlds that let them feel as if they were creating the future.

One of the funnier things about statistical prediction is that it forecasts a static world. Your best guess is to mimic past rates. By contrast, simulation can imagine different worlds, mechanistically mapping trajectories into the future based on our assumptions about the present.

However, simulations are not reality, and more sophistication is not good in and of itself. In fact, if small changes lead to unexpected consequences, a single poor modeling choice can too. The doom of The Limits to Growth did not play out. Nor did Forrester’s libertarian predictions about urban blight.

In the last class session, we grappled with two competing sales pitches for “silicon sampling.” Most people felt comfortable using silicon samples as part of some prediction loop. For example, you could create virtual training data for some recommendation system by asking LLMs to simulate answers to questions based on demographics. You might have it label examples while “pretending to be a plumber” and then “pretending to be a schoolteacher,” using those labels as synthetic training data. You can then validate the resulting system against standard KPIs with A/B tests or benchmarks. If the prediction system passes evaluation metrics, it doesn’t matter if it accurately models people.

On the other hand, it’s easy to believe that because LLMs perform well on these prediction tasks, they create realistic simulations of people. You can interactively inspect the answers LLMs give and gain trust in their fidelity. And then you can convince yourself that you can use these simulations to understand human behavior without talking to people. This is the same seductive simulation loop that Baker calls out. We’re more skeptical of simulation now than in 1970, likely because we’re so familiar with video games and how they aren’t real. But the prospect of simulating society is still enticing, and new well-funded research programs and startups are playing the same old tricks of immersive gameplay.

Subscribe now

By Ben Recht

TR26-294 | Some algebraic and geometric computations in CH via Chow forms | Robert Andrews, Abhibhav Garg, Rafael Mendes de Oliveira, Éric Schost

from ECCC Papers
The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding 1. whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, 2. whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, 3. membership in a given equidimensional component of a given algebraic variety, and 4. membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.
The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding 1. whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, 2. whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, 3. membership in a given equidimensional component of a given algebraic variety, and 4. membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.

TR26-293 | Lower Bounds for Adaptive Relaxed Locally Decodable Codes via the Capped Horvitz-Thompson Estimator | Guy Goldberg, Tom Gur, Sidhant Saraogi

from ECCC Papers
We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+\Omega(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.
We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+\Omega(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.

TR26-292 | Testing Implies Distance Estimation with Polynomial Loss | Cameron Seth, Gaia Carenini

from ECCC Papers
A tester decides whether a graph has a property or is far from having it. A distance estimator asks for more: it must approximately determine how many edges need to be changed. A property is said to be testable if there is a tester with a number of queries bounded independently of the graph's order for each conceivable distance. Similarly, if a property admits an estimator with a number of queries bounded independently of the graph's order, it is said to be estimable. Fischer and Newman proved that, in the dense graph model, every testable property is also estimable. We show that the passage from testing to estimation requires only a polynomial increase in query complexity. This resolves a long-standing open problem in property testing, previously tackled by Fiat and Ron, by Hoppen, Kohayakawa, Lang, Lefmann, and Stagni, and by Gishboliner, Kushnir, and Shapira.
A tester decides whether a graph has a property or is far from having it. A distance estimator asks for more: it must approximately determine how many edges need to be changed. A property is said to be testable if there is a tester with a number of queries bounded independently of the graph's order for each conceivable distance. Similarly, if a property admits an estimator with a number of queries bounded independently of the graph's order, it is said to be estimable. Fischer and Newman proved that, in the dense graph model, every testable property is also estimable. We show that the passage from testing to estimation requires only a polynomial increase in query complexity. This resolves a long-standing open problem in property testing, previously tackled by Fiat and Ron, by Hoppen, Kohayakawa, Lang, Lefmann, and Stagni, and by Gishboliner, Kushnir, and Shapira.

A Circuit Complexity Trichotomy For Regular Languages

from arXiv: Computational Complexity

Authors: Stefan Göller, Amaldev Manuel

We establish a trichotomy for the non-uniform, unbounded-fan-in circuit complexity of regular languages: the complexity of every regular language lies in exactly one of $O(1)$, $Ω(\log n)\cap O(\log n\log\log n)$, or $Ω(n)\cap O(n)$. Our main technical contribution is a characterization of the sublinear class in terms of algebra, word congruences, regular expressions, and logic. This class consists precisely of the regular languages computable by constant-depth circuits of size $O(\log^2 n)$, or, equivalently, by circuits of size $O(\log n\log\log n)$ and depth $O((\log\log n)^2)$. Consequently, every regular language of sublinear circuit complexity is definable in $\mathrm{FO}^2[Σ,<,\mathrm{mod}]$, which is contained in wire-linear $\mathbf{AC}^0$. For each of the three classes, deciding membership is $\mathbf{PSPACE}$-complete when the language is specified by a nondeterministic finite automaton. We also obtain a corresponding classification for regular languages with a neutral letter.

Authors: Stefan Göller, Amaldev Manuel

We establish a trichotomy for the non-uniform, unbounded-fan-in circuit complexity of regular languages: the complexity of every regular language lies in exactly one of $O(1)$, $Ω(\log n)\cap O(\log n\log\log n)$, or $Ω(n)\cap O(n)$. Our main technical contribution is a characterization of the sublinear class in terms of algebra, word congruences, regular expressions, and logic. This class consists precisely of the regular languages computable by constant-depth circuits of size $O(\log^2 n)$, or, equivalently, by circuits of size $O(\log n\log\log n)$ and depth $O((\log\log n)^2)$. Consequently, every regular language of sublinear circuit complexity is definable in $\mathrm{FO}^2[Σ,<,\mathrm{mod}]$, which is contained in wire-linear $\mathbf{AC}^0$. For each of the three classes, deciding membership is $\mathbf{PSPACE}$-complete when the language is specified by a nondeterministic finite automaton. We also obtain a corresponding classification for regular languages with a neutral letter.

Determinant boundaries and gaps in determinantal complexity

from arXiv: Computational Complexity

Authors: Karthik Sheshadri

Landsberg, Manivel, and Ressayre constructed determinant-boundary components whose defining forms, for odd n >= 3, are limits of n x n determinants but cannot themselves be represented by determinants of that size. Their border determinantal complexity (DC) is therefore smaller than their exact DC. We prove that the difference can grow linearly with n. For every n >= 10, we construct an explicit homogeneous form of degree n, concise in n^2 variables, whose orbit closure is a boundary component, whose border DC is n, and whose exact DC is at least 2n - 1. Thus n rows suffice when limits are allowed, whereas every exact representation requires at least 2n - 1 rows. The same bound holds for the unpadded odd-skew forms at every odd n >= 3. We also prove that every concise degree-n boundary form in n^2 variables has exact DC at least n + 2. We construct three families of pairwise distinct boundary components. Together with the traceless determinant, they give at least floor((n-1)/2) + floor(n/3) components for every n >= 3, with an additional term counted explicitly in the paper. The main lower bound follows from a general degree lemma. If an affine matrix represents an irreducible homogeneous polynomial f of degree d and has globally free rank-one cokernel over the coordinate ring of f = 0, its size is at least 2d - 1. The adjugate factors into two nonzero kernel vectors, each requiring a coordinate of degree at least d - 1. Comparing their product with the degree of a cofactor proves the bound. The lemma is sharp for individual representations, as witnessed by algebraic branching programs. Whether the exact-versus-border gap can grow superlinearly remains open.

Authors: Karthik Sheshadri

Landsberg, Manivel, and Ressayre constructed determinant-boundary components whose defining forms, for odd n >= 3, are limits of n x n determinants but cannot themselves be represented by determinants of that size. Their border determinantal complexity (DC) is therefore smaller than their exact DC. We prove that the difference can grow linearly with n. For every n >= 10, we construct an explicit homogeneous form of degree n, concise in n^2 variables, whose orbit closure is a boundary component, whose border DC is n, and whose exact DC is at least 2n - 1. Thus n rows suffice when limits are allowed, whereas every exact representation requires at least 2n - 1 rows. The same bound holds for the unpadded odd-skew forms at every odd n >= 3. We also prove that every concise degree-n boundary form in n^2 variables has exact DC at least n + 2. We construct three families of pairwise distinct boundary components. Together with the traceless determinant, they give at least floor((n-1)/2) + floor(n/3) components for every n >= 3, with an additional term counted explicitly in the paper. The main lower bound follows from a general degree lemma. If an affine matrix represents an irreducible homogeneous polynomial f of degree d and has globally free rank-one cokernel over the coordinate ring of f = 0, its size is at least 2d - 1. The adjugate factors into two nonzero kernel vectors, each requiring a coordinate of degree at least d - 1. Comparing their product with the degree of a cofactor proves the bound. The lemma is sharp for individual representations, as witnessed by algebraic branching programs. Whether the exact-versus-border gap can grow superlinearly remains open.

Tilted Convexity and an Exact Offline-Online Separation

from arXiv: Computational Complexity

Authors: Mohammad Pedramfar, Rajiv Khanna, Christopher John Quinn, Vaneet Aggarwal

Convexity has three equivalent descriptions: support by affine functions, monotonicity of the subdifferential, and increasing secant slopes. We weaken all three by one distortion with parameter $γ\in(0,1]$, which multiplies positive changes by $γ$ and divides negative ones by $γ$. The resulting classes are the convex functions at $γ=1$ and, for $γ<1$, are distinct and nested, with the parameter dropping by at most a square along the chain. For proper lower semicontinuous functions in these classes, the effective domain is convex, local minimizers are global, and the function is locally Lipschitz on the relative interior of its domain. The classes separate offline from online optimization over compact convex sets with efficient projection, with access to function values and bounded support vectors. A smooth member of any class can be minimized to additive accuracy $\varepsilon$ with $\mathrm{poly}(1/\varepsilon)$ first-order queries and projections. Online, with nonpositive losses and $α$-regret measured against $α$ times the best fixed point, a polynomial-time algorithm achieves $γ$-regret $O(\sqrt T)$ on the largest class. For every fixed, effectively specified $γ\in(0,1)$ and every $α\in(γ,1]$, no randomized polynomial-time algorithm has sublinear expected $α$-regret unless $\mathbf{NP}=\mathbf{RP}$, already for smooth concave polynomial losses in the smallest class, each minimized by one linear optimization. Thus, unless $\mathbf{NP}=\mathbf{RP}$, $γ$ is the exact online coefficient of each class, while the offline coefficient is one.

Authors: Mohammad Pedramfar, Rajiv Khanna, Christopher John Quinn, Vaneet Aggarwal

Convexity has three equivalent descriptions: support by affine functions, monotonicity of the subdifferential, and increasing secant slopes. We weaken all three by one distortion with parameter $γ\in(0,1]$, which multiplies positive changes by $γ$ and divides negative ones by $γ$. The resulting classes are the convex functions at $γ=1$ and, for $γ<1$, are distinct and nested, with the parameter dropping by at most a square along the chain. For proper lower semicontinuous functions in these classes, the effective domain is convex, local minimizers are global, and the function is locally Lipschitz on the relative interior of its domain. The classes separate offline from online optimization over compact convex sets with efficient projection, with access to function values and bounded support vectors. A smooth member of any class can be minimized to additive accuracy $\varepsilon$ with $\mathrm{poly}(1/\varepsilon)$ first-order queries and projections. Online, with nonpositive losses and $α$-regret measured against $α$ times the best fixed point, a polynomial-time algorithm achieves $γ$-regret $O(\sqrt T)$ on the largest class. For every fixed, effectively specified $γ\in(0,1)$ and every $α\in(γ,1]$, no randomized polynomial-time algorithm has sublinear expected $α$-regret unless $\mathbf{NP}=\mathbf{RP}$, already for smooth concave polynomial losses in the smallest class, each minimized by one linear optimization. Thus, unless $\mathbf{NP}=\mathbf{RP}$, $γ$ is the exact online coefficient of each class, while the offline coefficient is one.

Parity Decision Trees of Depth $O(4^d \log\|\widehat f\|_1)$ for Boolean Functions $f$ of $F_2$-Degree $d$

from arXiv: Computational Complexity

Authors: Shengyu Zhang

Let $f\colon F_2^n\to\{-1,+1\}$ have $F_2$-degree $d\ge2$. We prove that its deterministic parity decision tree depth satisfies \[ D_{\oplus} (f)= O(4^d \log_2\|\widehat f\|_1) = O(4^d \log_2 s(f)), \] where $\|\widehat f\|_1=\sum_ξ|\widehat f(ξ)|$ is the Fourier spectral norm and $s(f)$ is the Fourier sparsity. The bound is linear in the logarithm of the spectral norm for every fixed degree, with singly exponential dependence on $d$. This improves the previously best known bound of $O(2^{d^2/2} \log^{d-2} \|\widehat f\|_1)$ due to Tsang, Wong, Xie and Zhang (FOCS 2013). Our result establishes the log-rank conjecture for $\mathrm{XOR}$ functions of $F_2$-degree $O(\log\log s(f))$, extending the previously accessible range $d=O(1)$. The proof combines field lifting, multiplicative derivatives, a multiplicity-based generic-rank comparison, a Schur-complement vanishing argument, and dimension-preserving descent to $F_2$ to lower the degree on every coset of a common subspace.

Authors: Shengyu Zhang

Let $f\colon F_2^n\to\{-1,+1\}$ have $F_2$-degree $d\ge2$. We prove that its deterministic parity decision tree depth satisfies \[ D_{\oplus} (f)= O(4^d \log_2\|\widehat f\|_1) = O(4^d \log_2 s(f)), \] where $\|\widehat f\|_1=\sum_ξ|\widehat f(ξ)|$ is the Fourier spectral norm and $s(f)$ is the Fourier sparsity. The bound is linear in the logarithm of the spectral norm for every fixed degree, with singly exponential dependence on $d$. This improves the previously best known bound of $O(2^{d^2/2} \log^{d-2} \|\widehat f\|_1)$ due to Tsang, Wong, Xie and Zhang (FOCS 2013). Our result establishes the log-rank conjecture for $\mathrm{XOR}$ functions of $F_2$-degree $O(\log\log s(f))$, extending the previously accessible range $d=O(1)$. The proof combines field lifting, multiplicative derivatives, a multiplicity-based generic-rank comparison, a Schur-complement vanishing argument, and dimension-preserving descent to $F_2$ to lower the degree on every coset of a common subspace.

Subpolynomial-Query Proximal Sampling Beyond Log-Concavity

from arXiv: Computational Complexity

Authors: Wei Guo, Bo Yuan, Ye He, Jiaming Liang, Molei Tao, Yongxin Chen

We study the gradient query complexity of sampling from a density $π(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ has a $β$-Lipschitz gradient but need not be convex. Our approach uses the proximal sampler, which alternates Gaussian perturbation with sampling a position conditional on the noisy observation. We implement its observation chain using the high-order noisy conditional sampler of [OpenAI 2026], obtaining improved dimension dependence under functional inequalities. If $π$ satisfies a log-Sobolev inequality (LSI) with constant $α$, then for every fixed $ν>0$, there is a finite constant $C_ν$ depending only on $ν$ such that the algorithm returns a sample within total-variation distance $\varepsilon$ using at most $C_νκ(d+κ+\varepsilon^{-1}+\|\nabla V(x_0)\|^2/β)^ν$ queries, where $κ=β/α$ and $x_0$ is an arbitrary reference point for initialization. The analysis controls center-dependent local errors along exact proximal histories and removes the final observation noise by conditional decoding. Under a Poincaré inequality (PI), we obtain a corresponding guarantee from a warm start with a known bound on its Rényi divergence. An oracle lower bound shows that a stationary point alone, which suffices under LSI, is insufficient under PI.

Authors: Wei Guo, Bo Yuan, Ye He, Jiaming Liang, Molei Tao, Yongxin Chen

We study the gradient query complexity of sampling from a density $π(x)\propto\mathrm{e}^{-V(x)}$ on $\mathbb{R}^d$, where $V$ has a $β$-Lipschitz gradient but need not be convex. Our approach uses the proximal sampler, which alternates Gaussian perturbation with sampling a position conditional on the noisy observation. We implement its observation chain using the high-order noisy conditional sampler of [OpenAI 2026], obtaining improved dimension dependence under functional inequalities. If $π$ satisfies a log-Sobolev inequality (LSI) with constant $α$, then for every fixed $ν>0$, there is a finite constant $C_ν$ depending only on $ν$ such that the algorithm returns a sample within total-variation distance $\varepsilon$ using at most $C_νκ(d+κ+\varepsilon^{-1}+\|\nabla V(x_0)\|^2/β)^ν$ queries, where $κ=β/α$ and $x_0$ is an arbitrary reference point for initialization. The analysis controls center-dependent local errors along exact proximal histories and removes the final observation noise by conditional decoding. Under a Poincaré inequality (PI), we obtain a corresponding guarantee from a warm start with a known bound on its Rényi divergence. An oracle lower bound shows that a stationary point alone, which suffices under LSI, is insufficient under PI.

Exact Agreement Expansion up to the Top via Non-polarity

from arXiv: Computational Complexity

Authors: Tali Kaufman, Huy Tuan Pham

Agreement expansion asks whether local assignments to faces of a poset that usually agree on a sampled common face must agree with a single global function on most faces. Agreement expansion is exact (approximate) if local assignments need to exactly (approximately) agree with the global function. We suggest a new proof technique for agreement expansion that enables agreement expansion in settings which were previously beyond reach, such as: exact agreement expansion for Grassmannian posets (before only simplicial complexes support exact agreement expansion); agreement expansion based on assignments to top faces of the a complex (before agreement expansion is generally proven only for assignments to low dimensional faces); agreement expansion from one-sided spectral expansion (before, two sided spectral assumption was needed in general settings). Specifically, for down-closed simplicial and Grassmannian complexes, we prove an exact agreement theorem between any two layers assuming one-sided spectral expansion in links of singletons. As an application, we show that full bases of every rank-$k$ matroid admit exact agreement with soundness $O(k^2/[h(k-h)])$ when tested through $h$-element independent sets. We also obtain exact agreement on Grassmannian posets under one-sided spectral expansion, with stronger soundness under a local distance guarantee. Our key proof ideas center around non-polarity of the distributions of local assignments, probed via a random exposure of elements.

Authors: Tali Kaufman, Huy Tuan Pham

Agreement expansion asks whether local assignments to faces of a poset that usually agree on a sampled common face must agree with a single global function on most faces. Agreement expansion is exact (approximate) if local assignments need to exactly (approximately) agree with the global function. We suggest a new proof technique for agreement expansion that enables agreement expansion in settings which were previously beyond reach, such as: exact agreement expansion for Grassmannian posets (before only simplicial complexes support exact agreement expansion); agreement expansion based on assignments to top faces of the a complex (before agreement expansion is generally proven only for assignments to low dimensional faces); agreement expansion from one-sided spectral expansion (before, two sided spectral assumption was needed in general settings). Specifically, for down-closed simplicial and Grassmannian complexes, we prove an exact agreement theorem between any two layers assuming one-sided spectral expansion in links of singletons. As an application, we show that full bases of every rank-$k$ matroid admit exact agreement with soundness $O(k^2/[h(k-h)])$ when tested through $h$-element independent sets. We also obtain exact agreement on Grassmannian posets under one-sided spectral expansion, with stronger soundness under a local distance guarantee. Our key proof ideas center around non-polarity of the distributions of local assignments, probed via a random exposure of elements.

Dissecting Geometric Constraints in Grokking: Interaction-Dependent Acceleration in Modular Arithmetic

from arXiv: Computational Complexity

Authors: Zachary A. Hampton, Wasiq Khan, Bilal Khan

Spherical transformer configurations can accelerate grokking through simultaneous modifications of several model properties. The present study investigated the contributions of these modifications in the published Sphere+Fourier configuration using 960 controlled modular-arithmetic runs. A complete factorial analysis varied residual projection, readout weight normalization, logit scale, and weight decay. Across four matched settings with weight decay, the successful-run mean crossing times without projection were 8-37 times those observed with projection. However, residual projection was insufficient to induce grokking without additional interventions. Various other unprojected configurations also generalized. Since projection altered the magnitude entering the readout, these comparisons could not separate geometry from output scale. A separate 100-run fixed-horizon study demonstrated faster first crossing for the complete configuration across five train/test partitions. Faster crossing did not necessarily correspond to greater persistence over extended training. Control experiments demonstrated rapid grokking without Fourier initialization. Task and depth sweeps also identified conditions in which the configuration failed to reach the criterion by 50K epochs. Accordingly, the observed acceleration depended on interactions among the components and did not extend uniformly across the tested tasks or depths.

Authors: Zachary A. Hampton, Wasiq Khan, Bilal Khan

Spherical transformer configurations can accelerate grokking through simultaneous modifications of several model properties. The present study investigated the contributions of these modifications in the published Sphere+Fourier configuration using 960 controlled modular-arithmetic runs. A complete factorial analysis varied residual projection, readout weight normalization, logit scale, and weight decay. Across four matched settings with weight decay, the successful-run mean crossing times without projection were 8-37 times those observed with projection. However, residual projection was insufficient to induce grokking without additional interventions. Various other unprojected configurations also generalized. Since projection altered the magnitude entering the readout, these comparisons could not separate geometry from output scale. A separate 100-run fixed-horizon study demonstrated faster first crossing for the complete configuration across five train/test partitions. Faster crossing did not necessarily correspond to greater persistence over extended training. Control experiments demonstrated rapid grokking without Fourier initialization. Task and depth sweeps also identified conditions in which the configuration failed to reach the criterion by 50K epochs. Accordingly, the observed acceleration depended on interactions among the components and did not extend uniformly across the tested tasks or depths.

Reed-Muller Codes are Optimal Binary Linear 3-Query LCCs

from arXiv: Computational Complexity

Authors: Oliver Janzer, Peter Manohar

We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{Ω(\sqrt{k})}$. This improves on the prior best lower bound of $2^{Ω(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed-Muller codes, which achieve length $n = 2^{O(\sqrt{k})}$, is optimal up to constant factors in the exponent. Our proof builds on the approach of [KM24a], further refined in [Yan24,KM24b], that uses "chain derived equations" in combination with Kikuchi graphs. The crux of our argument is to show that a certain edge-coloured Kikuchi graph has a unique colour cycle: a cycle in which some colour is used exactly once. We do this by controlling the distribution of colours at each vertex in the graph using a random walk mixing argument and a novel hypergraph decomposition based on edge expansion of the hypergraph's "flattening" into a graph.

Authors: Oliver Janzer, Peter Manohar

We prove that any linear $3$-query locally correctable code $C \colon \mathbb{F}_2^k \to \mathbb{F}_2^n$ with constant correction radius must have length $n \geq 2^{Ω(\sqrt{k})}$. This improves on the prior best lower bound of $2^{Ω(\sqrt{k/\log k})}$, due to [AG24], and shows that the folklore construction of linear $3$-LCCs from Reed-Muller codes, which achieve length $n = 2^{O(\sqrt{k})}$, is optimal up to constant factors in the exponent. Our proof builds on the approach of [KM24a], further refined in [Yan24,KM24b], that uses "chain derived equations" in combination with Kikuchi graphs. The crux of our argument is to show that a certain edge-coloured Kikuchi graph has a unique colour cycle: a cycle in which some colour is used exactly once. We do this by controlling the distribution of colours at each vertex in the graph using a random walk mixing argument and a novel hypergraph decomposition based on edge expansion of the hypergraph's "flattening" into a graph.

Iterated Narrow Networks: Turing Completeness at Width 4 on Convex Input Spaces and the Complexity of Verification

from arXiv: Computational Complexity

Authors: Olivier Bournez, Johanne Cohen, Adrian Wurm

We study the iterates of a fixed ReLU network $N$ of width 4, that is, the discrete-time system $x_{k+1} = N(x_k)$ on $[0,1]^4$. Equivalently, $x_k$ is the output of a weight-tied network made of $k$ copies of $N$, or of a recurrent network with no external input, started at $x_0$. We prove that every deterministic Turing machine $M$ admits such a network $N_M$. The network can be constructed in polynomial time from $M$. Each iteration has depth $\mathrm{poly}(|M|)$. The iterates of $N_M$ simulate $M$ exactly on every encoded tape. The number of iterations is proportional to the running time of $M$. Moreover, the fourth coordinate irreversibly flags every point of the convex cube $[0,1]^4$ whose simulation reads an invalid digit. This flag is the certificate that makes hardness reductions sound on convex input regions, the regime of practical verifiers. Convex regions were previously inaccessible because of a continuity obstruction inherent to ReLU networks. The correspondence has three consequences. First, reachability for iterated width-4 ReLU networks spans the full complexity hierarchy along the single axis of the time bound: it is NP-complete for a unary bound, NEXP-complete for a binary bound and RE-complete (undecidable) with no bound. Second, two extra neurons of width suffice to extend the hardness to every common smooth activation: at width 6, the robust verification problem (a promise problem with $δ/2δ$ gap) is NP-complete for $\tanh$, sigmoid, GELU, Swish, ELU, and any polynomial-time-computable continuous non-polynomial activation with a polynomial-time-computable Lipschitz constant. Third, the whole spectrum transfers to discretized Neural ODEs in dimension 5.

Authors: Olivier Bournez, Johanne Cohen, Adrian Wurm

We study the iterates of a fixed ReLU network $N$ of width 4, that is, the discrete-time system $x_{k+1} = N(x_k)$ on $[0,1]^4$. Equivalently, $x_k$ is the output of a weight-tied network made of $k$ copies of $N$, or of a recurrent network with no external input, started at $x_0$. We prove that every deterministic Turing machine $M$ admits such a network $N_M$. The network can be constructed in polynomial time from $M$. Each iteration has depth $\mathrm{poly}(|M|)$. The iterates of $N_M$ simulate $M$ exactly on every encoded tape. The number of iterations is proportional to the running time of $M$. Moreover, the fourth coordinate irreversibly flags every point of the convex cube $[0,1]^4$ whose simulation reads an invalid digit. This flag is the certificate that makes hardness reductions sound on convex input regions, the regime of practical verifiers. Convex regions were previously inaccessible because of a continuity obstruction inherent to ReLU networks. The correspondence has three consequences. First, reachability for iterated width-4 ReLU networks spans the full complexity hierarchy along the single axis of the time bound: it is NP-complete for a unary bound, NEXP-complete for a binary bound and RE-complete (undecidable) with no bound. Second, two extra neurons of width suffice to extend the hardness to every common smooth activation: at width 6, the robust verification problem (a promise problem with $δ/2δ$ gap) is NP-complete for $\tanh$, sigmoid, GELU, Swish, ELU, and any polynomial-time-computable continuous non-polynomial activation with a polynomial-time-computable Lipschitz constant. Third, the whole spectrum transfers to discretized Neural ODEs in dimension 5.

Information Complexity of Inner Product and Matrix Multiplication via Shannon-to-Min-Entropy Conversion

from arXiv: Computational Complexity

Authors: Xiaohong Cai, Mingyang Zhu

We determine the information complexity of matrix multiplication over finite fields under independent uniform inputs, up to a universal additive constant. The binary inner-product case resolves the open question of Braverman, Garg, Pankratov, and Weinstein on the optimal dependence of information complexity on the error probability. For inputs in $\mathbb F_2^m$ and error at most $\varepsilon\in[0,1/2]$, the information complexity with output at Bob lies between $(1-2\varepsilon)m-c_0$ and $(1-2\varepsilon)m$, where $c_0<4.772$ is a universal constant. For independent uniform matrices $X\in\mathbb F_q^{t\times m}$ and $Y\in\mathbb F_q^{s\times m}$, where $t\le s$, computing $XY^{\mathsf T}$ with output at Bob and error at most $\varepsilon\in[0,1]$ has information complexity $(1-q^{ts}\varepsilon/(q^{ts}-1))^+mt\log_2q$, up to an additive gap of less than $9.544$ bits. This gap is independent of the matrix dimensions, field size, and error. At zero error, the information complexity is exactly $mt\log_2q$. The information lower bounds for both inner product and matrix multiplication follow from a new Shannon-to-min-entropy theorem for convolution on discrete groups. Let $Q_1,\ldots,Q_k$ be the finitely supported distributions of independent random variables taking values in a discrete group, and let $s$ be the largest support size. We show that if $k\ge\lceil\lfloor\log_2s\rfloor\ln2\rceil$, then the min-entropy of their ordered product is at least the average Shannon entropy of the $Q_i$ minus $c_0$ bits. This holds for arbitrary discrete groups, including infinite and noncommutative groups, even when the input distributions differ. The leading coefficient of the logarithmic threshold is optimal.

Authors: Xiaohong Cai, Mingyang Zhu

We determine the information complexity of matrix multiplication over finite fields under independent uniform inputs, up to a universal additive constant. The binary inner-product case resolves the open question of Braverman, Garg, Pankratov, and Weinstein on the optimal dependence of information complexity on the error probability. For inputs in $\mathbb F_2^m$ and error at most $\varepsilon\in[0,1/2]$, the information complexity with output at Bob lies between $(1-2\varepsilon)m-c_0$ and $(1-2\varepsilon)m$, where $c_0<4.772$ is a universal constant. For independent uniform matrices $X\in\mathbb F_q^{t\times m}$ and $Y\in\mathbb F_q^{s\times m}$, where $t\le s$, computing $XY^{\mathsf T}$ with output at Bob and error at most $\varepsilon\in[0,1]$ has information complexity $(1-q^{ts}\varepsilon/(q^{ts}-1))^+mt\log_2q$, up to an additive gap of less than $9.544$ bits. This gap is independent of the matrix dimensions, field size, and error. At zero error, the information complexity is exactly $mt\log_2q$. The information lower bounds for both inner product and matrix multiplication follow from a new Shannon-to-min-entropy theorem for convolution on discrete groups. Let $Q_1,\ldots,Q_k$ be the finitely supported distributions of independent random variables taking values in a discrete group, and let $s$ be the largest support size. We show that if $k\ge\lceil\lfloor\log_2s\rfloor\ln2\rceil$, then the min-entropy of their ordered product is at least the average Shannon entropy of the $Q_i$ minus $c_0$ bits. This holds for arbitrary discrete groups, including infinite and noncommutative groups, even when the input distributions differ. The leading coefficient of the logarithmic threshold is optimal.

ETH is Dead, Long Live SH

from arXiv: Computational Complexity

Authors: MIT Hardness Group, Josh Brunner, Erik D. Demaine, Fabian Frei, Timothy Gomez, Markus Hecher, Jeffery Li, Jayson Lynch

Impagliazzo and Paturi's 1999 Exponential Time Hypothesis (ETH) is false, but Stearns and Hunt's 1990 Satisfiability Hypothesis (SH) remains plausible. We prove that SH is equivalent to the recently proposed Weak ETH which rules out $O(2^{n^{1-\varepsilon}})L^{O(1)}$-time 3SAT algorithms for every constant $\varepsilon>0$, where $L$ is the input length. We show that SH (or Weak ETH) preserves the typical process for lower bounds: a polynomial-time reduction from a 3SAT formula of size $s=Θ(n+m)$ to an instance of size $N\le s^{b+o(1)}$, for fixed blowup exponent $b>0$, rules out $O(2^{N^{1/b-\varepsilon}})$-time algorithms for every constant $\varepsilon>0$. By comparison, for an $N=O(s^b)$ blowup, ETH would have ruled out $2^{o(N^{1/b})}$-time algorithms. The framework also yields parameterized lower bounds ruling out running times of the form $O(2^{k^{1/b-\varepsilon}})L^{O(1)}$. We also prove equivalence of the analogous counting hypotheses, #SH and Weak #ETH, and that they rule out $O(2^{(n+m)^{1-\varepsilon}})$-time algorithms for #2SAT for every constant $\varepsilon>0$.

Authors: MIT Hardness Group, Josh Brunner, Erik D. Demaine, Fabian Frei, Timothy Gomez, Markus Hecher, Jeffery Li, Jayson Lynch

Impagliazzo and Paturi's 1999 Exponential Time Hypothesis (ETH) is false, but Stearns and Hunt's 1990 Satisfiability Hypothesis (SH) remains plausible. We prove that SH is equivalent to the recently proposed Weak ETH which rules out $O(2^{n^{1-\varepsilon}})L^{O(1)}$-time 3SAT algorithms for every constant $\varepsilon>0$, where $L$ is the input length. We show that SH (or Weak ETH) preserves the typical process for lower bounds: a polynomial-time reduction from a 3SAT formula of size $s=Θ(n+m)$ to an instance of size $N\le s^{b+o(1)}$, for fixed blowup exponent $b>0$, rules out $O(2^{N^{1/b-\varepsilon}})$-time algorithms for every constant $\varepsilon>0$. By comparison, for an $N=O(s^b)$ blowup, ETH would have ruled out $2^{o(N^{1/b})}$-time algorithms. The framework also yields parameterized lower bounds ruling out running times of the form $O(2^{k^{1/b-\varepsilon}})L^{O(1)}$. We also prove equivalence of the analogous counting hypotheses, #SH and Weak #ETH, and that they rule out $O(2^{(n+m)^{1-\varepsilon}})$-time algorithms for #2SAT for every constant $\varepsilon>0$.

Exponential Lower Bounds for Products of Cubic Polynomials over Prime Fields

from arXiv: Computational Complexity

Authors: Bin Fu, Yunhui Fu

We study exact representations of explicit signed modular predicates on $\{-1,1\}^n$ as products of bounded-degree polynomials over prime fields, measured by total variable-occurrence cost. Our main result is an exponential lower bound for products of cubic polynomials. Over $\mathbb F_3$ the proof is self-contained: every cubic factor in a nonvanishing product is a unit on the sign cube, the unit identity forces bounded affine-observation complexity, and a simultaneous clustering argument converts this structural compression into an exponential occurrence lower bound. We also give an alternative reconstruction route that isolates the quadratic structure behind the cubic classification. We then extend the cubic theorem to every fixed prime field $\mathbb F_p$ of characteristic greater than three for odd target modulus $q$ coprime to $p$. The only non-elementary structural input in this extension is Karam's restricted-range rank theorem; the conversion from multiplication to parity and the $p$-tuple clustering argument are proved explicitly. Finally, we establish a coefficient-field-independent degree-lifting principle. If a lower bound is known at one fixed degree, restriction to a large independent set yields a quantitative lower bound at the next degree. In particular, the exponential cubic theorem implies $ L^{(p,q)}_4(n)=Ω_{p,q}\!\left(\frac{n^4}{(\log(2+n))^3}\right). $ For the characteristic-three core this already gives a lower bound for a restricted mixed-modulus circuit class.

Authors: Bin Fu, Yunhui Fu

We study exact representations of explicit signed modular predicates on $\{-1,1\}^n$ as products of bounded-degree polynomials over prime fields, measured by total variable-occurrence cost. Our main result is an exponential lower bound for products of cubic polynomials. Over $\mathbb F_3$ the proof is self-contained: every cubic factor in a nonvanishing product is a unit on the sign cube, the unit identity forces bounded affine-observation complexity, and a simultaneous clustering argument converts this structural compression into an exponential occurrence lower bound. We also give an alternative reconstruction route that isolates the quadratic structure behind the cubic classification. We then extend the cubic theorem to every fixed prime field $\mathbb F_p$ of characteristic greater than three for odd target modulus $q$ coprime to $p$. The only non-elementary structural input in this extension is Karam's restricted-range rank theorem; the conversion from multiplication to parity and the $p$-tuple clustering argument are proved explicitly. Finally, we establish a coefficient-field-independent degree-lifting principle. If a lower bound is known at one fixed degree, restriction to a large independent set yields a quantitative lower bound at the next degree. In particular, the exponential cubic theorem implies $ L^{(p,q)}_4(n)=Ω_{p,q}\!\left(\frac{n^4}{(\log(2+n))^3}\right). $ For the characteristic-three core this already gives a lower bound for a restricted mixed-modulus circuit class.

State Synthesis with an NP oracle

from arXiv: Computational Complexity

Authors: Aniruddha Sen

In this note, I describe a quantum algorithm that prepares an arbitrary state $|ψ\rangle$ to inverse exponential precision in polynomial time by making queries to an $\mathsf{NP}$ oracle, assuming one can directly query entries of the state vector corresponding to $|ψ\rangle$. The previous best upper bound on the complexity class required by an oracle for this task was $\mathsf{PP}$. I provide evidence that $\mathsf{NP}$ is the best possible upper bound by showing that any quantum algorithm using a $\mathsf{P}$ oracle, or even a $\mathsf{P/poly}$ oracle, requires exponentially many queries. I also discuss some implications of this result for the $\mathsf{QMA}$ search-to-decision problem and unitary synthesis.

Authors: Aniruddha Sen

In this note, I describe a quantum algorithm that prepares an arbitrary state $|ψ\rangle$ to inverse exponential precision in polynomial time by making queries to an $\mathsf{NP}$ oracle, assuming one can directly query entries of the state vector corresponding to $|ψ\rangle$. The previous best upper bound on the complexity class required by an oracle for this task was $\mathsf{PP}$. I provide evidence that $\mathsf{NP}$ is the best possible upper bound by showing that any quantum algorithm using a $\mathsf{P}$ oracle, or even a $\mathsf{P/poly}$ oracle, requires exponentially many queries. I also discuss some implications of this result for the $\mathsf{QMA}$ search-to-decision problem and unitary synthesis.

Pointwise Score Evaluation from Succinct Boolean Descriptions: SAT Equivalence for Strongly Log-Concave Gaussian Mixtures

from arXiv: Computational Complexity

Authors: Yinzhen Lv

Score-based generative models rely on access to accurate scores, but the cost of computing those scores depends on how the underlying distribution is represented. Can smoothness and strong log-concavity make score evaluation efficient when a distribution is specified by Boolean constraints? I investigate this question by separating geometric encoding from computational extraction. I construct a succinctly specified Gaussian mixture whose score at a fixed query location encodes whether a Boolean formula is satisfiable. The construction has rational component means and uniform strong log-concavity, yet deterministic polynomial-time approximation of the designated score coordinate to constant absolute accuracy is equivalent to polynomial-time solvability of Boolean satisfiability. This equivalence remains valid throughout a neighborhood whose radius is independent of dimension. I further characterize how numerical precision connects bounded witness enumeration to exact counting, and show that independent sample access turns the same target into a dimension-independent mean-estimation problem. An explicit parity construction also separates low-degree approximation from efficient computation. Together, these results show that favorable geometry alone does not resolve the computational cost of extracting scores from compact descriptions. They identify the precise extraction requirement in this formulation while leaving the existence of a general efficient evaluator open.

Authors: Yinzhen Lv

Score-based generative models rely on access to accurate scores, but the cost of computing those scores depends on how the underlying distribution is represented. Can smoothness and strong log-concavity make score evaluation efficient when a distribution is specified by Boolean constraints? I investigate this question by separating geometric encoding from computational extraction. I construct a succinctly specified Gaussian mixture whose score at a fixed query location encodes whether a Boolean formula is satisfiable. The construction has rational component means and uniform strong log-concavity, yet deterministic polynomial-time approximation of the designated score coordinate to constant absolute accuracy is equivalent to polynomial-time solvability of Boolean satisfiability. This equivalence remains valid throughout a neighborhood whose radius is independent of dimension. I further characterize how numerical precision connects bounded witness enumeration to exact counting, and show that independent sample access turns the same target into a dimension-independent mean-estimation problem. An explicit parity construction also separates low-degree approximation from efficient computation. Together, these results show that favorable geometry alone does not resolve the computational cost of extracting scores from compact descriptions. They identify the precise extraction requirement in this formulation while leaving the existence of a general efficient evaluator open.

Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence

from arXiv: Computational Complexity

Authors: Joseph M. Shunia

We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$, where $C_n=(\mathbb{Z}/n\mathbb{Z})[Y]/(Φ_r(Y))$. Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is $L^{32+o(1)}$, where $L=\lceil\log_2 n\rceil$, and verification after constructing $b$ costs $L^{6+o(1)}$. More generally, for a fixed real algebraic $1/2\leθ<1$, a common zero-free boundary $\Re s>θ$ for the relevant Dedekind zeta functions gives $L^{\max\{6,(1-θ)^{-1}\}+o(1)}$. This bound reaches $L^{6+o(1)}$ at $θ=5/6$, and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base-$2$ Fermat test, using the modulus $X^r-2$ with $r$ the least positive integer not dividing $n(n-1)$.

Authors: Joseph M. Shunia

We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$, where $C_n=(\mathbb{Z}/n\mathbb{Z})[Y]/(Φ_r(Y))$. Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is $L^{32+o(1)}$, where $L=\lceil\log_2 n\rceil$, and verification after constructing $b$ costs $L^{6+o(1)}$. More generally, for a fixed real algebraic $1/2\leθ<1$, a common zero-free boundary $\Re s>θ$ for the relevant Dedekind zeta functions gives $L^{\max\{6,(1-θ)^{-1}\}+o(1)}$. This bound reaches $L^{6+o(1)}$ at $θ=5/6$, and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base-$2$ Fermat test, using the modulus $X^r-2$ with $r$ the least positive integer not dividing $n(n-1)$.

Factor Closure in Noncommutative Algebraic Complexity

from arXiv: Computational Complexity

Authors: V. Arvind, Pushkar S. Joglekar

We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant outer factors $a,b$, has a circuit of size $O(sD^6)$. The proof uses a small weighted nondeterministic finite automaton (NFA) with replacement labels. Given $f=gh$, its labels depend on a word of maximum degree in $h$ and the coefficients of its suffixes. A pairing of accepting paths proves that the induced linear map $T$ satisfies $T(gh)=g$. Matrix substitution implements $T$ in a circuit. Applying the same substitution to a Boolean-sum verifier proves $\VNPnc$ closure. The construction also proves formula closure. The automaton has one copying state and, after leaving it, uses only scalar replacements. This structure gives formulas of size $O(s^2)$ for one-sided factors and $O(s^4)$ for central factors of a size-$s$ formula. The same map proves that polynomial-size noncommutative algebraic branching programs are closed under taking factors. Polynomial-size ABP factorizations over finite fields are already known from~\cite{AJ25}. These are arithmetic size bounds, with arbitrary field constants at unit cost, including over $\Q$. Over finite fields, combining the formula construction with~\cite{AJ22} gives randomized polynomial-time complete factorization of formula inputs, with factors output as formulas. For polynomial-degree circuit inputs, we obtain polynomial-length succinct ABP descriptions with polynomial-time edge queries. We also give two algorithms that output complete ordered irreducible factorizations as polynomial-size circuits: a deterministic $\FP^{\SigTwo}$ algorithm and a zero-error randomized algorithm with an NP oracle and expected polynomial running time.

Authors: V. Arvind, Pushkar S. Joglekar

We prove that $\VPnc$ is closed under taking factors over every field. If a noncommutative circuit of size $s$ computes a nonconstant polynomial of degree at most $D$, then each left or right factor has a circuit of size $O(sD^3)$. A central factor $g$ in a decomposition $f=agb$, with nonconstant outer factors $a,b$, has a circuit of size $O(sD^6)$. The proof uses a small weighted nondeterministic finite automaton (NFA) with replacement labels. Given $f=gh$, its labels depend on a word of maximum degree in $h$ and the coefficients of its suffixes. A pairing of accepting paths proves that the induced linear map $T$ satisfies $T(gh)=g$. Matrix substitution implements $T$ in a circuit. Applying the same substitution to a Boolean-sum verifier proves $\VNPnc$ closure. The construction also proves formula closure. The automaton has one copying state and, after leaving it, uses only scalar replacements. This structure gives formulas of size $O(s^2)$ for one-sided factors and $O(s^4)$ for central factors of a size-$s$ formula. The same map proves that polynomial-size noncommutative algebraic branching programs are closed under taking factors. Polynomial-size ABP factorizations over finite fields are already known from~\cite{AJ25}. These are arithmetic size bounds, with arbitrary field constants at unit cost, including over $\Q$. Over finite fields, combining the formula construction with~\cite{AJ22} gives randomized polynomial-time complete factorization of formula inputs, with factors output as formulas. For polynomial-degree circuit inputs, we obtain polynomial-length succinct ABP descriptions with polynomial-time edge queries. We also give two algorithms that output complete ordered irreducible factorizations as polynomial-size circuits: a deterministic $\FP^{\SigTwo}$ algorithm and a zero-error randomized algorithm with an NP oracle and expected polynomial running time.

The Runtime-Moment Frontier for Universal Sampling

from arXiv: Computational Complexity

Authors: Sebastian Ben Daniel

Can one sampler give every string at least a polynomial fraction of the probability that any source in a class gives it, while keeping a bounded moment of its own runtime? We answer this in terms of runtime moments. For every oracle and every rational $a>0$, one sampler with polynomial $a$th runtime moment dominates every source with polynomial $r$th moment with power $B(a,r)=1+(a-\min\{r,1\})_+/r$, and the Rényi divergence of order $B/(B-1)$ of the source from it is $O(\log n)$, a sharp quantitative form of Gurevich's domination on average. Relative to an oracle, $B(a,r)$ is optimal in both senses, with failures at every large input length. So the first moment is a threshold: one sampler of order one is universal for expected polynomial time, but none of higher order is universal even for polynomial-space sources with all polynomial moments. For expected-polynomial-time sources the threshold also holds relative to a random oracle and, without an oracle, under a sequential-hardness hypothesis on iterated functions or a parallel-sequentiality hypothesis on verifiable delay functions with short seeds. The frontier governs algorithms as well. For rational orders we determine the exact exponent by which polynomial runtime moments of deterministic deciders transfer from one fixed sampler to every source, and show, relative to oracles, that all correct solvers of a sparse language in $\mathsf{UP}\cap\mathsf{coUP}$ meet it. Where $B(a,r)=1$, actual running times transfer better than arbitrary costs, but only above a critical moment.

Authors: Sebastian Ben Daniel

Can one sampler give every string at least a polynomial fraction of the probability that any source in a class gives it, while keeping a bounded moment of its own runtime? We answer this in terms of runtime moments. For every oracle and every rational $a>0$, one sampler with polynomial $a$th runtime moment dominates every source with polynomial $r$th moment with power $B(a,r)=1+(a-\min\{r,1\})_+/r$, and the Rényi divergence of order $B/(B-1)$ of the source from it is $O(\log n)$, a sharp quantitative form of Gurevich's domination on average. Relative to an oracle, $B(a,r)$ is optimal in both senses, with failures at every large input length. So the first moment is a threshold: one sampler of order one is universal for expected polynomial time, but none of higher order is universal even for polynomial-space sources with all polynomial moments. For expected-polynomial-time sources the threshold also holds relative to a random oracle and, without an oracle, under a sequential-hardness hypothesis on iterated functions or a parallel-sequentiality hypothesis on verifiable delay functions with short seeds. The frontier governs algorithms as well. For rational orders we determine the exact exponent by which polynomial runtime moments of deterministic deciders transfer from one fixed sampler to every source, and show, relative to oracles, that all correct solvers of a sparse language in $\mathsf{UP}\cap\mathsf{coUP}$ meet it. Where $B(a,r)=1$, actual running times transfer better than arbitrary costs, but only above a critical moment.

Quantum Lower Bounds for Collision Problems

from arXiv: Computational Complexity

Authors: Pan Peng, Jingyu Wu

We establish tight lower bounds on the adaptive and non-adaptive quantum query complexities of Promised $k$-to-one Finding on an exactly $k$-to-one input string $x\in\mathbb Z_q^n$, for every fixed $k\ge2$ and every feasible alphabet size $q\ge n/k$. The lower bounds are $Ω_k(n^{1/2-1/(2(2^k-1))})$ for unrestricted adaptive algorithms and $Ω_k(n^{1-1/k})$ for one-round non-adaptive algorithms. The former matches the adaptive quantum algorithm of Liu and Zhandry [LZ19], whereas the latter matches classical birthday sampling even when the parallel queries are arbitrarily entangled. We also establish the non-adaptive lower bound $Ω_k(\sqrt{n/k})$ for $k$-to-one Testing when $q\ge n$, resolving an open question posed by Koiran, Landes, Portier, and Yao [KLPY10]. Our proofs generate the balanced input from a uniformly random permutation and use compressed permutation oracles to describe the information learned by the algorithm. The adaptive proof bounds the growth of partial collisions through successive queries. The one-round proof instead analyzes the state after the parallel queries directly, bounding the probability that the compressed database contains a full collision. In both proofs, a short verification step accounts for correct outputs that include unrecorded positions. For the decision problem, we also compare one-round states with independent and distinct output labels.

Authors: Pan Peng, Jingyu Wu

We establish tight lower bounds on the adaptive and non-adaptive quantum query complexities of Promised $k$-to-one Finding on an exactly $k$-to-one input string $x\in\mathbb Z_q^n$, for every fixed $k\ge2$ and every feasible alphabet size $q\ge n/k$. The lower bounds are $Ω_k(n^{1/2-1/(2(2^k-1))})$ for unrestricted adaptive algorithms and $Ω_k(n^{1-1/k})$ for one-round non-adaptive algorithms. The former matches the adaptive quantum algorithm of Liu and Zhandry [LZ19], whereas the latter matches classical birthday sampling even when the parallel queries are arbitrarily entangled. We also establish the non-adaptive lower bound $Ω_k(\sqrt{n/k})$ for $k$-to-one Testing when $q\ge n$, resolving an open question posed by Koiran, Landes, Portier, and Yao [KLPY10]. Our proofs generate the balanced input from a uniformly random permutation and use compressed permutation oracles to describe the information learned by the algorithm. The adaptive proof bounds the growth of partial collisions through successive queries. The one-round proof instead analyzes the state after the parallel queries directly, bounding the probability that the compressed database contains a full collision. In both proofs, a short verification step accounts for correct outputs that include unrecorded positions. For the decision problem, we also compare one-round states with independent and distinct output labels.

Improved NP-Hardness of Approximation for Max-3-Cut and for Max-k-Cut with Small k

from arXiv: Computational Complexity

Authors: Ziang Ni

We prove that Max-3-Cut is NP-hard to approximate within a factor of $27/29+\varepsilon \approx 0.931$. This improves the factor $16/17 \approx 0.941$ of Austrin, O'Donnell, Tan and Wright, the best NP-hardness result for Max-3-Cut since 2012. (Their result, unlike ours, also holds for 3-colourable graphs.) We also improve the NP-hardness of Max-$k$-Cut for every $k$ from 4 to 20; for instance, we obtain $45/47 \approx 0.957$ for Max-4-Cut and $33/34 \approx 0.971$ for Max-5-Cut, where the previous factors were 0.987 and 120/121. Our reductions are gadget reductions from Håstad's linear equations over $\mathbb{Z}_3$, with gadgets that have at most six auxiliary vertices. The main new idea is a negation trick: since the negation of an assignment is again an assignment, equations with right-hand sides $t$ and $-t$ can be handled together, and this makes the resulting predicates symmetric under all permutations of the colours. We complement it with a general soundness criterion for such reductions: in effect, the best strategy of the adversary is to use a single colour pattern everywhere. This criterion is the strongest analysis that uses Håstad's theorem as a black box, and, using LP duality, we show that under it no gadget reduction from these equations can beat $27/29$ for Max-3-Cut. Finally, we observe that a classical reduction from Max-3-Cut to Max-$k$-Cut, applied more carefully, already gives $82/85 \approx 0.965$ for Max-4-Cut on 4-colourable graphs, improving the previously stated $85/86$.

Authors: Ziang Ni

We prove that Max-3-Cut is NP-hard to approximate within a factor of $27/29+\varepsilon \approx 0.931$. This improves the factor $16/17 \approx 0.941$ of Austrin, O'Donnell, Tan and Wright, the best NP-hardness result for Max-3-Cut since 2012. (Their result, unlike ours, also holds for 3-colourable graphs.) We also improve the NP-hardness of Max-$k$-Cut for every $k$ from 4 to 20; for instance, we obtain $45/47 \approx 0.957$ for Max-4-Cut and $33/34 \approx 0.971$ for Max-5-Cut, where the previous factors were 0.987 and 120/121. Our reductions are gadget reductions from Håstad's linear equations over $\mathbb{Z}_3$, with gadgets that have at most six auxiliary vertices. The main new idea is a negation trick: since the negation of an assignment is again an assignment, equations with right-hand sides $t$ and $-t$ can be handled together, and this makes the resulting predicates symmetric under all permutations of the colours. We complement it with a general soundness criterion for such reductions: in effect, the best strategy of the adversary is to use a single colour pattern everywhere. This criterion is the strongest analysis that uses Håstad's theorem as a black box, and, using LP duality, we show that under it no gadget reduction from these equations can beat $27/29$ for Max-3-Cut. Finally, we observe that a classical reduction from Max-3-Cut to Max-$k$-Cut, applied more carefully, already gives $82/85 \approx 0.965$ for Max-4-Cut on 4-colourable graphs, improving the previously stated $85/86$.

The Exact Communication Threshold for Parallel Repetition

from arXiv: Computational Complexity

Authors: Daiki Suruga

The parallel repetition theorem is a fundamental tool in theoretical computer science, showing that the probability of winning all $n$ independent instances of a finite two-player game decays exponentially in $n$, provided that the game has no perfect strategy. The classical theorem assumes that the players cannot communicate, raising the question of how much communication can be allowed while preserving this exponential decay. For every finite total relation $g$ and arbitrary input distribution $μ$, we prove exponential decay at every fixed communication rate below a quantity called zero-error internal information complexity, denoted by $\mathrm{IC}_μ(g,0)$. Together with known results showing that $n\,\mathrm{IC}_μ(g,0)+o(n)$ bits of communication suffice to solve all $n$ instances with success probability tending to one, our bound determines the exact communication threshold. As a by-product, we obtain exact characterizations of amortized distributional and randomized communication complexity under the global-error criterion.

Authors: Daiki Suruga

The parallel repetition theorem is a fundamental tool in theoretical computer science, showing that the probability of winning all $n$ independent instances of a finite two-player game decays exponentially in $n$, provided that the game has no perfect strategy. The classical theorem assumes that the players cannot communicate, raising the question of how much communication can be allowed while preserving this exponential decay. For every finite total relation $g$ and arbitrary input distribution $μ$, we prove exponential decay at every fixed communication rate below a quantity called zero-error internal information complexity, denoted by $\mathrm{IC}_μ(g,0)$. Together with known results showing that $n\,\mathrm{IC}_μ(g,0)+o(n)$ bits of communication suffice to solve all $n$ instances with success probability tending to one, our bound determines the exact communication threshold. As a by-product, we obtain exact characterizations of amortized distributional and randomized communication complexity under the global-error criterion.

Nonnegative Rank Bounds for the Algebraic Monotone Calculus

from arXiv: Computational Complexity

Authors: Fedor Part

In Hrubes' Algebraic Monotone Calculus a DNF tautology $A$ is proved by a monotone arithmetic circuit computing $χ(A)-εχ(1)$ for some $ε>0$, where the coefficient of $χ(A)$ at $x^σ$ counts the terms of $A$ satisfied by $σ$, and $χ(1)=\prod_i(x_i^0+x_i^1)$. Proof size is circuit size, and $ε$ is existentially quantified, so lower bounds must hold at arbitrarily small $ε$. We prove lower bounds for bounded-depth fragments by two nonnegative rank measures of the target: monotone tensor rank and the nonnegative rank of a flattening. At depth three a proof is a nonnegative tensor decomposition. Restricting its terms to subcube indicators gives conical juntas, a system polynomially equivalent to Sherali-Adams in twin variables. For arbitrary terms we adapt Rothvoss' hyperplane separation to monotone tensor rank: Tseitin formulas on constant-degree expanders of large girth need proofs of size $2^{Ω(εn)}$ whenever $ε=ω(\log n/n)$. This cannot reach every $ε$: at shift $2^{-Θ(n\log n)}$ Tseitin acquires polynomial-size depth-three proofs, separating the two fragments exponentially. A constant-size index gadget gives bounds for every $ε\geq2^{-Ω(n)}$, converting a Sherali-Adams degree bound into a size bound. Depth three is essential: Tseitin has monotone proofs of size $O(n^3)$ and depth $O(\log n)$ at every $ε$. Beyond depth three we bound the nonnegative rank of a flattening by $L^{2^t}$ for a monotone formula with $L$ leaves and mixed-product depth $t$: the largest number of product gates on a root-to-leaf path whose two inputs each meet both sides of the bipartition. Approximate extension complexity then bounds perfect matching and odd factor instances, and planted CSP integrality gaps give an explicit family of planted Tseitin DNFs hard at inverse-polynomial shift.

Authors: Fedor Part

In Hrubes' Algebraic Monotone Calculus a DNF tautology $A$ is proved by a monotone arithmetic circuit computing $χ(A)-εχ(1)$ for some $ε>0$, where the coefficient of $χ(A)$ at $x^σ$ counts the terms of $A$ satisfied by $σ$, and $χ(1)=\prod_i(x_i^0+x_i^1)$. Proof size is circuit size, and $ε$ is existentially quantified, so lower bounds must hold at arbitrarily small $ε$. We prove lower bounds for bounded-depth fragments by two nonnegative rank measures of the target: monotone tensor rank and the nonnegative rank of a flattening. At depth three a proof is a nonnegative tensor decomposition. Restricting its terms to subcube indicators gives conical juntas, a system polynomially equivalent to Sherali-Adams in twin variables. For arbitrary terms we adapt Rothvoss' hyperplane separation to monotone tensor rank: Tseitin formulas on constant-degree expanders of large girth need proofs of size $2^{Ω(εn)}$ whenever $ε=ω(\log n/n)$. This cannot reach every $ε$: at shift $2^{-Θ(n\log n)}$ Tseitin acquires polynomial-size depth-three proofs, separating the two fragments exponentially. A constant-size index gadget gives bounds for every $ε\geq2^{-Ω(n)}$, converting a Sherali-Adams degree bound into a size bound. Depth three is essential: Tseitin has monotone proofs of size $O(n^3)$ and depth $O(\log n)$ at every $ε$. Beyond depth three we bound the nonnegative rank of a flattening by $L^{2^t}$ for a monotone formula with $L$ leaves and mixed-product depth $t$: the largest number of product gates on a root-to-leaf path whose two inputs each meet both sides of the bipartition. Approximate extension complexity then bounds perfect matching and odd factor instances, and planted CSP integrality gaps give an explicit family of planted Tseitin DNFs hard at inverse-polynomial shift.

Mutual-visibility colouring with two colours is NP-complete for every fixed diameter

from arXiv: Computational Complexity

Authors: Siddhesh Chaubal

A mutual-visibility set in a graph $G$ is a set $S$ of vertices such that any two vertices of $S$ are connected by a shortest path with no internal vertex in $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the minimum number of mutual-visibility sets needed to partition $V(G)$. Babu, Di Stefano and Lakshmanan showed that it is NP-complete to decide whether $χ_μ(G)\le 2$ for graphs of diameter four. We show that the problem remains NP-complete for graphs of diameter $d$, for every fixed $d\ge2$; diameter two is the smallest diameter for which the problem is non-trivial. The proof rests on a reformulation of $χ_μ(G)\le 2$ for diameter-two graphs as the $2$-colourability of an associated hypergraph, and on a reduction from $2$-colourability of $3$-uniform hypergraphs without short cycles.

Authors: Siddhesh Chaubal

A mutual-visibility set in a graph $G$ is a set $S$ of vertices such that any two vertices of $S$ are connected by a shortest path with no internal vertex in $S$, and the mutual-visibility chromatic number $χ_μ(G)$ is the minimum number of mutual-visibility sets needed to partition $V(G)$. Babu, Di Stefano and Lakshmanan showed that it is NP-complete to decide whether $χ_μ(G)\le 2$ for graphs of diameter four. We show that the problem remains NP-complete for graphs of diameter $d$, for every fixed $d\ge2$; diameter two is the smallest diameter for which the problem is non-trivial. The proof rests on a reformulation of $χ_μ(G)\le 2$ for diameter-two graphs as the $2$-colourability of an associated hypergraph, and on a reduction from $2$-colourability of $3$-uniform hypergraphs without short cycles.

Approximate clearing in financial networks with credit default swaps: hardness thresholds, a box algorithm, barriers, and exact solvency at bounded degree

from arXiv: Computational Complexity

Authors: Michal Čertík

We study approximate clearing in financial networks with credit default swaps. For weak notions (W) and (I), we prove PPAD-hardness below a threshold approximately $0.2530$ and give a polynomial-time interval algorithm above a threshold approximately $0.2834$. On networks whose CDS writers are solvent ex ante, the corresponding bounds are $(13-\sqrt{153})/4$ and $3-2\sqrt{2}$, approximately $0.1577$ and $0.1716$. The thresholds are sharp for bounded-radius algorithms: for (I) on writer-solvent networks, and for (I) and (W) in general. On non-degenerate networks, (W) and (I) have identical hardness and tractability threshold endpoints. For exact-solvency notion (D), PPAD-hardness holds at every rational error $\varepsilon<1$ on non-degenerate networks. On a restricted unit-notional class, let $d$ be the maximum number of CDS contracts held by a core bank, counted with multiplicity. For rational $\varepsilon\in[1/2,1)$, writing $ρ=(1+\varepsilon)/(1-\varepsilon)$, the problem is polynomial-time solvable when $d<(\lceilρ\rceil+1)ρ$ and PPAD-complete otherwise. At $\varepsilon=1/3$ the boundary is $d=5$ versus $d=6$; below $\varepsilon=1/9$, degree $2$ is already PPAD-complete, while degree $1$ is exactly solvable. These degree results concern (D) and do not close the (W)/(I) approximation windows. We also establish a barrier to Pure-Circuit reductions certified by interval propagation under stated interface and auxiliary-bank restrictions. A finite-precision interval algorithm reduces writer-solvent (I) to point selection in near-stalled boxes; selection is polynomial-time when the residual matrix is symmetric or monotone on each strongly connected component. The general and writer-solvent approximation windows remain open.

Authors: Michal Čertík

We study approximate clearing in financial networks with credit default swaps. For weak notions (W) and (I), we prove PPAD-hardness below a threshold approximately $0.2530$ and give a polynomial-time interval algorithm above a threshold approximately $0.2834$. On networks whose CDS writers are solvent ex ante, the corresponding bounds are $(13-\sqrt{153})/4$ and $3-2\sqrt{2}$, approximately $0.1577$ and $0.1716$. The thresholds are sharp for bounded-radius algorithms: for (I) on writer-solvent networks, and for (I) and (W) in general. On non-degenerate networks, (W) and (I) have identical hardness and tractability threshold endpoints. For exact-solvency notion (D), PPAD-hardness holds at every rational error $\varepsilon<1$ on non-degenerate networks. On a restricted unit-notional class, let $d$ be the maximum number of CDS contracts held by a core bank, counted with multiplicity. For rational $\varepsilon\in[1/2,1)$, writing $ρ=(1+\varepsilon)/(1-\varepsilon)$, the problem is polynomial-time solvable when $d<(\lceilρ\rceil+1)ρ$ and PPAD-complete otherwise. At $\varepsilon=1/3$ the boundary is $d=5$ versus $d=6$; below $\varepsilon=1/9$, degree $2$ is already PPAD-complete, while degree $1$ is exactly solvable. These degree results concern (D) and do not close the (W)/(I) approximation windows. We also establish a barrier to Pure-Circuit reductions certified by interval propagation under stated interface and auxiliary-bank restrictions. A finite-precision interval algorithm reduces writer-solvent (I) to point selection in near-stalled boxes; selection is polynomial-time when the residual matrix is symmetric or monotone on each strongly connected component. The general and writer-solvent approximation windows remain open.

Leaving Quantum Bias on the Table: Pushforwards as a Path to Quantum Advantage in Constrained Optimization

from arXiv: Computational Complexity

Authors: Chinonso Onah, Stuart Hadfield, Kristel Michielsen

Constrained quantum samplers often produce infeasible outputs that retain information about good feasible solutions. We develop a post-measurement framework that constructs problem-dependent classical feasibility maps from constraint-violation statistics and pushes the raw quantum sampling law forward onto the feasible set. The probability of each repaired solution is the total raw mass of its repair preimage, enlarging the successful sampling event from a target feasible set to its full repair preimage. We derive exact identities for repaired success and expected cost, and separate repair-basin geometry from sampler-dependent probability concentration. For noisy implementations, we distinguish leakage from the encoded subspace from concentration loss within it. For permutation constraints, Hungarian maximum-overlap repair provides an O(n^3) implementation, which we study on traveling-salesman problems (TSP). We derive conditional repaired-success bounds and quantify how basin aggregation reduces the absolute-transfer budget required by these bounds. Noiseless simulations show that optimal repair basins contain selectively overweighted infeasible contributors. For an n=10 TSP instance, pushforward repair recovered the optimum using approximately 10^3 shots, compared with 10^6 shots in the original CE-QAOA workflow without repair. On a 127-qubit IBM processor, using encodings of up to 100 qubits, repaired candidates match reference tours on smaller benchmarks and improve them on larger instances. These results establish concentration of quantum probability on useful repair preimages as a design principle for constrained quantum optimization.

Authors: Chinonso Onah, Stuart Hadfield, Kristel Michielsen

Constrained quantum samplers often produce infeasible outputs that retain information about good feasible solutions. We develop a post-measurement framework that constructs problem-dependent classical feasibility maps from constraint-violation statistics and pushes the raw quantum sampling law forward onto the feasible set. The probability of each repaired solution is the total raw mass of its repair preimage, enlarging the successful sampling event from a target feasible set to its full repair preimage. We derive exact identities for repaired success and expected cost, and separate repair-basin geometry from sampler-dependent probability concentration. For noisy implementations, we distinguish leakage from the encoded subspace from concentration loss within it. For permutation constraints, Hungarian maximum-overlap repair provides an O(n^3) implementation, which we study on traveling-salesman problems (TSP). We derive conditional repaired-success bounds and quantify how basin aggregation reduces the absolute-transfer budget required by these bounds. Noiseless simulations show that optimal repair basins contain selectively overweighted infeasible contributors. For an n=10 TSP instance, pushforward repair recovered the optimum using approximately 10^3 shots, compared with 10^6 shots in the original CE-QAOA workflow without repair. On a 127-qubit IBM processor, using encodings of up to 100 qubits, repaired candidates match reference tours on smaller benchmarks and improve them on larger instances. These results establish concentration of quantum probability on useful repair preimages as a design principle for constrained quantum optimization.

On the complexity of isomorphism problems for tensors, groups, and polynomials V: over commutative rings

from arXiv: Computational Complexity

Authors: Joshua A. Grochow, Youming Qiao, Katherine E. Stange, Xiaorui Sun

Tensors over commutative rings naturally appear in number theory, geometry, and group theory. For example, $2\times 2\times 2$ tensors over $\mathbb{Z}$ form the starting point of Bhargava's celebrated works generalising Gauss's composition law (Bhargava, Ann. Math., 2004). Symmetric tensors over $\mathbb{Z}$ are central to the classification of Calabi--Yau threefolds (Yau, Commun. Pure Appl. Math., 1978; Wall, Invent. Math., 1966), geometric objects of significance in string theory. Additionally, tensors over finite commutative rings closely correspond to finite nilpotent groups of class 2 (Baer, Trans. Am. Math. Soc., 1938). In this work, we study isomorphism testing of tensors over commutative rings from a complexity-theoretic viewpoint, and its applications. Some of our main results are: 1. Let $\mathrm{R}$ be a commutative ring. We introduce two complexity classes: $\mathsf{3TI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three modules over $\mathrm{R}$, and $\mathsf{3FTI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three free modules over $\mathrm{R}$. 2. We show that some classical problems considered by Grunewald and Segal (ibid.), and the problem of classifying Calabi--Yau threefolds, are $\mathsf{3FTI}_\mathbb{Z}$-complete. We also show that many natural problems are complete for $\mathsf{3TI}_{\mathbb{Z}/p^e\mathbb{Z}}$. 3. We show that testing isomorphism of tensors in $\mathbb{Z}^2\otimes\mathbb{Z}^2\otimes\mathbb{Z}^2$ is polynomial-time equivalent to the principal ideal problem in algorithmic number theory. 4. We present an $N^{O((\log N)^{8/9})}$-time algorithm for testing isomorphism of finite nilpotent groups of class $2$ and odd order $N$.

Authors: Joshua A. Grochow, Youming Qiao, Katherine E. Stange, Xiaorui Sun

Tensors over commutative rings naturally appear in number theory, geometry, and group theory. For example, $2\times 2\times 2$ tensors over $\mathbb{Z}$ form the starting point of Bhargava's celebrated works generalising Gauss's composition law (Bhargava, Ann. Math., 2004). Symmetric tensors over $\mathbb{Z}$ are central to the classification of Calabi--Yau threefolds (Yau, Commun. Pure Appl. Math., 1978; Wall, Invent. Math., 1966), geometric objects of significance in string theory. Additionally, tensors over finite commutative rings closely correspond to finite nilpotent groups of class 2 (Baer, Trans. Am. Math. Soc., 1938). In this work, we study isomorphism testing of tensors over commutative rings from a complexity-theoretic viewpoint, and its applications. Some of our main results are: 1. Let $\mathrm{R}$ be a commutative ring. We introduce two complexity classes: $\mathsf{3TI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three modules over $\mathrm{R}$, and $\mathsf{3FTI}_\mathrm{R}$ consisting of problems that are polynomial-time reducible to isomorphism problems of tensor products of three free modules over $\mathrm{R}$. 2. We show that some classical problems considered by Grunewald and Segal (ibid.), and the problem of classifying Calabi--Yau threefolds, are $\mathsf{3FTI}_\mathbb{Z}$-complete. We also show that many natural problems are complete for $\mathsf{3TI}_{\mathbb{Z}/p^e\mathbb{Z}}$. 3. We show that testing isomorphism of tensors in $\mathbb{Z}^2\otimes\mathbb{Z}^2\otimes\mathbb{Z}^2$ is polynomial-time equivalent to the principal ideal problem in algorithmic number theory. 4. We present an $N^{O((\log N)^{8/9})}$-time algorithm for testing isomorphism of finite nilpotent groups of class $2$ and odd order $N$.

FIXP-Completeness in Second-Price Auctions

from arXiv: Computational Complexity

Authors: Nobutaka Shimizu

Computing equilibria is a fundamental algorithmic problem in game theory and economics. Although general frameworks are available for proving membership in FIXP, the class capturing exact fixed-point computation, establishing FIXP-completeness often remains challenging. We prove FIXP-completeness of exact equilibrium computation in three second-price auction models: (1). pacing with return-on-investment constraints, with multiplier cap $1+δ$ for every fixed rational $δ>0$ [Filos-Ratsikas, Hansen, Høgh, and Hollender (2024)]; (2). budget-constrained generalized second-price sponsored shopping [Hansen, Li, and Nie (2026)]; (3). budget-constrained throttling [Liu (2026)]. These results resolve open questions on the exact complexity of automated bidding. Our technical ingredient is a FIXP-complete generalized-circuit problem using only clipped affine gates and the reciprocal $x\mapsto1/(1+τx)$, for any fixed positive rational $τ$. This gate basis provides a common framework for the three auction reductions: simulating these gates in each model establishes FIXP-hardness without directly implementing multiplication or squaring.

Authors: Nobutaka Shimizu

Computing equilibria is a fundamental algorithmic problem in game theory and economics. Although general frameworks are available for proving membership in FIXP, the class capturing exact fixed-point computation, establishing FIXP-completeness often remains challenging. We prove FIXP-completeness of exact equilibrium computation in three second-price auction models: (1). pacing with return-on-investment constraints, with multiplier cap $1+δ$ for every fixed rational $δ>0$ [Filos-Ratsikas, Hansen, Høgh, and Hollender (2024)]; (2). budget-constrained generalized second-price sponsored shopping [Hansen, Li, and Nie (2026)]; (3). budget-constrained throttling [Liu (2026)]. These results resolve open questions on the exact complexity of automated bidding. Our technical ingredient is a FIXP-complete generalized-circuit problem using only clipped affine gates and the reciprocal $x\mapsto1/(1+τx)$, for any fixed positive rational $τ$. This gate basis provides a common framework for the three auction reductions: simulating these gates in each model establishes FIXP-hardness without directly implementing multiplication or squaring.

The Complexity of Choosing Parity Projections

from arXiv: Computational Complexity

Authors: Karthik Sheshadri, Nicholas Smirnov

Cube attacks in cryptanalysis seek small sets of Boolean variables (cubes) whose parity sum is constant. We study the bounded constant derivative problem introduced by Smirnov (2026), which seeks a smallest cube with sum 1, and its sum-0 analogue. Despite their similarity, the problems differ sharply in parameterized complexity: the sum-1 problem admits a $2^{O(k^2)}\operatorname{poly}(N)$-time algorithm, where $k$ is cube size, whereas the sum-0 problem is not fixed-parameter tractable unless $\mathrm{FPT}=\mathrm{W}[1]$. Without a bound on cube size, both problems are $Σ_2^{\oplus\mathrm{P}}$-complete and NP-hard to approximate within $N^{1-\varepsilon}$ for every fixed $0<\varepsilon<1$. Our hardness proofs use a crown construction that embeds maximal monomials of parity-projected circuits into positive 2CNF. This yields completeness and inapproximability results, as well as a dichotomy for Boolean constraint languages of arity at most two: the bounded constant derivative problem is in P when all relations are affine and $Σ_2^{\oplus\mathrm{P}}$-complete otherwise. We also give algorithms parameterized by the number of quadratic rounds, treewidth, or ordered branching-program width, with lower bounds for the first two unless $\mathrm{NP}\subseteq\mathrm{SUBEXP}$. The treewidth lower bound holds for circuits; the case of 2CNF remains open.

Authors: Karthik Sheshadri, Nicholas Smirnov

Cube attacks in cryptanalysis seek small sets of Boolean variables (cubes) whose parity sum is constant. We study the bounded constant derivative problem introduced by Smirnov (2026), which seeks a smallest cube with sum 1, and its sum-0 analogue. Despite their similarity, the problems differ sharply in parameterized complexity: the sum-1 problem admits a $2^{O(k^2)}\operatorname{poly}(N)$-time algorithm, where $k$ is cube size, whereas the sum-0 problem is not fixed-parameter tractable unless $\mathrm{FPT}=\mathrm{W}[1]$. Without a bound on cube size, both problems are $Σ_2^{\oplus\mathrm{P}}$-complete and NP-hard to approximate within $N^{1-\varepsilon}$ for every fixed $0<\varepsilon<1$. Our hardness proofs use a crown construction that embeds maximal monomials of parity-projected circuits into positive 2CNF. This yields completeness and inapproximability results, as well as a dichotomy for Boolean constraint languages of arity at most two: the bounded constant derivative problem is in P when all relations are affine and $Σ_2^{\oplus\mathrm{P}}$-complete otherwise. We also give algorithms parameterized by the number of quadratic rounds, treewidth, or ordered branching-program width, with lower bounds for the first two unless $\mathrm{NP}\subseteq\mathrm{SUBEXP}$. The treewidth lower bound holds for circuits; the case of 2CNF remains open.

Quadratic Reconfiguration of Plane Spanning Paths on Two-Layer Point Sets

from arXiv: Computational Geometry

Authors: Samuel Avril, Yan Gerard, Guilherme D. da Fonseca

Consider a crossing-free polygonal path with $n$ vertices and a flip operation that removes an edge and adds another one, without changing the vertex set and keeping the path crossing free. It is conjectured that any two such paths can be transformed into each other by a sequence of flips. Recently, Kleist et al. (2024) proved the conjecture for the special cases of vertices arranged in two convex layers. However, the number of flips in their proof may be exponential. In this paper, we show that $O(n^2)$ flips suffice. More precisely, for a two-layer point set $S=L_0\cup L_1$, where $L_0$ and $L_1$ denote the outer and inner layers, respectively, our method uses $O(|L_0|+|L_1|^2)$ flips.

Authors: Samuel Avril, Yan Gerard, Guilherme D. da Fonseca

Consider a crossing-free polygonal path with $n$ vertices and a flip operation that removes an edge and adds another one, without changing the vertex set and keeping the path crossing free. It is conjectured that any two such paths can be transformed into each other by a sequence of flips. Recently, Kleist et al. (2024) proved the conjecture for the special cases of vertices arranged in two convex layers. However, the number of flips in their proof may be exponential. In this paper, we show that $O(n^2)$ flips suffice. More precisely, for a two-layer point set $S=L_0\cup L_1$, where $L_0$ and $L_1$ denote the outer and inner layers, respectively, our method uses $O(|L_0|+|L_1|^2)$ flips.

Non-simple Arrangements of Lines and Pseudolines with Many Triangles

from arXiv: Computational Geometry

Authors: Roman Parpalak, Denis Utkin

We study affine arrangements of an even number $n$ of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to $\lfloor n(n-5/2)/3\rfloor$. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of $n=6\cdot2^t+2$ ($8,14,26,50,\ldots$) straight lines with two triple points and $(n(n-5/2)+1)/3$ triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even $n\le12$. For $n=6,8,10,12,14$ the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for $n=8,12,14$ every optimal arrangement has multiple points. All optimal arrangements for even $n\le12$ are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value $n=11$, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For $n\equiv2\pmod 6$ this again gives one triangle more than the simple bound, in pseudolines up to $n=44$ and in straight lines for $n=20$ and $n=32$. Doubling these pseudoline arrangements gives members of the series $18\cdot2^t+2$ and $30\cdot2^t+2$ with the same excess, up to $n=146$ and $n=242$, respectively; those with $n=38$ and $n=74$ are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number $n$ of pseudolines has at most $\lfloor n(n-2)/3\rfloor-\lfloor n/6\rfloor$ triangles.

Authors: Roman Parpalak, Denis Utkin

We study affine arrangements of an even number $n$ of lines and pseudolines with many triangles. For simple ones, Blanc's bound limits the number of triangular faces to $\lfloor n(n-5/2)/3\rfloor$. Adapting the Bartholdi-Blanc-Loisel doubling to the even case, we construct an infinite series of arrangements of $n=6\cdot2^t+2$ ($8,14,26,50,\ldots$) straight lines with two triple points and $(n(n-5/2)+1)/3$ triangles, one more than the simple bound allows. We obtain exhaustive enumerations of triangle-maximal pseudoline arrangements, simple or not, for even $n\le12$. For $n=6,8,10,12,14$ the maximum number of triangles, for both lines and pseudolines, is $7,15,25,38,54$; for $n=8,12,14$ every optimal arrangement has multiple points. All optimal arrangements for even $n\le12$ are realizable by straight lines, except two arrangements of 10 pseudolines, which are not stretchable. For the odd value $n=11$, no arrangement with $33$ triangles exists even with multiple points. We also obtain even arrangements by adding a line to odd ones. For $n\equiv2\pmod 6$ this again gives one triangle more than the simple bound, in pseudolines up to $n=44$ and in straight lines for $n=20$ and $n=32$. Doubling these pseudoline arrangements gives members of the series $18\cdot2^t+2$ and $30\cdot2^t+2$ with the same excess, up to $n=146$ and $n=242$, respectively; those with $n=38$ and $n=74$ are realizable by straight lines. These results lead us to conjecture that an arrangement of an even number $n$ of pseudolines has at most $\lfloor n(n-2)/3\rfloor-\lfloor n/6\rfloor$ triangles.

Pair-Relative Partial Equivariance for Rotation-Robust Keypoint Descriptor Matching

from arXiv: Computational Geometry

Authors: Jongmin Lee

Learned keypoint descriptors trained on upright imagery often fail under large in-plane rotation. Steerers address this by applying a single exact SO(2) representation to the descriptor space, but this global group constraint can be too rigid for keypoint descriptors, whose angular responses need not follow a single group representation. We propose pair-relative partial equivariance, which measures equivariance under the empirical distribution of relative rotations in image pairs, and PaSteer, a relaxed descriptor-steering framework that allows the steering map to leave the exact representation manifold while retaining the lightweight, backbone-free design of steering. PaSteer instantiates this relaxation with two variants: an identity-preserving residual steerer (PaSteer-soft) and a free-form Fourier steerer (PaSteer-free). We show that the per-angle optimal linear steerer for a fixed non-equivariant descriptor generally does not obey the group composition law, explaining why exact representations can be suboptimal in the frozen-descriptor regime. On Roto-360, MegaDepth-1500, AIMS, and IMC-PT, PaSteer consistently improves rotation-robust matching over an exact steerer on frozen DeDoDe descriptors, especially under large relative rotations, with minimal upright degradation.

Authors: Jongmin Lee

Learned keypoint descriptors trained on upright imagery often fail under large in-plane rotation. Steerers address this by applying a single exact SO(2) representation to the descriptor space, but this global group constraint can be too rigid for keypoint descriptors, whose angular responses need not follow a single group representation. We propose pair-relative partial equivariance, which measures equivariance under the empirical distribution of relative rotations in image pairs, and PaSteer, a relaxed descriptor-steering framework that allows the steering map to leave the exact representation manifold while retaining the lightweight, backbone-free design of steering. PaSteer instantiates this relaxation with two variants: an identity-preserving residual steerer (PaSteer-soft) and a free-form Fourier steerer (PaSteer-free). We show that the per-angle optimal linear steerer for a fixed non-equivariant descriptor generally does not obey the group composition law, explaining why exact representations can be suboptimal in the frozen-descriptor regime. On Roto-360, MegaDepth-1500, AIMS, and IMC-PT, PaSteer consistently improves rotation-robust matching over an exact steerer on frozen DeDoDe descriptors, especially under large relative rotations, with minimal upright degradation.

On Guarding Line Arrangements

from arXiv: Computational Geometry

Authors: Valentin Brimkov

We study the problem of guarding a set of straight lines in the plane by selecting a minimum-cardinality set of points such that every line contains at least one selected point. This problem is known to be strongly NP-hard \cite{megiddo}. We investigate approximation algorithms for the problem and analyze their performance guarantees under various structural assumptions on the underlying line arrangement. In particular, we present approximation schemes based on set-cover techniques, relaxation methods, and properties of maximal lines. We establish constant-factor bounds on the approximation ratio for several classes of arrangements and derive relationships between the guarding number and the incidence parameters of the arrangement. Connections with Dirac-type incidence results are also explored.

Authors: Valentin Brimkov

We study the problem of guarding a set of straight lines in the plane by selecting a minimum-cardinality set of points such that every line contains at least one selected point. This problem is known to be strongly NP-hard \cite{megiddo}. We investigate approximation algorithms for the problem and analyze their performance guarantees under various structural assumptions on the underlying line arrangement. In particular, we present approximation schemes based on set-cover techniques, relaxation methods, and properties of maximal lines. We establish constant-factor bounds on the approximation ratio for several classes of arrangements and derive relationships between the guarding number and the incidence parameters of the arrangement. Connections with Dirac-type incidence results are also explored.

Graphs Have Structure: Declarative Manifolds, 3D Z-Order Monoid Roll-ups, and Zero-Simulation Visual Analytics in a Graph DBMS

from arXiv: Computational Geometry

Authors: Madhulatha Mandarapu, Sandeep Kunkunuru

Every real-world graph is embedded in an ambient structural manifold: proteins fold in Euclidean space ($R^3$), telecom networks sit on a planetary surface ($S^2 \times R$), and ontologies and calendars form discrete subsumption posets ($(V, \sqsubseteq)$). Yet property graph databases treat node coordinates as opaque payload scalars, leave spatial proximity edges unindexed, and force visualizers into $O(V^2)$ client-side force-directed simulations that destroy true geometry and truncate hierarchy roll-ups. We elevate ambient manifold structure to a first-class primitive in Samyama Graph. First, we introduce a declarative and auto-detected structural convention contract (samyama.structure/1) persisted in .sgsnap headers that binds node labels to Euclidean 3D, Geo-Spherical, Causal Spacetime, or Poset Hierarchy manifolds. Second, we show that both discrete posets (via Order-Embedded Hierarchy $[tin, tout]$ DFS intervals) and continuous 3D manifolds (via a 63-bit 3D Morton Z-order Structural-Geometric Index, SGI) reduce regional aggregation to 1D rank-interval monoid folds answered directly from index-resident Fenwick trees (SUM, COUNT) and Segment Trees (MIN, MAX). At $100,000$ 3D nodes, SGI builds in $28.14$ ms and answers Morton-interval SUM and MAX roll-ups in $0.048\,μ$s and $0.104\,μ$s---$8,345.1\times$ and $3,824.4\times$ faster than property scans. Third, we introduce 3D voxel-hashed virtual contact edges weighted by Topological--Geometric Stress $σ(u,v) = d_{graph}(u,v) / d_{manifold}(u,v)$ to isolate long-range tertiary folds, plus manifold-guided $A^*$ shortest-path search that cuts explored states by up to $3.80\times$ over Dijkstra. Fourth, we replace client layout simulation with server-computed 3D Cone-Tree/Ribbon coordinates and a non-WebGL 2.5D Isometric projection with painter's-order depth sorting and sub-millisecond Level-of-Detail (LoD) macro-cuts.

Authors: Madhulatha Mandarapu, Sandeep Kunkunuru

Every real-world graph is embedded in an ambient structural manifold: proteins fold in Euclidean space ($R^3$), telecom networks sit on a planetary surface ($S^2 \times R$), and ontologies and calendars form discrete subsumption posets ($(V, \sqsubseteq)$). Yet property graph databases treat node coordinates as opaque payload scalars, leave spatial proximity edges unindexed, and force visualizers into $O(V^2)$ client-side force-directed simulations that destroy true geometry and truncate hierarchy roll-ups. We elevate ambient manifold structure to a first-class primitive in Samyama Graph. First, we introduce a declarative and auto-detected structural convention contract (samyama.structure/1) persisted in .sgsnap headers that binds node labels to Euclidean 3D, Geo-Spherical, Causal Spacetime, or Poset Hierarchy manifolds. Second, we show that both discrete posets (via Order-Embedded Hierarchy $[tin, tout]$ DFS intervals) and continuous 3D manifolds (via a 63-bit 3D Morton Z-order Structural-Geometric Index, SGI) reduce regional aggregation to 1D rank-interval monoid folds answered directly from index-resident Fenwick trees (SUM, COUNT) and Segment Trees (MIN, MAX). At $100,000$ 3D nodes, SGI builds in $28.14$ ms and answers Morton-interval SUM and MAX roll-ups in $0.048\,μ$s and $0.104\,μ$s---$8,345.1\times$ and $3,824.4\times$ faster than property scans. Third, we introduce 3D voxel-hashed virtual contact edges weighted by Topological--Geometric Stress $σ(u,v) = d_{graph}(u,v) / d_{manifold}(u,v)$ to isolate long-range tertiary folds, plus manifold-guided $A^*$ shortest-path search that cuts explored states by up to $3.80\times$ over Dijkstra. Fourth, we replace client layout simulation with server-computed 3D Cone-Tree/Ribbon coordinates and a non-WebGL 2.5D Isometric projection with painter's-order depth sorting and sub-millisecond Level-of-Detail (LoD) macro-cuts.

Every $k$-Quasi-Planar Graph Has Linearly Many Edges

from arXiv: Computational Geometry

Authors: Ankan Sadhu

For every fixed integer $k\ge2$, we prove that every simple graph on $n$ vertices drawn in the plane with no $k$ pairwise crossing edges has at most $C_kn$ edges, where $C_k$ depends only on $k$. We allow any finite number of proper crossings between two edges, including edges with a common endpoint. For $n\ge3$, removing a small disk around each vertex turns the drawing into a family of arcs on a sphere with $n$ boundary circles. Each arc joins two different boundary circles, and different edges have different unordered pairs of endpoint circles. The arcs are therefore essential and pairwise nonhomotopic, with endpoints allowed to move along their boundary circles during a homotopy. Any family of pairwise intersecting lifts to the universal cover projects to distinct graph edges that cross pairwise. Its size is consequently at most $k-1$. Beeker and Lazarovich's theorem on surface patterns then bounds the number of arcs. Their estimate is linear in the size of a triangulation, and an explicit triangulation of the $n$-holed sphere has linearly many edges and triangles.

Authors: Ankan Sadhu

For every fixed integer $k\ge2$, we prove that every simple graph on $n$ vertices drawn in the plane with no $k$ pairwise crossing edges has at most $C_kn$ edges, where $C_k$ depends only on $k$. We allow any finite number of proper crossings between two edges, including edges with a common endpoint. For $n\ge3$, removing a small disk around each vertex turns the drawing into a family of arcs on a sphere with $n$ boundary circles. Each arc joins two different boundary circles, and different edges have different unordered pairs of endpoint circles. The arcs are therefore essential and pairwise nonhomotopic, with endpoints allowed to move along their boundary circles during a homotopy. Any family of pairwise intersecting lifts to the universal cover projects to distinct graph edges that cross pairwise. Its size is consequently at most $k-1$. Beeker and Lazarovich's theorem on surface patterns then bounds the number of arcs. Their estimate is linear in the size of a triangulation, and an explicit triangulation of the $n$-holed sphere has linearly many edges and triangles.