Theory of Computing Report

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.

Wednesday, November 04 2026, 01:00

Quantum Advantage has Few Complexity Implications

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.

Wednesday, November 04 2026, 01:00

Universal Hardness From Masking Channels

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.

Wednesday, November 04 2026, 01:00

Some algebraic and geometric computations in CH via Chow forms

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

On the Complexity of Exact Autobidding Equilibria

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.

Wednesday, November 04 2026, 01:00

On the Hardness of Approximate Quantum Circuit Membership and Optimization

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.

Wednesday, November 04 2026, 01:00

Monodromy Computation Model

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.

Wednesday, November 04 2026, 01:00

On the Role of Geometry in Solving Variational Inequalities

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.

Wednesday, November 04 2026, 01:00

The Complexity of ReLU Network Synthesis from a Parameter Multiset

Authors: Dimitrios Myrisiotis

We ask how hard it is to build a neural network when its parameter values are given in advance. An instance supplies a multiset of rational numbers, a target behavior, and a bound on the number of neurons, output included. The question is whether one can choose the graph of a ReLU network within the bound, and place the numbers on its edges and biases so that it meets the target. This separates the numbers from the architecture; permutation training instead fixes the hidden units and only rearranges the output weights. Matching two labeled examples exactly is already $\mathsf{NP}$-complete; we ask what changes when the target must hold on every input or on average. Agreeing with a Boolean circuit on every Boolean input is $Σ_2^{\mathsf P}$-complete, already for four neurons. Agreeing with a given ReLU network on every real input lies in $\mathsf{DP}$ for every fixed neuron bound and is $\mathsf{DP}$-complete for three neurons. Approximating a ReLU network on the unit cube within absolute error one everywhere is $Σ_2^{\mathsf P}$-complete, even with no hidden neurons and all numbers equal and positive. Deciding whether some network has expected squared loss at most a threshold is $\mathsf{NP}^{\mathsf{PP}}$-complete for uniform Boolean inputs. For uniform inputs on the unit cube and for standard Gaussian inputs with a positive threshold, the same holds for two neurons when a positive risk gap, given in binary and possibly exponentially small, separates yes from no instances. Without a gap, the cube version is complete for guessing a polynomial-length string, then comparing a sum of exponentially many rationals with a threshold, all computable in polynomial time from the input and the guess. Since the graph is chosen too, the reductions must rule out every other network built from the same numbers.

Authors: Dimitrios Myrisiotis

We ask how hard it is to build a neural network when its parameter values are given in advance. An instance supplies a multiset of rational numbers, a target behavior, and a bound on the number of neurons, output included. The question is whether one can choose the graph of a ReLU network within the bound, and place the numbers on its edges and biases so that it meets the target. This separates the numbers from the architecture; permutation training instead fixes the hidden units and only rearranges the output weights. Matching two labeled examples exactly is already $\mathsf{NP}$-complete; we ask what changes when the target must hold on every input or on average. Agreeing with a Boolean circuit on every Boolean input is $Σ_2^{\mathsf P}$-complete, already for four neurons. Agreeing with a given ReLU network on every real input lies in $\mathsf{DP}$ for every fixed neuron bound and is $\mathsf{DP}$-complete for three neurons. Approximating a ReLU network on the unit cube within absolute error one everywhere is $Σ_2^{\mathsf P}$-complete, even with no hidden neurons and all numbers equal and positive. Deciding whether some network has expected squared loss at most a threshold is $\mathsf{NP}^{\mathsf{PP}}$-complete for uniform Boolean inputs. For uniform inputs on the unit cube and for standard Gaussian inputs with a positive threshold, the same holds for two neurons when a positive risk gap, given in binary and possibly exponentially small, separates yes from no instances. Without a gap, the cube version is complete for guessing a polynomial-length string, then comparing a sum of exponentially many rationals with a threshold, all computable in polynomial time from the input and the guess. Since the graph is chosen too, the reductions must rule out every other network built from the same numbers.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Counterexamples to the Happy Edge Conjecture in Plane Spanning Trees

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].

Wednesday, November 04 2026, 01:00

Polyominoes Folding into Four and 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.

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.

Wednesday, November 04 2026, 01:00

Breaking the quantum barriers for Graph Collision

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.

Wednesday, November 04 2026, 01:00

Agnostic Learning by Imitating

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.

Wednesday, November 04 2026, 01:00

Finding Hypercliques Faster Than Brute Force

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.

Wednesday, November 04 2026, 01:00

Differentially Oblivious Priority Queues with Sublogarithmic Cost

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.

Wednesday, November 04 2026, 01:00

Abelian embeddings make useful promises: approximation of detrivialised equations

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 $\mathbb{Z}_p$.

Wednesday, November 04 2026, 01:00

Entropy-Weighted Soundness for Multi-Answer Birthday Graphs

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.

Wednesday, November 04 2026, 01:00

Binary Huffman Coding with Unequal Letter Costs Is NP-Complete

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Semidirect sums of tensors and the matrix multiplication exponent

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.

Wednesday, November 04 2026, 01:00

On the optimality of truncation among binary linear sketches

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.

Wednesday, November 04 2026, 01:00

A King for Almost Everyone

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].

Wednesday, November 04 2026, 01:00

Faster Planar Diameter using Pseudoline Envelopes

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 $\widetilde{O}(n^{3/2})$.

Wednesday, November 04 2026, 01:00

Geometric Bin Packing Beyond Convexity: Approximation Schemes without Augmentation

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.

Wednesday, November 04 2026, 01:00

Roundtrip All-Pairs Shortest Paths in Directed Graphs

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.

Wednesday, November 04 2026, 01:00

Gaussian Partitioning for Reliable Learning of Multi-Index Models

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.

Wednesday, November 04 2026, 01:00

Catalan Hashing: Efficient Linear Probing for Variable-Length Items

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.

Wednesday, November 04 2026, 01:00

Santa Claus and Weighted Nash Social Welfare with Subadditive Valuations

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Efficient Equilibrium Computation in Colonel Blotto Games

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.

Wednesday, November 04 2026, 01:00

Logarithmic Embeddings into Probabilistic Spanning Trees for Distances and Capacities

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.

Wednesday, November 04 2026, 01:00

Path Decompositions of Sparse Graphs via Gaussian Sweeps and Faster Exponential 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 $m$).

Wednesday, November 04 2026, 01:00

Recognizing Label-Connected 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.

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.

Wednesday, November 04 2026, 01:00

Quantum Query Lower Bounds from Classical Moment Matching

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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).

Wednesday, November 04 2026, 01:00

A Low-Diameter Nash-Williams--Tutte Theorem

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.

Wednesday, November 04 2026, 01:00

The Power of Interaction in Locally Edge-Private Subgraph Counting

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 $n$.

Wednesday, November 04 2026, 01:00

Matchings under One-Sided Preferences with Soft Quotas

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.

Wednesday, November 04 2026, 01:00

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

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].

Wednesday, November 04 2026, 01:00

Metric Facility Location under Batched Random Order

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Near-Linear Lower Bounds for Testing Directed Acyclicity

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.

Wednesday, November 04 2026, 01:00

The Power of Bidegeneracy for Hypergraph Data Structures

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.

Wednesday, November 04 2026, 01:00

Local Distributed Computation under Crash Failures

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Faster Regular Expression Matching via Two-Dimensional Tabulation

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Optimal Approximation for Constrained Correlation Clustering via 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.

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Optimal (Randomized) Compact Routing Schemes

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.

Wednesday, November 04 2026, 01:00

Approximation Algorithms for Multi-Criteria Network Design Problems

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.

Wednesday, November 04 2026, 01:00

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

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 $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Finding Integer Roots of Polynomials in Nearly Optimal Time

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.

Wednesday, November 04 2026, 01:00

Pure Differentially Private Query Release via Laplace Input Smoothing

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

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

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 $Θ(1/\varepsilon)$.

Wednesday, November 04 2026, 01:00

Dynamic Steiner Tree under Edge Updates

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.

Wednesday, November 04 2026, 01:00

XOR-Functions and Beyond in the Random Query Model

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.

Wednesday, November 04 2026, 01:00

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

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.

Wednesday, November 04 2026, 01:00

Metric TSP in Sublinear Time: Breaking Two Approximation

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 $Θ(n^2)$.

Wednesday, November 04 2026, 01:00

Testing Implies Distance Estimation with Polynomial Loss

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.

Wednesday, November 04 2026, 01:00

Online Metric Matching with 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.

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.

Wednesday, November 04 2026, 01:00

Incremental Open Addressing Has No Time-Space Tradeoff

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.

Wednesday, November 04 2026, 01:00

A Constant-Factor Approximation for Capacitated Facility Location with Outliers

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 $60/11+\eps$.

Wednesday, November 04 2026, 01:00

The Hyperclique Hypothesis is False

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.

Wednesday, November 04 2026, 01:00

A Family of Well-Behaved Multiflow Instances

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.

Wednesday, November 04 2026, 01:00

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.

Tuesday, November 03 2026, 23:28

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.

Tuesday, November 03 2026, 17:20

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.

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By Ben Recht

Tuesday, November 03 2026, 15:45

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.

Tuesday, November 03 2026, 15:38

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.

Tuesday, November 03 2026, 13:40

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.

Tuesday, November 03 2026, 11:52

A Circuit Complexity Trichotomy For Regular Languages

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.

Tuesday, November 03 2026, 01:00

Determinant boundaries and gaps in determinantal 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.

Tuesday, November 03 2026, 01:00

Tilted Convexity and an Exact Offline-Online Separation

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

Subpolynomial-Query Proximal Sampling Beyond Log-Concavity

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.

Tuesday, November 03 2026, 01:00

Exact Agreement Expansion up to the Top via Non-polarity

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

ETH is Dead, Long Live SH

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 $\varepsilon>0$.

Tuesday, November 03 2026, 01:00

Exponential Lower Bounds for Products of Cubic Polynomials over Prime Fields

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.

Tuesday, November 03 2026, 01:00

State Synthesis with an NP oracle

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence

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 $n(n-1)$.

Tuesday, November 03 2026, 01:00

Factor Closure in Noncommutative Algebraic 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.

Tuesday, November 03 2026, 01:00

The Runtime-Moment Frontier for Universal Sampling

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.

Tuesday, November 03 2026, 01:00

Quantum Lower Bounds for Collision Problems

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.

Tuesday, November 03 2026, 01:00

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

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 $85/86$.

Tuesday, November 03 2026, 01:00

The Exact Communication Threshold for Parallel Repetition

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.

Tuesday, November 03 2026, 01:00

Nonnegative Rank Bounds for the Algebraic Monotone Calculus

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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 $N$.

Tuesday, November 03 2026, 01:00

FIXP-Completeness in Second-Price Auctions

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.

Tuesday, November 03 2026, 01:00

The Complexity of Choosing Parity Projections

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

Non-simple Arrangements of Lines and Pseudolines with Many 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.

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

On Guarding Line Arrangements

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

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

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.

Tuesday, November 03 2026, 01:00

CuPH: Parallel Cubical Persistent Homology via Level-Synchronous Component Tracking

Authors: Guangyu Meng, Pengfei Gu, Erin Wolf Chambers, Danny Z. Chen

Persistent homology (PH) is a mathematically principled multi-scale topological descriptor of shape and connectivity in data. It is widely applied in image classification, segmentation, and shape analysis across medical and aerial imaging, providing structural information that pixel-wise deep learning may overlook. Standard cubical PH decomposes an image into cubical cells (vertices, edges, and squares) and reduces them sequentially in filtration order. This sequential reduction has $Θ(n)$ parallel depth for an $n$-pixel image, limiting within-image GPU parallelism and constraining its use in topology-aware deep learning. We introduce CuPH, a GPU-based parallel algorithm for computing cubical PH on 2D images that operates directly on pixels rather than cubical cells. CuPH buckets pixels by filtration values, activates them with a level-synchronous sweep, and tracks connected components with a lock-free union-find method. We prove that CuPH produces persistence diagrams identical to standard cubical PH on integer inputs, with a stability bound for floating-point inputs. CuPH reduces parallel depth from $Θ(n)$ to $O(L\log n)$ for an image with $L$ intensity levels. We implement the level sweep as a single CUDA kernel, eliminating cross-level kernel-launch overhead. Empirically, CuPH is $18.6$--$99.6\times$ faster than the existing GPU baseline (CMP). As a drop-in PH backend, CuPH accelerates topology-aware classification ($4.2$--$16.7\times$) and segmentation ($8.3$--$44\times$) while preserving accuracy. It also enables computing full-image PH losses on high-resolution images, improving segmentation Dice by $2.2$--$4.6$ points over patch-local PH baselines and accelerating training by $25$--$31\times$.

Authors: Guangyu Meng, Pengfei Gu, Erin Wolf Chambers, Danny Z. Chen

Persistent homology (PH) is a mathematically principled multi-scale topological descriptor of shape and connectivity in data. It is widely applied in image classification, segmentation, and shape analysis across medical and aerial imaging, providing structural information that pixel-wise deep learning may overlook. Standard cubical PH decomposes an image into cubical cells (vertices, edges, and squares) and reduces them sequentially in filtration order. This sequential reduction has $Θ(n)$ parallel depth for an $n$-pixel image, limiting within-image GPU parallelism and constraining its use in topology-aware deep learning. We introduce CuPH, a GPU-based parallel algorithm for computing cubical PH on 2D images that operates directly on pixels rather than cubical cells. CuPH buckets pixels by filtration values, activates them with a level-synchronous sweep, and tracks connected components with a lock-free union-find method. We prove that CuPH produces persistence diagrams identical to standard cubical PH on integer inputs, with a stability bound for floating-point inputs. CuPH reduces parallel depth from $Θ(n)$ to $O(L\log n)$ for an image with $L$ intensity levels. We implement the level sweep as a single CUDA kernel, eliminating cross-level kernel-launch overhead. Empirically, CuPH is $18.6$--$99.6\times$ faster than the existing GPU baseline (CMP). As a drop-in PH backend, CuPH accelerates topology-aware classification ($4.2$--$16.7\times$) and segmentation ($8.3$--$44\times$) while preserving accuracy. It also enables computing full-image PH losses on high-resolution images, improving segmentation Dice by $2.2$--$4.6$ points over patch-local PH baselines and accelerating training by $25$--$31\times$.

Tuesday, November 03 2026, 01:00

From Static Models to Editable CAD: A Design-Intent-Aware Framework for Reverse Parametrization

Authors: Yurim Lee, Sumin Lee, Namwoo Kang

Neutral CAD formats such as STEP typically preserve geometry but not feature histories or parameter dependencies. Thus, they are inherently limited for iterative design that requires modifications, a critical limitation for design optimization that relies on iterative processes. This study presents a design-intent-aware reverse-parametrization framework that converts history-free STEP parts into editable Grasshopper definitions and represents their assembly relations. Without task-specific training, the framework combines kernel-measured features, geometric relations, a compressed attributed adjacency graph, and multiview renderings. An LLM assigns dimensions independent, fixed, or dependent roles, confirms candidate joints, and interprets verification reports. Kernel- and rule-based modules perform measurement, parameter binding, geometry construction, and verification. Evaluation distinguishes default-shape preservation, design-intent reflection, and edit validity, with the latter examined through assembly-level editing examples involving checks for geometry regeneration, joint alignment, interference, and mechanism closure. As an example, five components of a mechanical mechanism were used, and across 25 runs, the mean global and analytic-feature-region intersection over unions (IoUs) were 0.984 and 0.983, respectively; all runs passed the combined shape-preservation criterion. In a separate request experiment, the requested dimension was selected as an independent variable in 19 of 20 runs. Product-level examples show coordinated changes in part geometry and assembly design while preserving assembly relations and avoiding inter-part interference. These results demonstrate editable reconstruction of history-free CAD models under the evaluated geometric and assembly constraints, indicating potential for subsequent design optimization.

Authors: Yurim Lee, Sumin Lee, Namwoo Kang

Neutral CAD formats such as STEP typically preserve geometry but not feature histories or parameter dependencies. Thus, they are inherently limited for iterative design that requires modifications, a critical limitation for design optimization that relies on iterative processes. This study presents a design-intent-aware reverse-parametrization framework that converts history-free STEP parts into editable Grasshopper definitions and represents their assembly relations. Without task-specific training, the framework combines kernel-measured features, geometric relations, a compressed attributed adjacency graph, and multiview renderings. An LLM assigns dimensions independent, fixed, or dependent roles, confirms candidate joints, and interprets verification reports. Kernel- and rule-based modules perform measurement, parameter binding, geometry construction, and verification. Evaluation distinguishes default-shape preservation, design-intent reflection, and edit validity, with the latter examined through assembly-level editing examples involving checks for geometry regeneration, joint alignment, interference, and mechanism closure. As an example, five components of a mechanical mechanism were used, and across 25 runs, the mean global and analytic-feature-region intersection over unions (IoUs) were 0.984 and 0.983, respectively; all runs passed the combined shape-preservation criterion. In a separate request experiment, the requested dimension was selected as an independent variable in 19 of 20 runs. Product-level examples show coordinated changes in part geometry and assembly design while preserving assembly relations and avoiding inter-part interference. These results demonstrate editable reconstruction of history-free CAD models under the evaluated geometric and assembly constraints, indicating potential for subsequent design optimization.

Tuesday, November 03 2026, 01:00

Hamiltonian Cycle Construction in Rectangular Grid Graphs with Three-Stacked Triangular Holes

Authors: Cecilia Chen, Dianna Xu

The Hamiltonian cycle problem is NP-complete for general grid graphs, while several structured classes admit efficient construction algorithms. In this paper, we investigate Hamiltonian cycle construction in rectangular grid graphs containing a three-stacked right-triangular hole. We explore two complementary approaches based on the geometric structure of the hole and the parity properties of rectangular subgraphs. The first approach constructs a Hamiltonian path in the complementary reversed staircase region, extends it into an inner Hamiltonian cycle, and merges it with a cycle in the surrounding graph. The second approach partitions the graph into six rectangular subgraphs and constructs a global Hamiltonian cycle by merging local cycles along compatible boundaries. When odd-sized rectangular subgraphs prevent direct cycle construction, we develop parity-based repartitioning techniques that transform selected configurations into even-sized rectangles. We provide constructions for adjacent and nonadjacent odd-sized configurations and investigate reductions for cases involving multiple odd-sized regions. These results establish a constructive framework for Hamiltonian cycles in grid graphs with staircase-shaped holes and identify the remaining cases requiring further analysis.

Authors: Cecilia Chen, Dianna Xu

The Hamiltonian cycle problem is NP-complete for general grid graphs, while several structured classes admit efficient construction algorithms. In this paper, we investigate Hamiltonian cycle construction in rectangular grid graphs containing a three-stacked right-triangular hole. We explore two complementary approaches based on the geometric structure of the hole and the parity properties of rectangular subgraphs. The first approach constructs a Hamiltonian path in the complementary reversed staircase region, extends it into an inner Hamiltonian cycle, and merges it with a cycle in the surrounding graph. The second approach partitions the graph into six rectangular subgraphs and constructs a global Hamiltonian cycle by merging local cycles along compatible boundaries. When odd-sized rectangular subgraphs prevent direct cycle construction, we develop parity-based repartitioning techniques that transform selected configurations into even-sized rectangles. We provide constructions for adjacent and nonadjacent odd-sized configurations and investigate reductions for cases involving multiple odd-sized regions. These results establish a constructive framework for Hamiltonian cycles in grid graphs with staircase-shaped holes and identify the remaining cases requiring further analysis.

Tuesday, November 03 2026, 01:00

Catenary Visibility: A Geometric Framework for Planning with Tethered Robots

Authors: B. Ábrego, J. M. Díaz-Báñez, S. Fernández, D. Flores-Peñaloza, V. Sánchez-Canales

This paper introduces \emph{catenary visibility}, a new geometric visibility concept for a robotic system consisting of a ground robot tethered to an aerial drone. The tether is modeled as an extensible catenary curve with bounded length, providing sustained power and significantly extending operational range. We define a point on the ground plane as \emph{catenary-visible} from a target point in 3D if there exists a collision-free catenary connecting them, given a set of obstacles. This paper introduces a geometric framework for visibility-aware planning of a tethered marsupial system, addressing the computational challenge of determining the feasible deployment locus: the collection of points in the ground plane that are catenary-visible from a given target in 3D space. We begin by solving the 2D subproblem, where the obstacles, target, and the ground vehicle's line of motion are all constrained to a single vertical plane. Our algorithm for this case computes the complete set of feasible launch points along the motion line and simultaneously determines the range of viable cable lengths for each. We also consider the query problem of finding all collision-free length ranges for a given location of the ground robot. Building upon this planar solution, we then generalize the approach to three dimensions, deriving a 3D solution for a constrained class of obstacles.

Authors: B. Ábrego, J. M. Díaz-Báñez, S. Fernández, D. Flores-Peñaloza, V. Sánchez-Canales

This paper introduces \emph{catenary visibility}, a new geometric visibility concept for a robotic system consisting of a ground robot tethered to an aerial drone. The tether is modeled as an extensible catenary curve with bounded length, providing sustained power and significantly extending operational range. We define a point on the ground plane as \emph{catenary-visible} from a target point in 3D if there exists a collision-free catenary connecting them, given a set of obstacles. This paper introduces a geometric framework for visibility-aware planning of a tethered marsupial system, addressing the computational challenge of determining the feasible deployment locus: the collection of points in the ground plane that are catenary-visible from a given target in 3D space. We begin by solving the 2D subproblem, where the obstacles, target, and the ground vehicle's line of motion are all constrained to a single vertical plane. Our algorithm for this case computes the complete set of feasible launch points along the motion line and simultaneously determines the range of viable cable lengths for each. We also consider the query problem of finding all collision-free length ranges for a given location of the ground robot. Building upon this planar solution, we then generalize the approach to three dimensions, deriving a 3D solution for a constrained class of obstacles.

Tuesday, November 03 2026, 01:00

Get Permanent Results Faster by Counting Stars

Authors: Radu Curticapean

We show that $n \times n$ permanents can be computed with $O^*{(2-\varepsilon)^n}$ arithmetic operations over every commutative ring, for $\varepsilon = 10^{-40}$. This solves a problem stated in Knuth's The Art of Computer Programming. Our algorithm implements the classical Binet--Minc identity with dynamic programming and careful bookkeeping. It also gives algebraic branching programs of size $O^*{(2-\varepsilon)^n}$.

Authors: Radu Curticapean

We show that $n \times n$ permanents can be computed with $O^*{(2-\varepsilon)^n}$ arithmetic operations over every commutative ring, for $\varepsilon = 10^{-40}$. This solves a problem stated in Knuth's The Art of Computer Programming. Our algorithm implements the classical Binet--Minc identity with dynamic programming and careful bookkeeping. It also gives algebraic branching programs of size $O^*{(2-\varepsilon)^n}$.

Tuesday, November 03 2026, 01:00

Online Algorithms Cannot Find Isolated Perceptron Solutions

Authors: Shuangping Li, Gabe Schoenbach

The binary perceptron problem asks for a length-$n$ sign vector in the intersection of $\lfloor αn\rfloor$-many independent random Gaussian half-spaces. A striking feature of this model is strong freezing: at every constraint density $α> 0$, all but a vanishing fraction of solutions are isolated -- $Ω(n)$-far from all others, in Hamming distance. Such solutions are believed to be algorithmically inaccessible. Nevertheless, efficient algorithms find solutions at some constraint densities $α$. One possible resolution of this apparent contradiction is that the solution space also contains rare well-connected structures, and that algorithms succeed by locating them. Gong, Huang, Li and Sellke (FOCS 2026) proved this for stable algorithms, by showing that such algorithms cannot locate isolated solutions with probability more than $\approx 0.84$. In this work, we study online algorithms, which receive the columns of the constraint matrix one by one and sign each irrevocably upon arrival. This is a widely used class of algorithms in discrepancy minimization, and it includes the algorithm of Jo and Lee (2026), which has one of the best provable guarantees for the binary perceptron. We prove that at any constraint density, any online algorithm outputs an isolated solution with probability $o(1)$. Our proof is based on the observation that the signs an online algorithm has already fixed can be completed to at most one isolated solution, so a successful algorithm must choose the unique correct sign for each of the last few columns. Using the exchangeability of these columns, we show that such an algorithm must match a suffix-parallel baseline, which signs each of them as if it were the next column to arrive. We then use an anti-concentration argument to show that such a baseline rarely produces a unique completion.

Authors: Shuangping Li, Gabe Schoenbach

The binary perceptron problem asks for a length-$n$ sign vector in the intersection of $\lfloor αn\rfloor$-many independent random Gaussian half-spaces. A striking feature of this model is strong freezing: at every constraint density $α> 0$, all but a vanishing fraction of solutions are isolated -- $Ω(n)$-far from all others, in Hamming distance. Such solutions are believed to be algorithmically inaccessible. Nevertheless, efficient algorithms find solutions at some constraint densities $α$. One possible resolution of this apparent contradiction is that the solution space also contains rare well-connected structures, and that algorithms succeed by locating them. Gong, Huang, Li and Sellke (FOCS 2026) proved this for stable algorithms, by showing that such algorithms cannot locate isolated solutions with probability more than $\approx 0.84$. In this work, we study online algorithms, which receive the columns of the constraint matrix one by one and sign each irrevocably upon arrival. This is a widely used class of algorithms in discrepancy minimization, and it includes the algorithm of Jo and Lee (2026), which has one of the best provable guarantees for the binary perceptron. We prove that at any constraint density, any online algorithm outputs an isolated solution with probability $o(1)$. Our proof is based on the observation that the signs an online algorithm has already fixed can be completed to at most one isolated solution, so a successful algorithm must choose the unique correct sign for each of the last few columns. Using the exchangeability of these columns, we show that such an algorithm must match a suffix-parallel baseline, which signs each of them as if it were the next column to arrive. We then use an anti-concentration argument to show that such a baseline rarely produces a unique completion.

Tuesday, November 03 2026, 01:00

Worst-Case to Average-Case Reductions for Basic Linear Algebra

Authors: Or Zamir

We study worst-case to average-case reductions for basic linear-algebraic problems over finite fields. A seminal work of Blum, Luby, and Rubinfeld established that any algorithm that successfully multiplies more than $75\%$ of matrix pairs over any finite field, implies an algorithm with essentially the same running time multiplying any pair of matrices. A sequence of follow-ups reduced the needed fraction of successful input pairs to any positive constant. Despite most of the other basic linear algebraic operations being computationally equivalent to matrix multiplication in the worst case, no such worst-case to average-case reductions were known for them. Recently, Vaikuntanathan and Zamir introduced a cryptographic primitive called trapdoor matrices and used them to prove such worst-case to average-case reductions for matrix inversion, solving a linear system, and computing the determinant, all conditioned on common cryptographic assumptions. Our main result is the first unconditional time-preserving worst-case to average-case reduction for matrix inversion. Any algorithm that correctly inverts any constant positive fraction of uniformly random invertible matrices implies a worst-case such algorithm. We also significantly strengthen the conditional reductions based on trapdoored matrices: We establish reductions for deciding invertibility and computing rank at the optimal success thresholds, and qualitatively improve the success threshold needed in the reduction for determinant computation---while previous reductions all required the success probability to be at least $1/2$, we show that any fixed positive average-case success probability suffices for exact worst-case determinant computation over infinitely many finite fields.

Authors: Or Zamir

We study worst-case to average-case reductions for basic linear-algebraic problems over finite fields. A seminal work of Blum, Luby, and Rubinfeld established that any algorithm that successfully multiplies more than $75\%$ of matrix pairs over any finite field, implies an algorithm with essentially the same running time multiplying any pair of matrices. A sequence of follow-ups reduced the needed fraction of successful input pairs to any positive constant. Despite most of the other basic linear algebraic operations being computationally equivalent to matrix multiplication in the worst case, no such worst-case to average-case reductions were known for them. Recently, Vaikuntanathan and Zamir introduced a cryptographic primitive called trapdoor matrices and used them to prove such worst-case to average-case reductions for matrix inversion, solving a linear system, and computing the determinant, all conditioned on common cryptographic assumptions. Our main result is the first unconditional time-preserving worst-case to average-case reduction for matrix inversion. Any algorithm that correctly inverts any constant positive fraction of uniformly random invertible matrices implies a worst-case such algorithm. We also significantly strengthen the conditional reductions based on trapdoored matrices: We establish reductions for deciding invertibility and computing rank at the optimal success thresholds, and qualitatively improve the success threshold needed in the reduction for determinant computation---while previous reductions all required the success probability to be at least $1/2$, we show that any fixed positive average-case success probability suffices for exact worst-case determinant computation over infinitely many finite fields.

Tuesday, November 03 2026, 01:00

Full-Rank Constructions of Explicit Lossless Expanders and Good Small-Set qLTCs

Authors: Yali Aharon-Holzman, Michael Ilan, Assaf Reiner, Amnon Ta-Shma

We construct explicit two-sided lossless expanders whose binary adjacency matrices have full rank, and use them to obtain explicit asymptotically good quantum LDPC codes whose specified X- and Z-check matrices have full row rank and satisfy small-set soundness, or confinement, on a linear range. Our lossless expanders are a variant of those of Hsieh, Lubotzky, Mohanty, Reiner, and Zhang, with a free action of a finite 2-group. Under such an action, injectivity and surjectivity of the adjacency matrix are detected on a fixed-size matrix indexed by vertex orbits. Adding a sparse factor to the gadget at the middle vertices of a single type then enforces full rank, with a controlled loss in expansion. To obtain the 2-group action, we observe that reducing the generators of the Lubotzky-Phillips-Sarnak Ramanujan graphs modulo powers of 2, instead of modulo an odd prime, gives Ramanujan Cayley graphs on finite 2-groups. This construction may be of independent interest. Plugging two such expanders into the balanced product of Lin and Hsieh gives the quantum codes, since full rank of the input graphs is inherited by the check matrices. Our construction supplies the code family sought by Hong, Guo, and Lucas. Applying their analysis, within their framework and under their dynamical assumptions, gives quantum memories whose free-energy density is analytic at every positive temperature and whose guaranteed lifetime is $exp(Ω(N))$ at sufficiently low fixed positive temperature, where N is the blocklength. This improves the previously guaranteed stretched-exponential timescale.

Authors: Yali Aharon-Holzman, Michael Ilan, Assaf Reiner, Amnon Ta-Shma

We construct explicit two-sided lossless expanders whose binary adjacency matrices have full rank, and use them to obtain explicit asymptotically good quantum LDPC codes whose specified X- and Z-check matrices have full row rank and satisfy small-set soundness, or confinement, on a linear range. Our lossless expanders are a variant of those of Hsieh, Lubotzky, Mohanty, Reiner, and Zhang, with a free action of a finite 2-group. Under such an action, injectivity and surjectivity of the adjacency matrix are detected on a fixed-size matrix indexed by vertex orbits. Adding a sparse factor to the gadget at the middle vertices of a single type then enforces full rank, with a controlled loss in expansion. To obtain the 2-group action, we observe that reducing the generators of the Lubotzky-Phillips-Sarnak Ramanujan graphs modulo powers of 2, instead of modulo an odd prime, gives Ramanujan Cayley graphs on finite 2-groups. This construction may be of independent interest. Plugging two such expanders into the balanced product of Lin and Hsieh gives the quantum codes, since full rank of the input graphs is inherited by the check matrices. Our construction supplies the code family sought by Hong, Guo, and Lucas. Applying their analysis, within their framework and under their dynamical assumptions, gives quantum memories whose free-energy density is analytic at every positive temperature and whose guaranteed lifetime is $exp(Ω(N))$ at sufficiently low fixed positive temperature, where N is the blocklength. This improves the previously guaranteed stretched-exponential timescale.

Tuesday, November 03 2026, 01:00

Fine-Grained Complexity Across the Computational Phase Transition

Authors: Zhe Ju, Jingcheng Liu, Yitong Yin

How does a computational problem transition from easy to hard? In the hard-core model, approximate counting and sampling are polynomial-time tractable up to tree uniqueness, but worst-case intractable at any constant relative distance above it unless $\mathsf{NP}=\mathsf{RP}$. We quantify the transition from polynomial to exponential mixing for classical Glauber dynamics. On $n$-vertex graphs of maximum degree $d\ge3$, at fugacity $λ=(1+δ)λ_{\mathrm c}(d)$ with sufficiently small $δ\ge0$ that may vanish with $n$, its worst-case mixing time has the form $\exp\{Θ(nδ^2/d)\}n^{1+Θ(1)}$. The upper bound is degree-uniform, and the matching lower bound holds for every fixed $d$. Annealing gives approximate counting with the same running-time upper bound at fixed accuracy. For general sampling and counting algorithms, we establish a robust fine-grained reduction from SAT without assuming a lower bound on SAT's running time. For any fixed $d\ge3$ and $0<γ<1$, a sampling or approximate-counting algorithm running in $\exp\{O((nδ^2/d)^{1-γ})\}\operatorname{poly}(n)$ time along any one suitable inverse-polynomial distance sequence $δ=δ(n)$, yields a randomized SAT algorithm running in $\exp\{\widetilde O(s^{1-γ})\}$ time on inputs of binary length $s$. This power-saving transfer identifies $nδ^2/d$ as the analogue of SAT input length for hard-core sampling and counting at relative distance $δ$ above criticality: beating the Glauber exponent by a fixed power through any sampling or counting method would yield a corresponding power improvement for SAT. At criticality, we establish $O(\sqrt{n/d})$ spectral independence, optimal in $n$ at fixed degree, and degree-uniform Glauber mixing in $\widetilde O(n(n/d)^{1+1/(d-2)})$ time, improving the previous $\widetilde O(n^{4+4/(d-2)})$ bound.

Authors: Zhe Ju, Jingcheng Liu, Yitong Yin

How does a computational problem transition from easy to hard? In the hard-core model, approximate counting and sampling are polynomial-time tractable up to tree uniqueness, but worst-case intractable at any constant relative distance above it unless $\mathsf{NP}=\mathsf{RP}$. We quantify the transition from polynomial to exponential mixing for classical Glauber dynamics. On $n$-vertex graphs of maximum degree $d\ge3$, at fugacity $λ=(1+δ)λ_{\mathrm c}(d)$ with sufficiently small $δ\ge0$ that may vanish with $n$, its worst-case mixing time has the form $\exp\{Θ(nδ^2/d)\}n^{1+Θ(1)}$. The upper bound is degree-uniform, and the matching lower bound holds for every fixed $d$. Annealing gives approximate counting with the same running-time upper bound at fixed accuracy. For general sampling and counting algorithms, we establish a robust fine-grained reduction from SAT without assuming a lower bound on SAT's running time. For any fixed $d\ge3$ and $0<γ<1$, a sampling or approximate-counting algorithm running in $\exp\{O((nδ^2/d)^{1-γ})\}\operatorname{poly}(n)$ time along any one suitable inverse-polynomial distance sequence $δ=δ(n)$, yields a randomized SAT algorithm running in $\exp\{\widetilde O(s^{1-γ})\}$ time on inputs of binary length $s$. This power-saving transfer identifies $nδ^2/d$ as the analogue of SAT input length for hard-core sampling and counting at relative distance $δ$ above criticality: beating the Glauber exponent by a fixed power through any sampling or counting method would yield a corresponding power improvement for SAT. At criticality, we establish $O(\sqrt{n/d})$ spectral independence, optimal in $n$ at fixed degree, and degree-uniform Glauber mixing in $\widetilde O(n(n/d)^{1+1/(d-2)})$ time, improving the previous $\widetilde O(n^{4+4/(d-2)})$ bound.

Tuesday, November 03 2026, 01:00

Circuit Lower Bounds for Extended Gate Bases from Subexponential Counting

Authors: Mengyang Wu, Minghao Wang

We study circuit lower bounds for extended gate bases using exact CNF counting and an extension-ring form of the contraction theorem of OpenAI. The deterministic counting theorem of Claude (Anthropic) yields an $\mathsf{E}^{\mathsf{NP}}$ language requiring $Ω(m\sqrt{\log m}/\log\log m)$ gates over unbounded AND/OR with binary XOR, infinitely often. Using that contraction theorem, a bounded binary base with a linear budget of arbitrary XOR and symmetric gates requires $Ω(m\sqrt{\log m})$ gates infinitely often. These results extend the binary-gate lower bound of OpenAI. Signed long-clause selectors reduce the first counting problem to the difference of two ordinary 3-CNF counts. Affine characters and unit-separated interpolation give the second counting interface, with logarithmic cost per symmetric-gate arity. A quantitative local-reduction transfer accounts for auxiliary variables, branch counts, and integer recovery.

Authors: Mengyang Wu, Minghao Wang

We study circuit lower bounds for extended gate bases using exact CNF counting and an extension-ring form of the contraction theorem of OpenAI. The deterministic counting theorem of Claude (Anthropic) yields an $\mathsf{E}^{\mathsf{NP}}$ language requiring $Ω(m\sqrt{\log m}/\log\log m)$ gates over unbounded AND/OR with binary XOR, infinitely often. Using that contraction theorem, a bounded binary base with a linear budget of arbitrary XOR and symmetric gates requires $Ω(m\sqrt{\log m})$ gates infinitely often. These results extend the binary-gate lower bound of OpenAI. Signed long-clause selectors reduce the first counting problem to the difference of two ordinary 3-CNF counts. Affine characters and unit-separated interpolation give the second counting interface, with logarithmic cost per symmetric-gate arity. A quantitative local-reduction transfer accounts for auxiliary variables, branch counts, and integer recovery.

Tuesday, November 03 2026, 01:00

Efficient Proper Agnostic Learning of Halfspaces

Authors: Anindya De, Elchanan Mossel, Shivam Nadimpalli, Joe Neeman

We give a proper membership-query algorithm for agnostically learning halfspaces under the uniform distribution on $\{\pm1\}^n$. Given query access to an arbitrary function $f:\{\pm1\}^n \to \{\pm1\}$, our algorithm runs in time $\mathrm{poly}(n)+2^{\mathrm{poly}(1/\varepsilon)}$ and outputs a halfspace with error at most $\mathrm{OPT}+\varepsilon$, where $\mathrm{OPT}$ is the error of the best-fitting halfspace for $f$. Previously, even with membership queries, the best known running time was $n^{\mathrm{poly}(1/\varepsilon)}$, achieved by the sample-based algorithm of Kalai, Klivans, Mansour, and Servedio (FOCS 2005). Our algorithm is the first agnostic halfspace learner under the uniform distribution on $\{\pm1\}^n$ whose running time is polynomial in $n$ with an exponent independent of $\varepsilon$. Combining our algorithm with a boosting argument yields a proper agnostic learner using only random examples. It achieves the same $\mathrm{OPT}+\varepsilon$ guarantee, with running time matching that of Kalai et al.\ up to an additive term depending only on $\varepsilon$. To our knowledge, this is the first sample-based proper agnostic learner in the Boolean setting with running time $2^{o(n)}$ for every fixed $\varepsilon>0$.

Authors: Anindya De, Elchanan Mossel, Shivam Nadimpalli, Joe Neeman

We give a proper membership-query algorithm for agnostically learning halfspaces under the uniform distribution on $\{\pm1\}^n$. Given query access to an arbitrary function $f:\{\pm1\}^n \to \{\pm1\}$, our algorithm runs in time $\mathrm{poly}(n)+2^{\mathrm{poly}(1/\varepsilon)}$ and outputs a halfspace with error at most $\mathrm{OPT}+\varepsilon$, where $\mathrm{OPT}$ is the error of the best-fitting halfspace for $f$. Previously, even with membership queries, the best known running time was $n^{\mathrm{poly}(1/\varepsilon)}$, achieved by the sample-based algorithm of Kalai, Klivans, Mansour, and Servedio (FOCS 2005). Our algorithm is the first agnostic halfspace learner under the uniform distribution on $\{\pm1\}^n$ whose running time is polynomial in $n$ with an exponent independent of $\varepsilon$. Combining our algorithm with a boosting argument yields a proper agnostic learner using only random examples. It achieves the same $\mathrm{OPT}+\varepsilon$ guarantee, with running time matching that of Kalai et al.\ up to an additive term depending only on $\varepsilon$. To our knowledge, this is the first sample-based proper agnostic learner in the Boolean setting with running time $2^{o(n)}$ for every fixed $\varepsilon>0$.

Tuesday, November 03 2026, 01:00

An $\mathsf{FPRAS}$ for Mixed Volumes

Authors: Xiaoyu Chen, Kuikui Liu

We design the first fully polynomial-time randomized approximation scheme ($\mathsf{FPRAS}$) to estimate the mixed volume of $n$ arbitrary compact convex sets in $\mathbb{R}^{n}$ specified by weak membership oracles. This disproves a longstanding conjecture of Gurvits (2009). $\textbf{AI Disclosure}$ The main ideas were generated entirely by GPT 6 Astra. The purpose of this paper is to provide a human-written exposition of the algorithm.

Authors: Xiaoyu Chen, Kuikui Liu

We design the first fully polynomial-time randomized approximation scheme ($\mathsf{FPRAS}$) to estimate the mixed volume of $n$ arbitrary compact convex sets in $\mathbb{R}^{n}$ specified by weak membership oracles. This disproves a longstanding conjecture of Gurvits (2009). $\textbf{AI Disclosure}$ The main ideas were generated entirely by GPT 6 Astra. The purpose of this paper is to provide a human-written exposition of the algorithm.

Tuesday, November 03 2026, 01:00

A Decomposition Theorem and a Simple Algorithm for Contraction Fixed Points

Authors: Xi Chen, Yuhao Li, Mihalis Yannakakis

We prove a decomposition theorem for the problem of computing an approximate fixed point of a contraction map $f\colon [0,1]^k \to [0,1]^k$ under the $\ell_{\infty}$-norm. After a preliminary version of the decomposition theorem appeared on arXiv [CLY26], Feodorov and Haslebacher [FH26] gave a striking application of the theorem, showing that it can be used to convert any $\operatorname{poly}(k,\log(1/\varepsilon))$-query algorithm with running time $(k\log(1/\varepsilon))^{O(k)}$ into an algorithm with running time $(k\log(1/\varepsilon))^{O(\sqrt{k})}$. They instantiated this framework with an $O(k^2\log(1/\varepsilon))$-query algorithm with the desired running time, thereby obtaining the first subexponential-time algorithm for contraction fixed points under the $\ell_{\infty}$-norm and, via well-known reductions, the first deterministic subexponential-time algorithm for simple stochastic games and mean-payoff games. In addition to the decomposition theorem, as our second main result in this paper, we show that the $O(k\log(1/\varepsilon))$-query algorithm of [CLY25] can be implemented in time $(k\log(1/\varepsilon))^{O(k)}$; our algorithm may be viewed as conceptually simpler compared to the algorithm of [FH26]. A key observation here is that the function constructed in [CLY25], whose fixed points are used as the next query points, turns out to be nonexpansive under the $\ell_{\infty}$-norm, so the brute-force procedure can be replaced by known algorithms for contraction fixed points.

Authors: Xi Chen, Yuhao Li, Mihalis Yannakakis

We prove a decomposition theorem for the problem of computing an approximate fixed point of a contraction map $f\colon [0,1]^k \to [0,1]^k$ under the $\ell_{\infty}$-norm. After a preliminary version of the decomposition theorem appeared on arXiv [CLY26], Feodorov and Haslebacher [FH26] gave a striking application of the theorem, showing that it can be used to convert any $\operatorname{poly}(k,\log(1/\varepsilon))$-query algorithm with running time $(k\log(1/\varepsilon))^{O(k)}$ into an algorithm with running time $(k\log(1/\varepsilon))^{O(\sqrt{k})}$. They instantiated this framework with an $O(k^2\log(1/\varepsilon))$-query algorithm with the desired running time, thereby obtaining the first subexponential-time algorithm for contraction fixed points under the $\ell_{\infty}$-norm and, via well-known reductions, the first deterministic subexponential-time algorithm for simple stochastic games and mean-payoff games. In addition to the decomposition theorem, as our second main result in this paper, we show that the $O(k\log(1/\varepsilon))$-query algorithm of [CLY25] can be implemented in time $(k\log(1/\varepsilon))^{O(k)}$; our algorithm may be viewed as conceptually simpler compared to the algorithm of [FH26]. A key observation here is that the function constructed in [CLY25], whose fixed points are used as the next query points, turns out to be nonexpansive under the $\ell_{\infty}$-norm, so the brute-force procedure can be replaced by known algorithms for contraction fixed points.

Tuesday, November 03 2026, 01:00

A Numerical Refinement of an Exact Discrete Fourier Transform Bound

Authors: Zhihao Yao, Sichao Liu

We refine the numerical bounds in Kartal's instantiation of Sussman's exact discrete Fourier transform construction. For every integer n >= 1, the resulting algorithm computes the length-n transform with work O(n (log n)^(1 - delta)), where delta = 0.0030201659. The bound holds in an exact complex-arithmetic RAM model with a supplied canonical root of unity. This final exponent saving exceeds the earlier certified parameter C_0 = 0.00302016 by 5.9 x 10^(-9). We retain the underlying circuit, tighten rational bounds for sums of powers of its block and target ranks, and increase the fixed tensor power from 7760 to 11640. These bounds establish the inequalities for Sussman's recursion at the intermediate certificate parameter C = 0.00302016595. The gap C - delta absorbs the overhead of recursion and Fourier reduction. The resulting transform theorem is proved in Lean and checked against the reference problem statement.

Authors: Zhihao Yao, Sichao Liu

We refine the numerical bounds in Kartal's instantiation of Sussman's exact discrete Fourier transform construction. For every integer n >= 1, the resulting algorithm computes the length-n transform with work O(n (log n)^(1 - delta)), where delta = 0.0030201659. The bound holds in an exact complex-arithmetic RAM model with a supplied canonical root of unity. This final exponent saving exceeds the earlier certified parameter C_0 = 0.00302016 by 5.9 x 10^(-9). We retain the underlying circuit, tighten rational bounds for sums of powers of its block and target ranks, and increase the fixed tensor power from 7760 to 11640. These bounds establish the inequalities for Sussman's recursion at the intermediate certificate parameter C = 0.00302016595. The gap C - delta absorbs the overhead of recursion and Fourier reduction. The resulting transform theorem is proved in Lean and checked against the reference problem statement.

Tuesday, November 03 2026, 01:00

The Movement Complexity of Sorting

Authors: Sichen Wang

The complexity of sorting is usually measured by the number of comparisons. This measure ignores the movement of the read heads between comparisons, which is a real cost on many machines. We study this cost on its own. Keys lie on a $d$-dimensional grid, two heads move over the grid and compare the keys beneath them, and the only cost is the total movement of the heads. The algorithm plans its route in $r$ rounds and sees the outcomes of each round only after running it. For every $d$ and $r$ we determine the expected movement needed to sort $n$ keys: it is $n^β$ up to polylogarithmic factors, where $β=1+D^{r-1}/(D^{r}-(D-1)^{r})$ and $D=2d$, falling from $n^{2}$ at $r=1$ to $n^{1+1/(2d)}$ with unlimited rounds. Our algorithm makes every round move about the same amount, and a matching lower bound holds against every algorithm.

Authors: Sichen Wang

The complexity of sorting is usually measured by the number of comparisons. This measure ignores the movement of the read heads between comparisons, which is a real cost on many machines. We study this cost on its own. Keys lie on a $d$-dimensional grid, two heads move over the grid and compare the keys beneath them, and the only cost is the total movement of the heads. The algorithm plans its route in $r$ rounds and sees the outcomes of each round only after running it. For every $d$ and $r$ we determine the expected movement needed to sort $n$ keys: it is $n^β$ up to polylogarithmic factors, where $β=1+D^{r-1}/(D^{r}-(D-1)^{r})$ and $D=2d$, falling from $n^{2}$ at $r=1$ to $n^{1+1/(2d)}$ with unlimited rounds. Our algorithm makes every round move about the same amount, and a matching lower bound holds against every algorithm.

Tuesday, November 03 2026, 01:00

The Value of Steiner Hubs in Two-Hop Distance Labeling

Authors: Sichen Wang

Two-hop distance labelings were introduced in 2002 by Cohen, Halperin, Kaplan, and Zwick with hubs of two kinds: vertices of the graph, or, in their Steiner variant, arbitrary objects carrying arbitrary real values. On no family is the value of that freedom known. We determine it on $\ell_1$ products of betweenness-free metrics, those with no point between two others, a class containing every Hamming graph. The freedom is worth nothing when the factors have at most three letters, while from four letters on the largest saving is polynomial in the size of the product, attained on products of positively weighted stars. The $d$-dimensional hypercube is the two-letter case, and labeling it takes $(5/2+o(1))^d$ entries even with arbitrary hubs, matching the vertex-hub optimum of Goldberg, Razenshteyn, and Savchenko. Every lower bound here comes from one semidefinite program on a single factor, whose value $Λ$ never exceeds the rate that vertex hubs already achieve on the Hamming graph; where $Λ$ meets that ceiling, the freedom is worth nothing, and the ceiling is met exactly up to three letters.

Authors: Sichen Wang

Two-hop distance labelings were introduced in 2002 by Cohen, Halperin, Kaplan, and Zwick with hubs of two kinds: vertices of the graph, or, in their Steiner variant, arbitrary objects carrying arbitrary real values. On no family is the value of that freedom known. We determine it on $\ell_1$ products of betweenness-free metrics, those with no point between two others, a class containing every Hamming graph. The freedom is worth nothing when the factors have at most three letters, while from four letters on the largest saving is polynomial in the size of the product, attained on products of positively weighted stars. The $d$-dimensional hypercube is the two-letter case, and labeling it takes $(5/2+o(1))^d$ entries even with arbitrary hubs, matching the vertex-hub optimum of Goldberg, Razenshteyn, and Savchenko. Every lower bound here comes from one semidefinite program on a single factor, whose value $Λ$ never exceeds the rate that vertex hubs already achieve on the Hamming graph; where $Λ$ meets that ceiling, the freedom is worth nothing, and the ceiling is met exactly up to three letters.

Tuesday, November 03 2026, 01:00

Compressed Oracle Duality: A simple framework for knowing what quantum adversaries know

Authors: Matthew Ding, Mark Zhandry

Given a quantum adversary that queries a random oracle, how do we characterize the information that they "know"? The Compressed Oracle method of Zhandry (CRYPTO 2019) is one method that records the queries that the adversary makes; this recording serves as a representation of the information that is learned. However, a significant limitation of the Compressed Oracle method is that it may continue to store information that the adversary has "forgotten." This issue is especially pertinent in the space-bounded setting, where the adversary may uncompute information to free up space, while the information may still reside in the compressed oracle. In this work, we present a simple and interpretable framework called "Compressed Oracle Duality," which extends the Compressed Oracle method for space-bounded algorithms. By applying an isometry to the memory of a quantum adversary, we obtain a pair of compressed oracles with identical marginal distributions and classical correlations which correspond to the knowledge the adversary retains. In fact, we show that the outcome of measuring the new compressed oracle database corresponds exactly to applying the pretty-good measurement on the adversary's memory. As an application of our framework for collision search, we give a simplified proof of the recent $T^2S = Ω(N\log N)$ quantum time-space tradeoff for label-symmetric algorithms due to Magniez-Zur (arXiv:2609.10808); we also prove a stronger classical tradeoff of $TS=Ω(N\log N)$ assuming label symmetry. The proofs under our framework do not require representation theory and instead bound the mutual information between the compressed oracle pairs. This highlights both how the label symmetry assumption bypasses known proof barriers as well as the challenges that remain in achieving unconditional quantum time-space tradeoffs.

Authors: Matthew Ding, Mark Zhandry

Given a quantum adversary that queries a random oracle, how do we characterize the information that they "know"? The Compressed Oracle method of Zhandry (CRYPTO 2019) is one method that records the queries that the adversary makes; this recording serves as a representation of the information that is learned. However, a significant limitation of the Compressed Oracle method is that it may continue to store information that the adversary has "forgotten." This issue is especially pertinent in the space-bounded setting, where the adversary may uncompute information to free up space, while the information may still reside in the compressed oracle. In this work, we present a simple and interpretable framework called "Compressed Oracle Duality," which extends the Compressed Oracle method for space-bounded algorithms. By applying an isometry to the memory of a quantum adversary, we obtain a pair of compressed oracles with identical marginal distributions and classical correlations which correspond to the knowledge the adversary retains. In fact, we show that the outcome of measuring the new compressed oracle database corresponds exactly to applying the pretty-good measurement on the adversary's memory. As an application of our framework for collision search, we give a simplified proof of the recent $T^2S = Ω(N\log N)$ quantum time-space tradeoff for label-symmetric algorithms due to Magniez-Zur (arXiv:2609.10808); we also prove a stronger classical tradeoff of $TS=Ω(N\log N)$ assuming label symmetry. The proofs under our framework do not require representation theory and instead bound the mutual information between the compressed oracle pairs. This highlights both how the label symmetry assumption bypasses known proof barriers as well as the challenges that remain in achieving unconditional quantum time-space tradeoffs.

Tuesday, November 03 2026, 01:00

Limits and Constants of OpenAI's Claimed ETH Refutation

Authors: Fabio Dias, Yvens Lima

A recent OpenAI preprint claims a deterministic algorithm that counts the satisfying assignments of a $k$-CNF formula in time $2^{O_k(n\log\log n/\sqrt{\log n})}$, and $2^{O(n/\sqrt{\log n})}$ at bounded clause density, which would refute the Exponential Time Hypothesis. The headline bound is proved by tensor contraction; the body develops an interpolation algorithm with the weaker exponent $n(\log\log n)^2/\sqrt{\log n}$. This note does four things. First, it explains the interpolation algorithm in plain terms. Second, it asks how large $n$ must be for the bounds to bite. The preprint states no constants, so we treat the hidden constant $C$ as a parameter: the general bound beats the $2^n$ of brute force only beyond $n\approx 10^{32}$ for $C=2$, $10^{126}$ for $C=3$ and $10^{3517}$ for $C=10$. We then extract the constants from the tensor-contraction proof: for every input with $m$ clauses at bounded density, the running time it certifies stays above $2^n$ unless $\log_2(n+m)>10^9$, and on random 3-CNF at the satisfiability threshold unless $\log_2 n>5\times10^{11}$. So the bounds are subexponential as $n\to\infty$, but the proof certifies no speed-up over brute force for any input of bounded density that could ever be written down. Third, for the clause-product form of the interpolation algorithm we prove limits in a cost model matching the preprint's evaluation. By a short algebraic lemma, polynomial cost needs components of $O(\log^3 n)$ variables with per-point evaluation, or primal treewidth $O(\log^2 n)$ with dynamic programming over the groups. So the scheme costs $2^{Ω(\sqrt{\mathrm{tw}})}$ at treewidth $\mathrm{tw}$, $2^{Ω(\sqrt n)}$ with high probability on random 3-CNF above clause density 3, and $2^{Ω(n^{1/3})}$ even on trivially countable path formulas. We explain why the preprint's sharper routes escape these limits. Fourth, we report experiments.

Authors: Fabio Dias, Yvens Lima

A recent OpenAI preprint claims a deterministic algorithm that counts the satisfying assignments of a $k$-CNF formula in time $2^{O_k(n\log\log n/\sqrt{\log n})}$, and $2^{O(n/\sqrt{\log n})}$ at bounded clause density, which would refute the Exponential Time Hypothesis. The headline bound is proved by tensor contraction; the body develops an interpolation algorithm with the weaker exponent $n(\log\log n)^2/\sqrt{\log n}$. This note does four things. First, it explains the interpolation algorithm in plain terms. Second, it asks how large $n$ must be for the bounds to bite. The preprint states no constants, so we treat the hidden constant $C$ as a parameter: the general bound beats the $2^n$ of brute force only beyond $n\approx 10^{32}$ for $C=2$, $10^{126}$ for $C=3$ and $10^{3517}$ for $C=10$. We then extract the constants from the tensor-contraction proof: for every input with $m$ clauses at bounded density, the running time it certifies stays above $2^n$ unless $\log_2(n+m)>10^9$, and on random 3-CNF at the satisfiability threshold unless $\log_2 n>5\times10^{11}$. So the bounds are subexponential as $n\to\infty$, but the proof certifies no speed-up over brute force for any input of bounded density that could ever be written down. Third, for the clause-product form of the interpolation algorithm we prove limits in a cost model matching the preprint's evaluation. By a short algebraic lemma, polynomial cost needs components of $O(\log^3 n)$ variables with per-point evaluation, or primal treewidth $O(\log^2 n)$ with dynamic programming over the groups. So the scheme costs $2^{Ω(\sqrt{\mathrm{tw}})}$ at treewidth $\mathrm{tw}$, $2^{Ω(\sqrt n)}$ with high probability on random 3-CNF above clause density 3, and $2^{Ω(n^{1/3})}$ even on trivially countable path formulas. We explain why the preprint's sharper routes escape these limits. Fourth, we report experiments.

Tuesday, November 03 2026, 01:00

Adjacency Labeling Schemes of Size $\log n+O(\log\log^* n)$ for Graphs of Bounded Treewidth

Authors: Noy Rotbart

Gavoille and Labourel (ESA 2007) gave an adjacency labeling scheme of size $\log n+O(k\log\log n)$ for graphs of treewidth at most $k$, and asked whether $\log n+O(k)$ bits suffice. We give a labeling scheme of size $\log n+O(\log\log^* n)$ for every fixed $k$, the first improvement of the second-order term for any $k\ge 2$. Equivalently, there is an induced-universal graph with $n(\log^* n)^{O_k(1)}$ vertices for this family. We work with laminar families of intervals, in which a vertex finds each earlier neighbor as the owner of an interval of a given nesting depth around it. As long as every subtree occupies consecutive positions, the nesting depth must be logarithmic, and $\log\log n$ bits per neighbor follow. We give up contiguity. Heavy subtrees are moved out, which brings the depth down to $O(\log^* n)$; the intervals broken by the moves are repaired with $2^{O(k^3)}$ additional intervals per vertex; and the bits needed to find these intervals are paid for by the moved subtrees.

Authors: Noy Rotbart

Gavoille and Labourel (ESA 2007) gave an adjacency labeling scheme of size $\log n+O(k\log\log n)$ for graphs of treewidth at most $k$, and asked whether $\log n+O(k)$ bits suffice. We give a labeling scheme of size $\log n+O(\log\log^* n)$ for every fixed $k$, the first improvement of the second-order term for any $k\ge 2$. Equivalently, there is an induced-universal graph with $n(\log^* n)^{O_k(1)}$ vertices for this family. We work with laminar families of intervals, in which a vertex finds each earlier neighbor as the owner of an interval of a given nesting depth around it. As long as every subtree occupies consecutive positions, the nesting depth must be logarithmic, and $\log\log n$ bits per neighbor follow. We give up contiguity. Heavy subtrees are moved out, which brings the depth down to $O(\log^* n)$; the intervals broken by the moves are repaired with $2^{O(k^3)}$ additional intervals per vertex; and the bits needed to find these intervals are paid for by the moved subtrees.

Tuesday, November 03 2026, 01:00

A GPU Engine for Sparse Matrix-Vector Products on $k^2$-Tree-Compressed Graphs

Authors: Francesco Tosoni, Gabriele Mencagli

The $k^2$-tree is a compact representation of Boolean matrices: it stores the recursive hierarchy of their non-empty submatrices, with neither values nor coordinates, and serves rows and columns alike. Its traversal, however, locates the children of each node by rank queries, a chain of dependent irregular loads that leaves a GPU waiting on memory. We implement sparse matrix-vector multiplication (SpMV) on $k^2$-trees for the GPU. Since the traversal does not depend on the vector, the positions of the children are static: we descend the tree one level at a time and obtain them from a parallel prefix sum or from a sampled directory of about 0.1 bits per node. Descending in tiles bounds the temporary memory by the depth of the tree, so the device memory of a product is known before it runs. The same traversal computes right and left products, for one vector or a batch, over the plus-times, Boolean and tropical semirings. We evaluate the engine on an NVIDIA GB10 and an H100 with fourteen Wikidata, Software Heritage, web and social graphs of up to 1.2 billion edges, against cuSPARSE, a grammar-compressed GPU engine and a Boolean CSR. Our results show that the tree is 2.3-27x smaller than the CSR of cuSPARSE and that, with its last internal level cached, a product takes 0.7-2.4x the time of cuSPARSE on the right and 0.27-2.6x on the left. On a 1.2-billion-edge graph, a product needs 1.1 GB on the device, 9x less than cuSPARSE. The engine is memory bounded: the $k^2$-tree is the format of choice when device memory, not time, is the constraint.

Authors: Francesco Tosoni, Gabriele Mencagli

The $k^2$-tree is a compact representation of Boolean matrices: it stores the recursive hierarchy of their non-empty submatrices, with neither values nor coordinates, and serves rows and columns alike. Its traversal, however, locates the children of each node by rank queries, a chain of dependent irregular loads that leaves a GPU waiting on memory. We implement sparse matrix-vector multiplication (SpMV) on $k^2$-trees for the GPU. Since the traversal does not depend on the vector, the positions of the children are static: we descend the tree one level at a time and obtain them from a parallel prefix sum or from a sampled directory of about 0.1 bits per node. Descending in tiles bounds the temporary memory by the depth of the tree, so the device memory of a product is known before it runs. The same traversal computes right and left products, for one vector or a batch, over the plus-times, Boolean and tropical semirings. We evaluate the engine on an NVIDIA GB10 and an H100 with fourteen Wikidata, Software Heritage, web and social graphs of up to 1.2 billion edges, against cuSPARSE, a grammar-compressed GPU engine and a Boolean CSR. Our results show that the tree is 2.3-27x smaller than the CSR of cuSPARSE and that, with its last internal level cached, a product takes 0.7-2.4x the time of cuSPARSE on the right and 0.27-2.6x on the left. On a 1.2-billion-edge graph, a product needs 1.1 GB on the device, 9x less than cuSPARSE. The engine is memory bounded: the $k^2$-tree is the format of choice when device memory, not time, is the constraint.

Tuesday, November 03 2026, 01:00

Near-Optimal Space and Time Cut Sparsification in Turnstile Streams

Authors: Honghao Lin, Vahab Mirrokni, David P. Woodruff

A seminal result of Ahn, Guha, and McGregor (PODS'12) showed that a cut sparsifier of an unweighted graph can be computed from a near-linear number of linear measurements. Later work extended this to spectral sparsifiers (Kapralov et al., FOCS'14) and to near-linear recovery time (Kapralov et al., SODA'20). For weighted graphs, these sketches rely on grouping edges by weight, which general turnstile streams do not allow. Chen, Khanna, and Li (FOCS'22) removed this limitation, using $\tilde{O}(n/\varepsilon^3)$ bits of space. We improve the space to $\tilde{O}(n/\varepsilon^2)$ bits. Our algorithm makes one pass over a general turnstile stream of a weighted graph, processes each update in $\tilde{O}(1)$ time, and recovers, in $\tilde{O}(n/\varepsilon^2)$ time, a reweighted subgraph with $O(n\varepsilon^{-2}\log^2 n)$ edges that preserves every cut within a factor of $1\pm\varepsilon$. Its space, update time, and recovery time are all optimal up to polylogarithmic factors.

Authors: Honghao Lin, Vahab Mirrokni, David P. Woodruff

A seminal result of Ahn, Guha, and McGregor (PODS'12) showed that a cut sparsifier of an unweighted graph can be computed from a near-linear number of linear measurements. Later work extended this to spectral sparsifiers (Kapralov et al., FOCS'14) and to near-linear recovery time (Kapralov et al., SODA'20). For weighted graphs, these sketches rely on grouping edges by weight, which general turnstile streams do not allow. Chen, Khanna, and Li (FOCS'22) removed this limitation, using $\tilde{O}(n/\varepsilon^3)$ bits of space. We improve the space to $\tilde{O}(n/\varepsilon^2)$ bits. Our algorithm makes one pass over a general turnstile stream of a weighted graph, processes each update in $\tilde{O}(1)$ time, and recovers, in $\tilde{O}(n/\varepsilon^2)$ time, a reweighted subgraph with $O(n\varepsilon^{-2}\log^2 n)$ edges that preserves every cut within a factor of $1\pm\varepsilon$. Its space, update time, and recovery time are all optimal up to polylogarithmic factors.

Tuesday, November 03 2026, 01:00

Should We Combine Our Data? Algorithms and Limits for Privately Assessing Data Sharing

Authors: Mahdi Haghifam, Kumar Kshitij Patel, Avrim Blum, Giulia Fanti, Juba Ziani

In this paper, we study when two parties should pool their data to train a machine learning model. Two agents, Alice and Bob, each hold a sensitive labeled dataset and want to train a classifier from a given hypothesis class $\mathcal{H}$. Even when each party's dataset can be perfectly classified on its own, the two parties may follow conflicting labeling standards or have selection biases that affect their features and label sets, such that no single classifier in $\mathcal{H}$ fits their combined data. Before committing to data sharing, Alice and Bob therefore want to test whether a single classifier in $\mathcal{H}$ can jointly fit both datasets---without revealing their individual data points to one another. We formalize this problem from a learning-theoretic viewpoint via $\operatorname{opt}_\mathcal{H}$, the minimum \emph{number of mistakes} made by any hypothesis in $\mathcal{H}$ on their pooled data. Through interactive communication, the agents must distinguish $\operatorname{opt}_\mathcal{H}=0$ (compatible datasets) from $\operatorname{opt}_\mathcal{H}\geq t$ (conflicting datasets) under pure $\varepsilon$-two-party differential privacy. A trusted curator could do this with $t=O(1/\varepsilon)$, and we ask how small $t$ can be without one. For thresholds on the real line over a domain of size $m$, we show that the optimal $t$ is $\tildeΘ(\log\log m/\varepsilon)$, exponentially smaller than the $Θ(\log m/\varepsilon)$ samples needed to privately learn thresholds, even with a curator. For halfspaces in $\mathbb{R}^d$, the dependence on $m$ jumps to $\log(m)$: we give a protocol with $t=\tilde{O}(d^3\log m/\varepsilon)$ and show that some domains require $t=Ω(d\log(m/d)/\varepsilon)$.

Authors: Mahdi Haghifam, Kumar Kshitij Patel, Avrim Blum, Giulia Fanti, Juba Ziani

In this paper, we study when two parties should pool their data to train a machine learning model. Two agents, Alice and Bob, each hold a sensitive labeled dataset and want to train a classifier from a given hypothesis class $\mathcal{H}$. Even when each party's dataset can be perfectly classified on its own, the two parties may follow conflicting labeling standards or have selection biases that affect their features and label sets, such that no single classifier in $\mathcal{H}$ fits their combined data. Before committing to data sharing, Alice and Bob therefore want to test whether a single classifier in $\mathcal{H}$ can jointly fit both datasets---without revealing their individual data points to one another. We formalize this problem from a learning-theoretic viewpoint via $\operatorname{opt}_\mathcal{H}$, the minimum \emph{number of mistakes} made by any hypothesis in $\mathcal{H}$ on their pooled data. Through interactive communication, the agents must distinguish $\operatorname{opt}_\mathcal{H}=0$ (compatible datasets) from $\operatorname{opt}_\mathcal{H}\geq t$ (conflicting datasets) under pure $\varepsilon$-two-party differential privacy. A trusted curator could do this with $t=O(1/\varepsilon)$, and we ask how small $t$ can be without one. For thresholds on the real line over a domain of size $m$, we show that the optimal $t$ is $\tildeΘ(\log\log m/\varepsilon)$, exponentially smaller than the $Θ(\log m/\varepsilon)$ samples needed to privately learn thresholds, even with a curator. For halfspaces in $\mathbb{R}^d$, the dependence on $m$ jumps to $\log(m)$: we give a protocol with $t=\tilde{O}(d^3\log m/\varepsilon)$ and show that some domains require $t=Ω(d\log(m/d)/\varepsilon)$.

Tuesday, November 03 2026, 01:00

Online Matching with Arbitrary Stochastic Rewards

Authors: Rajan Udwani

Online matching with stochastic rewards is a generalization of online bipartite matching in which an attempt to match an online arrival to an offline resource may fail with some probability. Each arrival allows at most one attempt and the success probabilities of its edges are revealed when it arrives. Greedy is 0.5-competitive, which is the best possible for deterministic algorithms. While better competitive ratios are known in several special cases, beating 0.5 for arbitrary probabilities has remained open. We show that, against the adaptive offline benchmark and for arbitrary probabilities, a randomized Perturbed Greedy algorithm is at least 0.591-competitive in the unweighted setting and at least 0.531-competitive with vertex weights. The perturbation functions combine Greedy with scaled versions of the classic exponential function.

Authors: Rajan Udwani

Online matching with stochastic rewards is a generalization of online bipartite matching in which an attempt to match an online arrival to an offline resource may fail with some probability. Each arrival allows at most one attempt and the success probabilities of its edges are revealed when it arrives. Greedy is 0.5-competitive, which is the best possible for deterministic algorithms. While better competitive ratios are known in several special cases, beating 0.5 for arbitrary probabilities has remained open. We show that, against the adaptive offline benchmark and for arbitrary probabilities, a randomized Perturbed Greedy algorithm is at least 0.591-competitive in the unweighted setting and at least 0.531-competitive with vertex weights. The perturbation functions combine Greedy with scaled versions of the classic exponential function.

Tuesday, November 03 2026, 01:00

Square-Root Pass Semi-Streaming Algorithms for Digraph Problems

Authors: Prantar Ghosh, Amirhossein Janfaza

We study directed graph problems in the semi-streaming model where we are restricted to $\widetilde{O}(n):=O(n\cdot \text{polylog}(n))$ space for processing the edge stream of an $n$-node graph and can make one or a few passes to solve the given problem. A longstanding open problem in multipass streaming is determining the number of semi-streaming passes needed to solve directed $s,t$-reachability: there is an exponential gap---$Ω(\log n/\log \log n)$ vs $n^{1/2+o(1)}$---between the best-known lower and upper bounds. Several related fundamental digraph problems, e.g., topological sorting, acyclicity testing, and decomposition into strongly connected components (SCCs), are known to have the same semi-streaming pass lower bound. However, no $o(n/\text{polylog}(n))$-pass upper bound was known for any of them. Our main result achieves an $n^{1/2+o(1)}$-pass semi-streaming algorithm for each of these problems. As a corollary, we get algorithms with the same pass and space usage for a broad class of DAG problems that admit a certain form of dynamic program over a topological ordering. This class includes weighted single-source shortest paths, longest paths, max-outdegree+1 coloring, and constructing a lexicographically-first DFS (depth-first search) forest. Our algorithm for the last problem can be seen as a generalization of our results on topological sorting and acyclicity testing: recall that standard textbook algorithms for these problems use DFS as a subroutine. Our algorithms are optimal up to an $n^{o(1)}$ factor in passes under the reasonable conjecture that $s,t$-reachability requires $Ω(\sqrt{n})$ semi-streaming passes. However, using the fact that our algorithms can be parameterized by the number of passes taken by topological sorting or single-source reachability algorithms, we prove stronger lower bounds for some of the problems even under weaker conjectures.

Authors: Prantar Ghosh, Amirhossein Janfaza

We study directed graph problems in the semi-streaming model where we are restricted to $\widetilde{O}(n):=O(n\cdot \text{polylog}(n))$ space for processing the edge stream of an $n$-node graph and can make one or a few passes to solve the given problem. A longstanding open problem in multipass streaming is determining the number of semi-streaming passes needed to solve directed $s,t$-reachability: there is an exponential gap---$Ω(\log n/\log \log n)$ vs $n^{1/2+o(1)}$---between the best-known lower and upper bounds. Several related fundamental digraph problems, e.g., topological sorting, acyclicity testing, and decomposition into strongly connected components (SCCs), are known to have the same semi-streaming pass lower bound. However, no $o(n/\text{polylog}(n))$-pass upper bound was known for any of them. Our main result achieves an $n^{1/2+o(1)}$-pass semi-streaming algorithm for each of these problems. As a corollary, we get algorithms with the same pass and space usage for a broad class of DAG problems that admit a certain form of dynamic program over a topological ordering. This class includes weighted single-source shortest paths, longest paths, max-outdegree+1 coloring, and constructing a lexicographically-first DFS (depth-first search) forest. Our algorithm for the last problem can be seen as a generalization of our results on topological sorting and acyclicity testing: recall that standard textbook algorithms for these problems use DFS as a subroutine. Our algorithms are optimal up to an $n^{o(1)}$ factor in passes under the reasonable conjecture that $s,t$-reachability requires $Ω(\sqrt{n})$ semi-streaming passes. However, using the fact that our algorithms can be parameterized by the number of passes taken by topological sorting or single-source reachability algorithms, we prove stronger lower bounds for some of the problems even under weaker conjectures.

Tuesday, November 03 2026, 01:00

Exact Sum-of-Radii Clustering under Any Constant Stability

Authors: Shyam Ravichandran, Lev Reyzin

We establish a separation between the stability thresholds of Minimum Sum-of-Radii (MSR) clustering and two closely related objectives. Known hardness results show that, on general metrics, exact recovery for Minimum Sum-of-Diameters and symmetric $k$-center is hard for every fixed $1<γ<2$, unless NP=RP. We show that MSR admits deterministic polynomial-time exact recovery for every fixed $γ>1$, extending Gao and Reyzin's result for $γ\ge2$. All three results use the same Bilu--Linial stability model: the optimal partition remains uniquely optimal when each pairwise distance is multiplied by any factor in $[1,γ]$, without requiring the perturbed distances to satisfy the triangle inequality. The number of clusters is part of the input. For $1<γ<2$, our algorithm runs in $n^{O((γ-1)^{-5})}$ time. The main structural insight is that a cheap merge in one part of the instance would allow a profitable split elsewhere under a suitable perturbation, contradicting stability. The resulting exchange inequality controls local cluster density, which gives linear growth and hence bounded treewidth in local interaction graphs. We use random partitions to obtain these regions, solve them by dynamic programming, and then derandomize the construction. For rational input and fixed rational $γ>1$, recognizing stability in this model is coNP-complete. Verifying that a supplied partition remains uniquely optimal under all allowed perturbations is coNP-complete for both arbitrary symmetric and metric perturbations.

Authors: Shyam Ravichandran, Lev Reyzin

We establish a separation between the stability thresholds of Minimum Sum-of-Radii (MSR) clustering and two closely related objectives. Known hardness results show that, on general metrics, exact recovery for Minimum Sum-of-Diameters and symmetric $k$-center is hard for every fixed $1<γ<2$, unless NP=RP. We show that MSR admits deterministic polynomial-time exact recovery for every fixed $γ>1$, extending Gao and Reyzin's result for $γ\ge2$. All three results use the same Bilu--Linial stability model: the optimal partition remains uniquely optimal when each pairwise distance is multiplied by any factor in $[1,γ]$, without requiring the perturbed distances to satisfy the triangle inequality. The number of clusters is part of the input. For $1<γ<2$, our algorithm runs in $n^{O((γ-1)^{-5})}$ time. The main structural insight is that a cheap merge in one part of the instance would allow a profitable split elsewhere under a suitable perturbation, contradicting stability. The resulting exchange inequality controls local cluster density, which gives linear growth and hence bounded treewidth in local interaction graphs. We use random partitions to obtain these regions, solve them by dynamic programming, and then derandomize the construction. For rational input and fixed rational $γ>1$, recognizing stability in this model is coNP-complete. Verifying that a supplied partition remains uniquely optimal under all allowed perturbations is coNP-complete for both arbitrary symmetric and metric perturbations.

Tuesday, November 03 2026, 01:00

The Optimizer's Windfall: Bounding Hindsight Gain under Random Restrictions

Authors: Xiaohui Bei, Wenhao Wu

Suppose an optimization problem is randomly restricted, independently on each variable, and re-optimized after the restriction is revealed. For multilinear objectives, every fixed solution is unbiased after rescaling, but re-optimizing introduces an upward bias. We bound this excess, the hindsight gain, over products of arbitrary compact local domains in Euclidean unit balls. For interactions $A=(A_e)_e$ among $r$ distinct variables and independent random weights $ζ_i$, the gain is at most $O_r(\|A\|_F M^{(r-1)/2}σ_{\max})$, where $\|A\|_F^2$ is the sum of squared coefficients, $M=\sum_i\mathbb{E}ζ_i^2$, and $σ_{\max}^2=\max_i\mathrm{Var}(ζ_i)$; only second moments are needed. Under uniform retention with probability $p$, we sharpen this to $O_r(p^{r-1/2}\sqrt{1-p}\,n^{(r-1)/2}\|A\|_F)$, which is tight for every fixed $r$ and $p\in(0,1)$. Our proof combines a comparison principle along a single support direction with tensor projection, vector contraction, and repeated thinning. Applications: (i) For fixed arity and alphabet size, solving the subproblem induced by $O(\varepsilon^{-2})$ sampled variables estimates a dense Max-$r$CSP optimum within $\pm\varepsilon n^r$, and this sample size is optimal. (ii) The same approach estimates MaxCut and the distance to bipartiteness within $\pm\varepsilon n^2$ using $O(\varepsilon^{-4})$ nonadaptive queries and $O(\varepsilon^{-2})$ examined vertices, both optimal. (iii) We answer a question of Yufei Zhao: vertex percolation at rate $p$ preserves the max-cut fraction whenever $p\,e(G)/v(G)\to\infty$, and this condition is sharp. (iv) For bounded $r$-local quantum Hamiltonians of fixed local dimension, $O(\varepsilon^{-3})$ uniformly sampled particles suffice to estimate the ground-state energy within $\pm\varepsilon n^r$, improving the previous $O(\varepsilon^{-6}\log(1/\varepsilon))$ bound.

Authors: Xiaohui Bei, Wenhao Wu

Suppose an optimization problem is randomly restricted, independently on each variable, and re-optimized after the restriction is revealed. For multilinear objectives, every fixed solution is unbiased after rescaling, but re-optimizing introduces an upward bias. We bound this excess, the hindsight gain, over products of arbitrary compact local domains in Euclidean unit balls. For interactions $A=(A_e)_e$ among $r$ distinct variables and independent random weights $ζ_i$, the gain is at most $O_r(\|A\|_F M^{(r-1)/2}σ_{\max})$, where $\|A\|_F^2$ is the sum of squared coefficients, $M=\sum_i\mathbb{E}ζ_i^2$, and $σ_{\max}^2=\max_i\mathrm{Var}(ζ_i)$; only second moments are needed. Under uniform retention with probability $p$, we sharpen this to $O_r(p^{r-1/2}\sqrt{1-p}\,n^{(r-1)/2}\|A\|_F)$, which is tight for every fixed $r$ and $p\in(0,1)$. Our proof combines a comparison principle along a single support direction with tensor projection, vector contraction, and repeated thinning. Applications: (i) For fixed arity and alphabet size, solving the subproblem induced by $O(\varepsilon^{-2})$ sampled variables estimates a dense Max-$r$CSP optimum within $\pm\varepsilon n^r$, and this sample size is optimal. (ii) The same approach estimates MaxCut and the distance to bipartiteness within $\pm\varepsilon n^2$ using $O(\varepsilon^{-4})$ nonadaptive queries and $O(\varepsilon^{-2})$ examined vertices, both optimal. (iii) We answer a question of Yufei Zhao: vertex percolation at rate $p$ preserves the max-cut fraction whenever $p\,e(G)/v(G)\to\infty$, and this condition is sharp. (iv) For bounded $r$-local quantum Hamiltonians of fixed local dimension, $O(\varepsilon^{-3})$ uniformly sampled particles suffice to estimate the ground-state energy within $\pm\varepsilon n^r$, improving the previous $O(\varepsilon^{-6}\log(1/\varepsilon))$ bound.

Tuesday, November 03 2026, 01:00

Expander Routing under Adversarial Deletions via Temporal Aggregation

Authors: Bernhard Haeupler, Anton Paramonov

In expander routing, we seek to establish paths in the expander graph between given source-target pairs while minimizing both dilation (maximum path length) and congestion (maximum edge load). We study this problem in a decremental dynamic setting where the underlying graph remains an expander despite ongoing edge deletions, and source-target pairs can be added or removed. Our goal is to maintain paths under these updates while keeping dilation and congestion low and performing low computation per update. We present a randomized algorithm that maintains paths with polylogarithmic dilation and congestion against an adaptive adversary, while incurring only polylogarithmic amortized computation per update, all with high probability. Our approach leverages a recent technique called Temporal Aggregation to maintain a collection of low-congestion, low-dilation random walks resilient to adaptive adversaries.

Authors: Bernhard Haeupler, Anton Paramonov

In expander routing, we seek to establish paths in the expander graph between given source-target pairs while minimizing both dilation (maximum path length) and congestion (maximum edge load). We study this problem in a decremental dynamic setting where the underlying graph remains an expander despite ongoing edge deletions, and source-target pairs can be added or removed. Our goal is to maintain paths under these updates while keeping dilation and congestion low and performing low computation per update. We present a randomized algorithm that maintains paths with polylogarithmic dilation and congestion against an adaptive adversary, while incurring only polylogarithmic amortized computation per update, all with high probability. Our approach leverages a recent technique called Temporal Aggregation to maintain a collection of low-congestion, low-dilation random walks resilient to adaptive adversaries.

Tuesday, November 03 2026, 01:00

Stochastic Matching with Polynomial Degree

Authors: Soheil Behnezhad

We study the stochastic matching problem, where a graph $G$ is known in advance, but each edge survives independently with probability $p$. The goal is to select a subgraph $H$ of small maximum degree before seeing the surviving edges, while preserving a $(1-ε)$ fraction of the expected maximum matching size. This problem has received significant attention due to its numerous applications in kidney exchange, labor markets, and online dating. In this paper, we show there is always a subgraph $H \subseteq G$ achieving a $(1-ε)$-approximation while having a maximum degree of at most $O\left(\frac{\log(1/ε)}{ε^4} \cdot \frac{1}{p}\right).$ This resolves a major conjecture of the literature, first stated by [BDH, STOC'20], that $poly(\frac{1}{εp})$ degrees suffice. The best previous bound was $(1/p)^{\exp((1/ε)\log(1/ε))}$ by [ABGR, STOC'25]. The key to this result is a lemma of independent interest that any distribution over the matchings of a stochastic graph can be transformed to another distribution which matches each vertex with exactly the same probability, but exhibits extremely low total pairwise correlation.

Authors: Soheil Behnezhad

We study the stochastic matching problem, where a graph $G$ is known in advance, but each edge survives independently with probability $p$. The goal is to select a subgraph $H$ of small maximum degree before seeing the surviving edges, while preserving a $(1-ε)$ fraction of the expected maximum matching size. This problem has received significant attention due to its numerous applications in kidney exchange, labor markets, and online dating. In this paper, we show there is always a subgraph $H \subseteq G$ achieving a $(1-ε)$-approximation while having a maximum degree of at most $O\left(\frac{\log(1/ε)}{ε^4} \cdot \frac{1}{p}\right).$ This resolves a major conjecture of the literature, first stated by [BDH, STOC'20], that $poly(\frac{1}{εp})$ degrees suffice. The best previous bound was $(1/p)^{\exp((1/ε)\log(1/ε))}$ by [ABGR, STOC'25]. The key to this result is a lemma of independent interest that any distribution over the matchings of a stochastic graph can be transformed to another distribution which matches each vertex with exactly the same probability, but exhibits extremely low total pairwise correlation.

Tuesday, November 03 2026, 01:00

Calibrated Multiaccuracy: Deterministic and Randomized Sample Complexity

Authors: Jiuyao Lu, Georgy Noarov, Aaron Roth

Calibrated multiaccuracy asks that a predictor be globally calibrated and multiaccurate with respect to a bounded test family $G$; that is, its residuals should be uncorrelated with every test $g(x)\in G$. This weakening of multicalibration suffices for many applications. Its sample complexity was not known to improve on that of full multicalibration, which is $\widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{3}}\right)$ when $|G|$ is at least a sufficiently large power of $\log(1/\varepsilon)$. We resolve the sample complexity of $\varepsilon$-calibrated multiaccuracy, showing that it scales as $$ \widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{5/2}}\right), $$ separating it from the complexity of multicalibration. The upper bound is obtained by an algorithm that outputs a randomized predictor. For deterministic predictors, we establish the tight rate $$ \widetildeΘ\left( \frac{\log|G|}{\varepsilon^2} + \frac{(\log|G|)^{1/3}}{\varepsilon^{8/3}} \right) $$ whenever $\log|G|=Ω(\log(1/\varepsilon))$. When $\log|G|\le\varepsilon^{-1+κ}$ for any fixed $κ\in(0,1)$, this rate is polynomially larger than the rate for randomized predictors. Thus, unlike for multicalibration, calibrated multiaccuracy exhibits a genuine sample-complexity separation between deterministic and randomized predictors. The matching upper and lower bounds show that the unusual cube-root dependence on $\log|G|$ is intrinsic. At $\log|G|\asymp\varepsilon^{-1}$, the minimax rates for randomized and deterministic calibrated multiaccuracy both coincide with the minimax rate for multicalibration.

Authors: Jiuyao Lu, Georgy Noarov, Aaron Roth

Calibrated multiaccuracy asks that a predictor be globally calibrated and multiaccurate with respect to a bounded test family $G$; that is, its residuals should be uncorrelated with every test $g(x)\in G$. This weakening of multicalibration suffices for many applications. Its sample complexity was not known to improve on that of full multicalibration, which is $\widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{3}}\right)$ when $|G|$ is at least a sufficiently large power of $\log(1/\varepsilon)$. We resolve the sample complexity of $\varepsilon$-calibrated multiaccuracy, showing that it scales as $$ \widetildeΘ\left(\frac{\log|G|}{\varepsilon^2} + \frac{1}{\varepsilon^{5/2}}\right), $$ separating it from the complexity of multicalibration. The upper bound is obtained by an algorithm that outputs a randomized predictor. For deterministic predictors, we establish the tight rate $$ \widetildeΘ\left( \frac{\log|G|}{\varepsilon^2} + \frac{(\log|G|)^{1/3}}{\varepsilon^{8/3}} \right) $$ whenever $\log|G|=Ω(\log(1/\varepsilon))$. When $\log|G|\le\varepsilon^{-1+κ}$ for any fixed $κ\in(0,1)$, this rate is polynomially larger than the rate for randomized predictors. Thus, unlike for multicalibration, calibrated multiaccuracy exhibits a genuine sample-complexity separation between deterministic and randomized predictors. The matching upper and lower bounds show that the unusual cube-root dependence on $\log|G|$ is intrinsic. At $\log|G|\asymp\varepsilon^{-1}$, the minimax rates for randomized and deterministic calibrated multiaccuracy both coincide with the minimax rate for multicalibration.

Tuesday, November 03 2026, 01:00

Approximating Submodular Orienteering via Rooted Subgraph Decomposition

Authors: Lars Rohwedder

The Submodular Orienteering problem asks for an $s$-$t$ walk in a directed graph that maximizes a monotone submodular function over the visited vertices subject to a bound on its length. We give a deterministic polynomial-time approximation algorithm with subpolynomial approximation rate $2^{O(\sqrt{\log n\,\log\log n})}$. This improves over the $O(n^\varepsilon)$ approximation rate that was previously known for every fixed $\varepsilon>0$. We prove a decomposition lemma that partitions a directed graph into rooted subgraphs such that every path intersects only polylogarithmically many such subgraphs. Our algorithm uses this decomposition to construct its recursive subproblems.

Authors: Lars Rohwedder

The Submodular Orienteering problem asks for an $s$-$t$ walk in a directed graph that maximizes a monotone submodular function over the visited vertices subject to a bound on its length. We give a deterministic polynomial-time approximation algorithm with subpolynomial approximation rate $2^{O(\sqrt{\log n\,\log\log n})}$. This improves over the $O(n^\varepsilon)$ approximation rate that was previously known for every fixed $\varepsilon>0$. We prove a decomposition lemma that partitions a directed graph into rooted subgraphs such that every path intersects only polylogarithmically many such subgraphs. Our algorithm uses this decomposition to construct its recursive subproblems.

Tuesday, November 03 2026, 01:00

Optimal Sample Complexity for Max-Cut

Authors: Itamar Bender, Robert Krauthgamer

We determine the optimal sample size for constructing value coresets for Max-Cut via uniform vertex sampling. That is, sampling $O(\varepsilon^{-2})$ vertices uniformly at random in $n$-vertex graphs with edge weights in $[0,1]$ yields a reweighted subgraph whose Max-Cut value is an additive $\varepsilon n^2$-approximation to that of the original graph, with constant success probability. This improves the previously known $O(\varepsilon^{-4})$ upper bound and matches the known lower bound. The same guarantee holds also for importance sampling, where vertices are sampled proportionally to their weighted degrees, and we further prove a matching lower bound for this sampling method. Our analysis extends directly to metric Max-Cut, showing that importance sampling of $O(\varepsilon^{-2})$ vertices yields $(1\pm\varepsilon)$-approximation to the Max-Cut value, and to Max-Cut in graphs with edge weights in $[0,1]$ and average weighted degree $\bar d$, achieving the same $(1\pm\varepsilon)$-approximation by either uniform or importance sampling of $O(\varepsilon^{-2}n/\bar d)$ vertices. We additionally show how to use this coreset to recover a cut of expected value at least $OPT-\varepsilon n^2$, by making $O(n\varepsilon^{-2})$ additional edge-weight queries. Finally, our result extends to Max-$r$-Cut, giving an additive $\varepsilon n^2$-approximation using a sample of $O(\varepsilon^{-2}\log(2r))$ vertices.

Authors: Itamar Bender, Robert Krauthgamer

We determine the optimal sample size for constructing value coresets for Max-Cut via uniform vertex sampling. That is, sampling $O(\varepsilon^{-2})$ vertices uniformly at random in $n$-vertex graphs with edge weights in $[0,1]$ yields a reweighted subgraph whose Max-Cut value is an additive $\varepsilon n^2$-approximation to that of the original graph, with constant success probability. This improves the previously known $O(\varepsilon^{-4})$ upper bound and matches the known lower bound. The same guarantee holds also for importance sampling, where vertices are sampled proportionally to their weighted degrees, and we further prove a matching lower bound for this sampling method. Our analysis extends directly to metric Max-Cut, showing that importance sampling of $O(\varepsilon^{-2})$ vertices yields $(1\pm\varepsilon)$-approximation to the Max-Cut value, and to Max-Cut in graphs with edge weights in $[0,1]$ and average weighted degree $\bar d$, achieving the same $(1\pm\varepsilon)$-approximation by either uniform or importance sampling of $O(\varepsilon^{-2}n/\bar d)$ vertices. We additionally show how to use this coreset to recover a cut of expected value at least $OPT-\varepsilon n^2$, by making $O(n\varepsilon^{-2})$ additional edge-weight queries. Finally, our result extends to Max-$r$-Cut, giving an additive $\varepsilon n^2$-approximation using a sample of $O(\varepsilon^{-2}\log(2r))$ vertices.

Tuesday, November 03 2026, 01:00

Asymptotically Optimal Weighted Edge Colorings via Local Bin Packing

Authors: Rishikesh Gajjala, Arindam Khan

We study weighted edge coloring, where each edge of a graph $G$ has a weight in $(0,1]$, and colors are assigned so that the incident weight of each color at every vertex is at most one. The objective is to minimize the number of colors, denoted by $χ'_w(G)$. This generalizes classical edge coloring (all weights are one) and bin packing ($G$ is a star). Two natural lower bounds are the maximum weighted degree $Δ_w(G)$ and the maximum local bin-packing number $\mathrm{bp}(G)$: the maximum, over vertices, of the minimum number of unit bins needed to pack the incident weights. We have $\lceilΔ_w(G)\rceil\le\mathrm{bp}(G)\leχ'_w(G)$. How closely these local lower bounds approximate the global optimum is a longstanding open problem. Our main result establishes an asymptotic weighted analogue of Vizing's theorem: every simple graph admits a coloring using at most $(1+o(1))\mathrm{bp}(G)$ colors. Thus local bin packings can be made globally consistent with asymptotically vanishing relative loss, even in nonbipartite graphs. Our local-to-global rounding framework coordinates feasible local packing distributions into a partial coloring, leaving only $o(\mathrm{bp}(G))$ uncolored weight at every vertex. For bipartite multigraphs, we prove $χ'_w(G)\le(2+o(1))Δ_w(G)$, establishing the Chung--Ross conjecture [SICOMP'91] asymptotically. For loopless multigraphs, we obtain the exact bound $χ'_w(G)\le3\lceilΔ_w(G)\rceil$. Both leading coefficients are optimal, giving weighted analogues of the classical Kőnig--Shannon bounds. Further upper and lower bounds show that the classical factor-$3/2$ separation does not persist with the local bin-packing benchmark. We also disprove the exact weighted-degree conjecture of Feige and Singh [ESA'08]. Our upper bounds have polynomial-time algorithms, and their existence statements are formalized in Lean 4.

Authors: Rishikesh Gajjala, Arindam Khan

We study weighted edge coloring, where each edge of a graph $G$ has a weight in $(0,1]$, and colors are assigned so that the incident weight of each color at every vertex is at most one. The objective is to minimize the number of colors, denoted by $χ'_w(G)$. This generalizes classical edge coloring (all weights are one) and bin packing ($G$ is a star). Two natural lower bounds are the maximum weighted degree $Δ_w(G)$ and the maximum local bin-packing number $\mathrm{bp}(G)$: the maximum, over vertices, of the minimum number of unit bins needed to pack the incident weights. We have $\lceilΔ_w(G)\rceil\le\mathrm{bp}(G)\leχ'_w(G)$. How closely these local lower bounds approximate the global optimum is a longstanding open problem. Our main result establishes an asymptotic weighted analogue of Vizing's theorem: every simple graph admits a coloring using at most $(1+o(1))\mathrm{bp}(G)$ colors. Thus local bin packings can be made globally consistent with asymptotically vanishing relative loss, even in nonbipartite graphs. Our local-to-global rounding framework coordinates feasible local packing distributions into a partial coloring, leaving only $o(\mathrm{bp}(G))$ uncolored weight at every vertex. For bipartite multigraphs, we prove $χ'_w(G)\le(2+o(1))Δ_w(G)$, establishing the Chung--Ross conjecture [SICOMP'91] asymptotically. For loopless multigraphs, we obtain the exact bound $χ'_w(G)\le3\lceilΔ_w(G)\rceil$. Both leading coefficients are optimal, giving weighted analogues of the classical Kőnig--Shannon bounds. Further upper and lower bounds show that the classical factor-$3/2$ separation does not persist with the local bin-packing benchmark. We also disprove the exact weighted-degree conjecture of Feige and Singh [ESA'08]. Our upper bounds have polynomial-time algorithms, and their existence statements are formalized in Lean 4.

Tuesday, November 03 2026, 01:00

Faster Exact Algorithms for 3- and 4-Coloring: Deterministic Improvements and Further Randomized Speedups

Authors: Ryosuke Yamano, Tetsuo Shibuya

We study exact exponential-time algorithms for 3- and 4-coloring. For 3-coloring, the best previously published running-time bound is $O^*(1.3289^n)$, due to Eppstein (SODA 2001). We improve this bound to $O^*(1.3204^n)$ deterministically and $O^*(1.3159^n)$ using randomization. For 4-coloring, the best previously known running-time bound is $O^*(1.7159^n)$, due to Wu, Gu, Jiang, Shao, and Xu (ESA 2024). We improve this bound to $O^*(1.6975^n)$ deterministically and $O^*(1.6933^n)$ using randomization. Our algorithms for both problems combine constraint satisfaction algorithms with dynamic programming over tree and path decompositions, exploiting their complementary strengths on different graph structures. For 6-coloring, the best published randomized running-time bound is exponentially smaller than the best published deterministic bound. For 3- and 4-coloring, however, no such improvement through randomization was previously known. To the best of our knowledge, our results provide the first randomized running-time bounds for these two problems that exponentially improve upon the best previously published deterministic bounds. This raises the question of whether randomization offers an inherent exponential advantage for 3- and 4-coloring.

Authors: Ryosuke Yamano, Tetsuo Shibuya

We study exact exponential-time algorithms for 3- and 4-coloring. For 3-coloring, the best previously published running-time bound is $O^*(1.3289^n)$, due to Eppstein (SODA 2001). We improve this bound to $O^*(1.3204^n)$ deterministically and $O^*(1.3159^n)$ using randomization. For 4-coloring, the best previously known running-time bound is $O^*(1.7159^n)$, due to Wu, Gu, Jiang, Shao, and Xu (ESA 2024). We improve this bound to $O^*(1.6975^n)$ deterministically and $O^*(1.6933^n)$ using randomization. Our algorithms for both problems combine constraint satisfaction algorithms with dynamic programming over tree and path decompositions, exploiting their complementary strengths on different graph structures. For 6-coloring, the best published randomized running-time bound is exponentially smaller than the best published deterministic bound. For 3- and 4-coloring, however, no such improvement through randomization was previously known. To the best of our knowledge, our results provide the first randomized running-time bounds for these two problems that exponentially improve upon the best previously published deterministic bounds. This raises the question of whether randomization offers an inherent exponential advantage for 3- and 4-coloring.

Tuesday, November 03 2026, 01:00

Revisiting (Bijective) Burrows-Wheeler Transform Space-Time Trade-Off Constructions

Authors: Dominik Köppl, Tomohiro I

The bijective Burrows-Wheeler transform (BBWT) replaces the delimiter of the ordinary BWT by a Lyndon factorization and sorts factor conjugates in $ω$-order. It admits quadratic-time constant-workspace algorithms and linear-time linear-space algorithms. We extend the small-space BWT trade-off of Crochemore et al. [JDA,'15] to the BBWT and accelerate both constructions. For $τ\in [3..n]$, let $σ_τ\le\min(σ,τ+1)$ be the maximum construction-batch alphabet size. We construct either transform in $ O\!\left( \frac{n^2}τ \frac{\logτ}{\log\logτ} \right) $ time and $O(τσ_τ)$ additional space, assuming $τσ_τ=O(n)$. This improves the transferred $O((n^2/τ)\logτ)$ bound by a $\log\logτ$ factor. If $σ_τ\in w^{O(1)}$, where $w\inΩ(\log n)$ is the word size, the time improves to $ O\!\left( \frac{n^2}τ + \frac{n\logτ}{\log\logτ} \right). $ For inversion, the same bounds hold with the global alphabet size $σ$ in place of $σ_τ$.

Authors: Dominik Köppl, Tomohiro I

The bijective Burrows-Wheeler transform (BBWT) replaces the delimiter of the ordinary BWT by a Lyndon factorization and sorts factor conjugates in $ω$-order. It admits quadratic-time constant-workspace algorithms and linear-time linear-space algorithms. We extend the small-space BWT trade-off of Crochemore et al. [JDA,'15] to the BBWT and accelerate both constructions. For $τ\in [3..n]$, let $σ_τ\le\min(σ,τ+1)$ be the maximum construction-batch alphabet size. We construct either transform in $ O\!\left( \frac{n^2}τ \frac{\logτ}{\log\logτ} \right) $ time and $O(τσ_τ)$ additional space, assuming $τσ_τ=O(n)$. This improves the transferred $O((n^2/τ)\logτ)$ bound by a $\log\logτ$ factor. If $σ_τ\in w^{O(1)}$, where $w\inΩ(\log n)$ is the word size, the time improves to $ O\!\left( \frac{n^2}τ + \frac{n\logτ}{\log\logτ} \right). $ For inversion, the same bounds hold with the global alphabet size $σ$ in place of $σ_τ$.

Tuesday, November 03 2026, 01:00

Hardness Results for Approximate Distance Labeling in Sparse Graphs

Authors: Tsvi Kopelowitz, Ariel Korin, Liam Roditty

Distance labeling schemes assign short labels to the vertices of a graph so that distances between pairs of vertices can be recovered, exactly or approximately, from the two corresponding labels alone. For sparse graphs, Kosowski, Uznański, and Viennot [PODC'2019] showed that, assuming the hardness of the Sum-Index communication problem, exact distance labeling for sparse unweighted graphs requires near-linear label size. Their construction also gives lower bounds for stretch $1+o(1)$, but does not rule out substantially smaller labels for stretch $1+\varepsilon$, where $\varepsilon>0$ is fixed. We prove lower bounds for this $(1+\varepsilon)$-stretch regime through a density/stretch tradeoff. For every constant integer $k\ge 1$, we show that any approximate distance labeling scheme with stretch strictly smaller than $1+1/k$ for unweighted graphs with $m=O(n^{1+1/k})$ edges requires maximum label size at least $\mathrm{SumIndex}(n)/2^{O_k(\sqrt{\log n})}$ for infinitely many values of $n$. Thus, under the conjecture that $\mathrm{SumIndex}(n)=n^{1-o(1)}$, the maximum label size must be $n^{1-o(1)}$. The main new ingredient is an unweighted graph construction based on $3$-AP-free sets, which replaces the weighted uniqueness mechanism of the previous Sum-Index reduction by an additive uniqueness mechanism and keeps the relevant critical paths short enough to obtain a constant multiplicative distance gap. The same construction also yields near-tight lower bounds for related $2$-hop structures, including hub labeling schemes, shortcut sets, and approximate distance hopsets.

Authors: Tsvi Kopelowitz, Ariel Korin, Liam Roditty

Distance labeling schemes assign short labels to the vertices of a graph so that distances between pairs of vertices can be recovered, exactly or approximately, from the two corresponding labels alone. For sparse graphs, Kosowski, Uznański, and Viennot [PODC'2019] showed that, assuming the hardness of the Sum-Index communication problem, exact distance labeling for sparse unweighted graphs requires near-linear label size. Their construction also gives lower bounds for stretch $1+o(1)$, but does not rule out substantially smaller labels for stretch $1+\varepsilon$, where $\varepsilon>0$ is fixed. We prove lower bounds for this $(1+\varepsilon)$-stretch regime through a density/stretch tradeoff. For every constant integer $k\ge 1$, we show that any approximate distance labeling scheme with stretch strictly smaller than $1+1/k$ for unweighted graphs with $m=O(n^{1+1/k})$ edges requires maximum label size at least $\mathrm{SumIndex}(n)/2^{O_k(\sqrt{\log n})}$ for infinitely many values of $n$. Thus, under the conjecture that $\mathrm{SumIndex}(n)=n^{1-o(1)}$, the maximum label size must be $n^{1-o(1)}$. The main new ingredient is an unweighted graph construction based on $3$-AP-free sets, which replaces the weighted uniqueness mechanism of the previous Sum-Index reduction by an additive uniqueness mechanism and keeps the relevant critical paths short enough to obtain a constant multiplicative distance gap. The same construction also yields near-tight lower bounds for related $2$-hop structures, including hub labeling schemes, shortcut sets, and approximate distance hopsets.

Tuesday, November 03 2026, 01:00

Sketching in Regression: Sketchy in Practice?

Authors: Somarddha Das, Soupayan Das Gupta, Koustav Goswami

The Johnson-Lindenstrauss (JL) lemma and its extensions provide tools for solving the $L_2$ regression problem in high-dimensional settings with $n>d$, by transforming a subspace of $\mathbb{R}^{n}$ to a subspace of $\mathbb{R}^{m}$, (where $m

Authors: Somarddha Das, Soupayan Das Gupta, Koustav Goswami

The Johnson-Lindenstrauss (JL) lemma and its extensions provide tools for solving the $L_2$ regression problem in high-dimensional settings with $n>d$, by transforming a subspace of $\mathbb{R}^{n}$ to a subspace of $\mathbb{R}^{m}$, (where $m

Tuesday, November 03 2026, 01:00

Counting Feedback Vertex Sets of All Sizes in Subexponential Time, Conditionally

Authors: Mengyang Wu, Minghao Wang

Assuming a Boolean tensor contraction hypothesis (Theorem D.1 of OpenAI's recent preprint refuting the Exponential Time Hypothesis), we study exact feedback vertex set counting in general graphs. For an $n$-vertex, $m$-edge simple graph, we compute the complete size distribution deterministically in $\mathrm{poly}(n+m)\,2^{O(n(\log\log n)^2/\sqrt{\log n})}$ time and polynomial working space. The algorithm combines a whole-edge forest identity, subset-index scalar networks, sparse parity filtering, and disjoint high-degree branching. A binomial inversion recovers the multiplicities introduced by exceptional-set guesses. The counting interface supports fixed vertices, minimum solutions, and unranking. Component markers yield induced-tree and induced-path counts within the same bound. Sparse Hamilton, forest, and Steiner edge-tree counting follow from the scalar encoding.

Authors: Mengyang Wu, Minghao Wang

Assuming a Boolean tensor contraction hypothesis (Theorem D.1 of OpenAI's recent preprint refuting the Exponential Time Hypothesis), we study exact feedback vertex set counting in general graphs. For an $n$-vertex, $m$-edge simple graph, we compute the complete size distribution deterministically in $\mathrm{poly}(n+m)\,2^{O(n(\log\log n)^2/\sqrt{\log n})}$ time and polynomial working space. The algorithm combines a whole-edge forest identity, subset-index scalar networks, sparse parity filtering, and disjoint high-degree branching. A binomial inversion recovers the multiplicities introduced by exceptional-set guesses. The counting interface supports fixed vertices, minimum solutions, and unranking. Component markers yield induced-tree and induced-path counts within the same bound. Sparse Hamilton, forest, and Steiner edge-tree counting follow from the scalar encoding.

Tuesday, November 03 2026, 01:00

Large independent sets via neighborhood block dynamics, Glauber dynamics, and local occupancy

Authors: Ewan Davies, Brian Tan

Local occupancy can be used to bound the expected size of a hard-core independent set. The framework of locally stationary distributions shows that for triangle-free graphs, despite slow mixing at the relevant fugacity parameter, standard Glauber dynamics can be used to find sets close to the local occupancy guarantee. We improve the application of this framework to the hard-core model by replacing a conditional mixing argument with a more direct approach based on a short sequence of single-site updates that, from every initial configuration, achieves the required occupancy inequality. In general graphs, naive conditional mixing estimates or certain problematic structures in the neighborhood can in principle incur running time cost exponential in the maximum degree $Δ$. We design short update sequences that, via our relaxed criterion, yield running times polynomial in $Δ$ for graphs with a hereditary bound $b$ on the density of $K_{t,t}$ subgraphs in neighborhoods. Our methods handle the case where $t$ and $b$ can grow with the maximum degree, and the resulting algorithm attains the random graph shattering threshold when they grow sufficiently slowly. That is, we give improved local occupancy parameters and a short update sequence analysis which together show that Glauber dynamics finds independent sets of size $(1-o(1))n(\logΔ)/Δ$ in time polynomial in $n$ and $Δ$ whenever $t$ and $b$ are constant or grow suitably slowly. In a similar regime, we also establish that the (correspondence) chromatic number is at most $(1+o(1))Δ/\logΔ$ which improves upon prior results by a factor of at least $4$ in the case $t\ge 4$.

Authors: Ewan Davies, Brian Tan

Local occupancy can be used to bound the expected size of a hard-core independent set. The framework of locally stationary distributions shows that for triangle-free graphs, despite slow mixing at the relevant fugacity parameter, standard Glauber dynamics can be used to find sets close to the local occupancy guarantee. We improve the application of this framework to the hard-core model by replacing a conditional mixing argument with a more direct approach based on a short sequence of single-site updates that, from every initial configuration, achieves the required occupancy inequality. In general graphs, naive conditional mixing estimates or certain problematic structures in the neighborhood can in principle incur running time cost exponential in the maximum degree $Δ$. We design short update sequences that, via our relaxed criterion, yield running times polynomial in $Δ$ for graphs with a hereditary bound $b$ on the density of $K_{t,t}$ subgraphs in neighborhoods. Our methods handle the case where $t$ and $b$ can grow with the maximum degree, and the resulting algorithm attains the random graph shattering threshold when they grow sufficiently slowly. That is, we give improved local occupancy parameters and a short update sequence analysis which together show that Glauber dynamics finds independent sets of size $(1-o(1))n(\logΔ)/Δ$ in time polynomial in $n$ and $Δ$ whenever $t$ and $b$ are constant or grow suitably slowly. In a similar regime, we also establish that the (correspondence) chromatic number is at most $(1+o(1))Δ/\logΔ$ which improves upon prior results by a factor of at least $4$ in the case $t\ge 4$.

Tuesday, November 03 2026, 01:00

Generalized Flows over Time: Complexity, Structure and Algorithm

Authors: Siyuan Chen, Shengminjie Chen, Suixiang Gao, Zheyu Jiang, Dun Ma, Wenguo Yang

Generalized flows over time constitute a general network flow model that incorporates both transit times and flow losses or gains. We study their complexity, structure, and approximation. We prove that the generalized maximum flow over time problem remains NP-hard under standard binary encoding, even on lossy two-terminal series-parallel networks. The maximum temporally repeated restriction is APX-hard with general gains and NP-hard on lossy networks. We then consider the minimum source supply needed to satisfy a fixed sink demand within a given horizon, and prove that an optimal solution always exists without intermediate storage. On lossy networks, such a solution can additionally use only simple source--sink paths. These structural properties yield a fully polynomial-time approximation scheme via time condensation, with analogous schemes for generalized maximum flows over time and quickest flows with a source-supply budget.

Authors: Siyuan Chen, Shengminjie Chen, Suixiang Gao, Zheyu Jiang, Dun Ma, Wenguo Yang

Generalized flows over time constitute a general network flow model that incorporates both transit times and flow losses or gains. We study their complexity, structure, and approximation. We prove that the generalized maximum flow over time problem remains NP-hard under standard binary encoding, even on lossy two-terminal series-parallel networks. The maximum temporally repeated restriction is APX-hard with general gains and NP-hard on lossy networks. We then consider the minimum source supply needed to satisfy a fixed sink demand within a given horizon, and prove that an optimal solution always exists without intermediate storage. On lossy networks, such a solution can additionally use only simple source--sink paths. These structural properties yield a fully polynomial-time approximation scheme via time condensation, with analogous schemes for generalized maximum flows over time and quickest flows with a source-supply budget.

Tuesday, November 03 2026, 01:00

Submodularity Is Barely Testable

Authors: Diptaksho Palit, Yuichi Yoshida

In 2010, Seshadhri and Vondrák asked: is submodularity testable? For functions $f : \{ 0, 1 \}^n \to \mathbb{R}$, they gave a tester that makes $2^{O(\sqrt{n} \log n)}$ queries. We show that every tester, even an adaptive one with two-sided error, must make $2^{Ω(\sqrt{n})}$ queries, thereby concluding: submodularity is barely testable.

Authors: Diptaksho Palit, Yuichi Yoshida

In 2010, Seshadhri and Vondrák asked: is submodularity testable? For functions $f : \{ 0, 1 \}^n \to \mathbb{R}$, they gave a tester that makes $2^{O(\sqrt{n} \log n)}$ queries. We show that every tester, even an adaptive one with two-sided error, must make $2^{Ω(\sqrt{n})}$ queries, thereby concluding: submodularity is barely testable.

Tuesday, November 03 2026, 01:00

Faster graphic matroid parity: Breaking the $nm$ time complexity barrier

Authors: Tatsuya Terao

In the graphic matroid parity problem, we are given an undirected multigraph $G=(V,E)$ with $n$ vertices, together with a partition of $E$ into $m$ pairs of distinct edges, and the goal is to find a maximum-cardinality collection of pairs whose union is a forest. This problem generalizes both graph matching and graphic matroid intersection. Graph matching admits the celebrated $\tilde{O}(\sqrt{n}m)$-time algorithm of Micali--Vazirani~[FOCS'80], and graphic matroid intersection admits $o(nm)$-time algorithms due to Gabow--Stallmann~[ICALP'85] and Gabow--Xu~[FOCS'89]. In contrast, for graphic matroid parity, Gabow--Stallmann~[FOCS'84 \& Combinatorica'86] gave an $O(n^2 m)$-time algorithm, and its $\tilde{O}(nm)$-time implementation by Gabow--Stallmann~[ICALP'85] had long remained the best known bound, except in very dense regimes where $m=ω(n^3)$, in which case the algebraic $\tilde{O}(n^4)$-time algorithm of Cheung--Lau--Leung~[SODA'11 \& TALG'14] is faster. We present a randomized $\tilde{O}(n^{3/2} \sqrt{m} + \sqrt{n} m)$-time algorithm for graphic matroid parity. This is the first algorithm to break the $O(nm)$-time complexity barrier for this problem outside the very dense regime. Moreover, it improves over the previous bounds except in very sparse regimes where $m=\tilde{O}(n)$ and in very dense regimes where $m = \tildeΩ(n^{3.5})$. In particular, when $m=Θ(n^2)$, our algorithm runs in $\tilde{O}(n^{2.5})$ time, which nearly matches the $\tilde{O}(n^{2.5})$-time bounds of Micali--Vazirani for graph matching and of Gabow--Xu for graphic matroid intersection in this regime.

Authors: Tatsuya Terao

In the graphic matroid parity problem, we are given an undirected multigraph $G=(V,E)$ with $n$ vertices, together with a partition of $E$ into $m$ pairs of distinct edges, and the goal is to find a maximum-cardinality collection of pairs whose union is a forest. This problem generalizes both graph matching and graphic matroid intersection. Graph matching admits the celebrated $\tilde{O}(\sqrt{n}m)$-time algorithm of Micali--Vazirani~[FOCS'80], and graphic matroid intersection admits $o(nm)$-time algorithms due to Gabow--Stallmann~[ICALP'85] and Gabow--Xu~[FOCS'89]. In contrast, for graphic matroid parity, Gabow--Stallmann~[FOCS'84 \& Combinatorica'86] gave an $O(n^2 m)$-time algorithm, and its $\tilde{O}(nm)$-time implementation by Gabow--Stallmann~[ICALP'85] had long remained the best known bound, except in very dense regimes where $m=ω(n^3)$, in which case the algebraic $\tilde{O}(n^4)$-time algorithm of Cheung--Lau--Leung~[SODA'11 \& TALG'14] is faster. We present a randomized $\tilde{O}(n^{3/2} \sqrt{m} + \sqrt{n} m)$-time algorithm for graphic matroid parity. This is the first algorithm to break the $O(nm)$-time complexity barrier for this problem outside the very dense regime. Moreover, it improves over the previous bounds except in very sparse regimes where $m=\tilde{O}(n)$ and in very dense regimes where $m = \tildeΩ(n^{3.5})$. In particular, when $m=Θ(n^2)$, our algorithm runs in $\tilde{O}(n^{2.5})$ time, which nearly matches the $\tilde{O}(n^{2.5})$-time bounds of Micali--Vazirani for graph matching and of Gabow--Xu for graphic matroid intersection in this regime.

Tuesday, November 03 2026, 01:00

Near-Quadratic Error-Free Graph Matching above Otter's Threshold via Approximate Local Tree Correlation Testing

Authors: Jiale Cheng, Ziao Wang, Lei Ying

This paper presents a near-quadratic-time algorithm that correctly matches almost all nodes of two correlated Erdős-Rényi graphs without any incorrect matches. Each graph has average degree $λ$ and its edges are subsampled from a common parent Erdős-Rényi graph with $n$ nodes, with subsampling probability $s$. The nodes of one graph are then relabeled by an unknown permutation. The algorithm is built on a test for correlated versus independent Galton-Watson trees with approximated likelihood ratios. For every constant $s>\sqrt{C_{\mathrm{Otter}}}$, where $C_{\mathrm{Otter}}\approx0.338$ is Otter's tree-counting constant, and every $λ=ω(1)$ with $λ=(\log n)^{O(1)}$, the algorithm outputs a partial matching of size $n-o(n)$ that contains no incorrect matches with high probability. If, in addition, $λs\ge(1+ε)\log n$ for a constant $ε>0$, a seeded completion step recovers the permutation exactly. In both cases, the time complexity of the algorithm is $n^{2+o(1)}$ with high probability, whereas previous polynomial-time algorithms with the same guarantees either have a time-complexity bound whose exponent goes to $\infty$ as $s$ approaches $\sqrt{C_{\mathrm{Otter}}}$ from above, or apply only when $s$ is close to $1$.

Authors: Jiale Cheng, Ziao Wang, Lei Ying

This paper presents a near-quadratic-time algorithm that correctly matches almost all nodes of two correlated Erdős-Rényi graphs without any incorrect matches. Each graph has average degree $λ$ and its edges are subsampled from a common parent Erdős-Rényi graph with $n$ nodes, with subsampling probability $s$. The nodes of one graph are then relabeled by an unknown permutation. The algorithm is built on a test for correlated versus independent Galton-Watson trees with approximated likelihood ratios. For every constant $s>\sqrt{C_{\mathrm{Otter}}}$, where $C_{\mathrm{Otter}}\approx0.338$ is Otter's tree-counting constant, and every $λ=ω(1)$ with $λ=(\log n)^{O(1)}$, the algorithm outputs a partial matching of size $n-o(n)$ that contains no incorrect matches with high probability. If, in addition, $λs\ge(1+ε)\log n$ for a constant $ε>0$, a seeded completion step recovers the permutation exactly. In both cases, the time complexity of the algorithm is $n^{2+o(1)}$ with high probability, whereas previous polynomial-time algorithms with the same guarantees either have a time-complexity bound whose exponent goes to $\infty$ as $s$ approaches $\sqrt{C_{\mathrm{Otter}}}$ from above, or apply only when $s$ is close to $1$.

Tuesday, November 03 2026, 01:00

Faster Algorithms for Disjoint Paths in DAGs and Two Disjoint Shortest Paths

Authors: Yang Cao, Haoran Qi, Hanzhi Wang

For $k$ vertex-disjoint paths with prescribed endpoints in a directed acyclic graph (DAG), the $O(mn^{k-1})$-time pebbling algorithm of Fortune, Hopcroft, and Wyllie [TCS 1980] has remained the fastest for every $k\geq3$ for over four decades, even when the paths must be shortest. We break this bound for every $k\geq3$ with randomized algorithms running in expected $O_k(m+n^{k-3+ω+\varepsilon})$ time for every $\varepsilon>0$, where $ω<2.372$ is the matrix multiplication exponent; this is $O_k(m+n^{k-0.62})$. For two disjoint shortest paths, we find the paths as fast as we detect them: in expected linear time on DAGs and on undirected graphs with polynomially bounded integer weights, and in expected $O(\min\{mn,n^{ω+\varepsilon}\})$ time on directed graphs whose cycles have positive weight, which on dense graphs is $o(mn)$ even for detection. All algorithms evaluate a polynomial whose monomials are the solutions: a new first-collision circuit speeds up the evaluation in DAGs, and a correlated extraction step at cuts crossed once by every path finds a solution at the cost of a constant number of evaluations.

Authors: Yang Cao, Haoran Qi, Hanzhi Wang

For $k$ vertex-disjoint paths with prescribed endpoints in a directed acyclic graph (DAG), the $O(mn^{k-1})$-time pebbling algorithm of Fortune, Hopcroft, and Wyllie [TCS 1980] has remained the fastest for every $k\geq3$ for over four decades, even when the paths must be shortest. We break this bound for every $k\geq3$ with randomized algorithms running in expected $O_k(m+n^{k-3+ω+\varepsilon})$ time for every $\varepsilon>0$, where $ω<2.372$ is the matrix multiplication exponent; this is $O_k(m+n^{k-0.62})$. For two disjoint shortest paths, we find the paths as fast as we detect them: in expected linear time on DAGs and on undirected graphs with polynomially bounded integer weights, and in expected $O(\min\{mn,n^{ω+\varepsilon}\})$ time on directed graphs whose cycles have positive weight, which on dense graphs is $o(mn)$ even for detection. All algorithms evaluate a polynomial whose monomials are the solutions: a new first-collision circuit speeds up the evaluation in DAGs, and a correlated extraction step at cuts crossed once by every path finds a solution at the cost of a constant number of evaluations.

Tuesday, November 03 2026, 01:00

Faster Semi-Streaming and Dynamic Approximate Maximum Weight Matchings in General Graphs: A Unified Framework

Authors: Jiale Chen

We study the $(1-\varepsilon)$-approximate maximum weight matching problem in general graphs with $n$ vertices and integer edge weights in $[1,W]$. For $W=n^{O(1)}$, we obtain faster algorithms in the semi-streaming and dynamic settings: (1) Semi-streaming: Deterministic $O(\varepsilon^{-2})$-pass and $O(\varepsilon^{-2}\log(1/\varepsilon))$-pass algorithms for unweighted and weighted matching. (2) Incremental: A deterministic algorithm for weighted matching processing $m$ edge insertions in total time $O((m\varepsilon^{-3}+n\varepsilon^{-6})\log^3 n)$. (3) Decremental: A randomized algorithm for weighted matching processing all edge deletions from an $m$-edge graph in total time $O((m\varepsilon^{-8}+n\varepsilon^{-47})\log^{O(1)}n)$ with high probability. (4) Fully dynamic: A deterministic reduction from weighted matching in general graphs to unweighted matching in bipartite graphs incurring $O(\varepsilon^{-4}\log^{O(1)}(1/\varepsilon))$ multiplicative update-time overhead, independent of $n$. Underlying these results is a unified framework that reduces the problem to repeatedly solving two simpler matching subproblems: a $(1-\varepsilon)$-approximate maximum weight matching on a sparse subgraph with $O(n/\varepsilon)$ edges, and a constant approximation on the remaining edges. The framework builds on dual entropy regularization, a smoothing technique underlying efficient solvers for packing linear programs. To handle the exponentially many odd-set constraints, we adapt the smoothing technique to regularize only the vertex duals and develop a new cut partition that preserves odd-set feasibility during optimization.

Authors: Jiale Chen

We study the $(1-\varepsilon)$-approximate maximum weight matching problem in general graphs with $n$ vertices and integer edge weights in $[1,W]$. For $W=n^{O(1)}$, we obtain faster algorithms in the semi-streaming and dynamic settings: (1) Semi-streaming: Deterministic $O(\varepsilon^{-2})$-pass and $O(\varepsilon^{-2}\log(1/\varepsilon))$-pass algorithms for unweighted and weighted matching. (2) Incremental: A deterministic algorithm for weighted matching processing $m$ edge insertions in total time $O((m\varepsilon^{-3}+n\varepsilon^{-6})\log^3 n)$. (3) Decremental: A randomized algorithm for weighted matching processing all edge deletions from an $m$-edge graph in total time $O((m\varepsilon^{-8}+n\varepsilon^{-47})\log^{O(1)}n)$ with high probability. (4) Fully dynamic: A deterministic reduction from weighted matching in general graphs to unweighted matching in bipartite graphs incurring $O(\varepsilon^{-4}\log^{O(1)}(1/\varepsilon))$ multiplicative update-time overhead, independent of $n$. Underlying these results is a unified framework that reduces the problem to repeatedly solving two simpler matching subproblems: a $(1-\varepsilon)$-approximate maximum weight matching on a sparse subgraph with $O(n/\varepsilon)$ edges, and a constant approximation on the remaining edges. The framework builds on dual entropy regularization, a smoothing technique underlying efficient solvers for packing linear programs. To handle the exponentially many odd-set constraints, we adapt the smoothing technique to regularize only the vertex duals and develop a new cut partition that preserves odd-set feasibility during optimization.

Tuesday, November 03 2026, 01:00

Paying for Fractional Error: Tight Constant-Round Approximation of Covering Problems with Short Broadcasts

Authors: Saeed Amiri, Sebastian Siebertz, Wojciech Wawrzyniak

We present a general framework for deterministic constant-round approximation of covering problems in sparse networks in the broadcast CONGEST model, where every vertex broadcasts the same $O(\log n)$-bit message to all its neighbors. The framework combines two independent tools. The first is a solver that computes a $(1+δ)$-approximate solution of any local covering linear program with bounded positive integer costs on every class that excludes a fixed minor. The second is a rounding principle: a rounding rule that saves a fixed fraction of its own output, for every feasible input, pays for a solver error fixed in advance, so that an approximate fractional solution yields an exact coefficient. For ordinary domination, we achieve a $2a+1$ approximation against the fractional optimum $γ_f(G)$ for every fixed rational $a\ge1$, whenever every minor has a ratio of edges to vertices of at most $a$. On planar graphs this is seven, matching the lower bound of Hilke, Lenzen, and Suomela. The coefficients are also optimal at bounded treewidth and pathwidth, for $K_t$-minor-free graphs with $3\le t\le9$, on triangle-free planar graphs (five), and on planar graphs of large girth, where it approaches the optimal three. On every fixed surface the exact bound is $7γ(G)$, and we show that a bound of $7γ_f(G)$ is impossible on some fixed surfaces even sequentially. For total domination, on graphs without isolated vertices and under the same minor-density assumption, the exact coefficient is $2a$ against the fractional optimum, including the optimal six on planar graphs. Red-blue domination, and hence set cover and hitting set on planar incidence graphs, has the optimal limiting coefficient four. Our method also gives randomized constant-round and deterministic $O(\log^* n)$-round $(1+\varepsilon)$-approximation schemes, with the randomized cost guarantee holding in expectation.

Authors: Saeed Amiri, Sebastian Siebertz, Wojciech Wawrzyniak

We present a general framework for deterministic constant-round approximation of covering problems in sparse networks in the broadcast CONGEST model, where every vertex broadcasts the same $O(\log n)$-bit message to all its neighbors. The framework combines two independent tools. The first is a solver that computes a $(1+δ)$-approximate solution of any local covering linear program with bounded positive integer costs on every class that excludes a fixed minor. The second is a rounding principle: a rounding rule that saves a fixed fraction of its own output, for every feasible input, pays for a solver error fixed in advance, so that an approximate fractional solution yields an exact coefficient. For ordinary domination, we achieve a $2a+1$ approximation against the fractional optimum $γ_f(G)$ for every fixed rational $a\ge1$, whenever every minor has a ratio of edges to vertices of at most $a$. On planar graphs this is seven, matching the lower bound of Hilke, Lenzen, and Suomela. The coefficients are also optimal at bounded treewidth and pathwidth, for $K_t$-minor-free graphs with $3\le t\le9$, on triangle-free planar graphs (five), and on planar graphs of large girth, where it approaches the optimal three. On every fixed surface the exact bound is $7γ(G)$, and we show that a bound of $7γ_f(G)$ is impossible on some fixed surfaces even sequentially. For total domination, on graphs without isolated vertices and under the same minor-density assumption, the exact coefficient is $2a$ against the fractional optimum, including the optimal six on planar graphs. Red-blue domination, and hence set cover and hitting set on planar incidence graphs, has the optimal limiting coefficient four. Our method also gives randomized constant-round and deterministic $O(\log^* n)$-round $(1+\varepsilon)$-approximation schemes, with the randomized cost guarantee holding in expectation.

Tuesday, November 03 2026, 01:00

Multiplicative Weights for Mixed Block Constraints and Maximum Multicommodity Flow

Authors: Vladimir Kolmogorov

We study multiplicative-weights (MWU) algorithms for block-separable feasibility problems containing both packing and covering constraints. Each iteration optimizes a weighted linear objective over one block, either chosen uniformly at random or visited in a fixed cyclic order. We show that the cyclic algorithm returns an $O(\varepsilon)$-approximate solution after $O(kρ\varepsilon^{-2}\log m)$ oracle calls where $k$ is the number of blocks and $ρ$ is the width of the oracle. The same holds for the randomized version with high probability. Previously, such bounds were shown when the coupling constraints have the same sign [Radzik'95; Villavicencio-Grigoriadis'96]. These analyses critically depended on monotonicity arguments that are not available in the mixed case. As an application, we obtain a deterministic $(1-\varepsilon)$-approximation for maximum multicommodity flow (M-MCF) using $O(k\varepsilon^{-2}\log m\log(km/\varepsilon))$ approximate single-commodity min-cost-flow calls. For planar graphs, and more generally for graph families admitting a near-linear-time min-cost-flow algorithm, this gives total running time $\widetilde O(km\varepsilon^{-2})$. Compared with the $O(km^{1+o(1)}\varepsilon^{-1})$ algorithm of [Chen-Graur-Sidford'25], this improves the dependence on $m$ for these graph families at the cost of a worse dependence on $\varepsilon$. Finally, we give an exponentiation-free finite-precision implementation of the cyclic algorithm. It represents each internal weight by a logarithmic-bit mantissa and exponent, while every price supplied to the block oracle is rational and has a polynomially bounded common denominator.

Authors: Vladimir Kolmogorov

We study multiplicative-weights (MWU) algorithms for block-separable feasibility problems containing both packing and covering constraints. Each iteration optimizes a weighted linear objective over one block, either chosen uniformly at random or visited in a fixed cyclic order. We show that the cyclic algorithm returns an $O(\varepsilon)$-approximate solution after $O(kρ\varepsilon^{-2}\log m)$ oracle calls where $k$ is the number of blocks and $ρ$ is the width of the oracle. The same holds for the randomized version with high probability. Previously, such bounds were shown when the coupling constraints have the same sign [Radzik'95; Villavicencio-Grigoriadis'96]. These analyses critically depended on monotonicity arguments that are not available in the mixed case. As an application, we obtain a deterministic $(1-\varepsilon)$-approximation for maximum multicommodity flow (M-MCF) using $O(k\varepsilon^{-2}\log m\log(km/\varepsilon))$ approximate single-commodity min-cost-flow calls. For planar graphs, and more generally for graph families admitting a near-linear-time min-cost-flow algorithm, this gives total running time $\widetilde O(km\varepsilon^{-2})$. Compared with the $O(km^{1+o(1)}\varepsilon^{-1})$ algorithm of [Chen-Graur-Sidford'25], this improves the dependence on $m$ for these graph families at the cost of a worse dependence on $\varepsilon$. Finally, we give an exponentiation-free finite-precision implementation of the cyclic algorithm. It represents each internal weight by a logarithmic-bit mantissa and exponent, while every price supplied to the block oracle is rational and has a polynomially bounded common denominator.

Tuesday, November 03 2026, 01:00

Approximation Algorithms for Maximum-Weight Degree-Bounded Forests

Authors: Theophile Thiery, Vera Traub, Nicola Widmer

In the maximum-weight degree-bounded forest problem, we are given a graph with nonnegative edge weights and a degree upper bound on each vertex. The objective is to find a maximum-weight forest satisfying the degree bounds. We show that this problem admits a simple $2$-approximation via the greedy algorithm. This already improves over the best-known guarantees obtained via a reduction to weighted $3$-matroid parity, and sets a baseline to improve upon. Our main result is an $\frac{8}{5}$-approximation algorithm for the maximum-weight degree-bounded forest problem. For the special case of uniform degree bounds, our algorithm achieves an approximation ratio of $\frac{3}{2}$. It computes two sets of edges, each of which we then restrict to a feasible solution in an optimal way, and finally returns the better of the two resulting solutions. To obtain the initial two edge sets, we first relax the degree bounds and leverage an algorithm by Lau and Singh [JACM'15] for degree-bounded spanning trees, and second we relax the forest constraints and employ an algorithm for maximum-weight $b$-matchings.

Authors: Theophile Thiery, Vera Traub, Nicola Widmer

In the maximum-weight degree-bounded forest problem, we are given a graph with nonnegative edge weights and a degree upper bound on each vertex. The objective is to find a maximum-weight forest satisfying the degree bounds. We show that this problem admits a simple $2$-approximation via the greedy algorithm. This already improves over the best-known guarantees obtained via a reduction to weighted $3$-matroid parity, and sets a baseline to improve upon. Our main result is an $\frac{8}{5}$-approximation algorithm for the maximum-weight degree-bounded forest problem. For the special case of uniform degree bounds, our algorithm achieves an approximation ratio of $\frac{3}{2}$. It computes two sets of edges, each of which we then restrict to a feasible solution in an optimal way, and finally returns the better of the two resulting solutions. To obtain the initial two edge sets, we first relax the degree bounds and leverage an algorithm by Lau and Singh [JACM'15] for degree-bounded spanning trees, and second we relax the forest constraints and employ an algorithm for maximum-weight $b$-matchings.

Tuesday, November 03 2026, 01:00

Efficient Lempel-Ziv Prediction with the Run-Length BWT

Authors: Ohad Avnery

Lempel-Ziv Plurality (LZP) is an algorithm of Kosoy that predicts the next symbol of a string by using its Lempel-Ziv factorization, and makes at most $O(z\log n)$ mistakes on every string of length $n$ with $z$ LZ77 phrases. We give efficient online implementations of LZP and of a new variant of it. Our first implementation maintains a run-length encoded Burrows-Wheeler transform (RLBWT) of the reversed history. It implements LZP, uses $O(z\log z\log^2 n)$ bits, and has worst-case update and prediction times $O(\log z+\log\log n)$ and $O(σ(\log z+\log\log n))$, respectively, where $σ$ is the size of the alphabet. This is an exponential improvement over Kosoy's implementation, whose update time is $O(z\log^7 n)$. We then introduce the Lempel-Ziv Majority (LZM) predictor, which has the same mistake bound and update time, and predicts in $O(1)$ time independently of $σ$. These two predictors accept new symbols at any time, without knowing the alphabet in advance. For an alphabet known in advance, a different data structure gives $O(σ(1+\log z/\log\log n))$ time per symbol, which is constant when $σ=O(1)$ and $z=\mathrm{polylog}(n)$.

Authors: Ohad Avnery

Lempel-Ziv Plurality (LZP) is an algorithm of Kosoy that predicts the next symbol of a string by using its Lempel-Ziv factorization, and makes at most $O(z\log n)$ mistakes on every string of length $n$ with $z$ LZ77 phrases. We give efficient online implementations of LZP and of a new variant of it. Our first implementation maintains a run-length encoded Burrows-Wheeler transform (RLBWT) of the reversed history. It implements LZP, uses $O(z\log z\log^2 n)$ bits, and has worst-case update and prediction times $O(\log z+\log\log n)$ and $O(σ(\log z+\log\log n))$, respectively, where $σ$ is the size of the alphabet. This is an exponential improvement over Kosoy's implementation, whose update time is $O(z\log^7 n)$. We then introduce the Lempel-Ziv Majority (LZM) predictor, which has the same mistake bound and update time, and predicts in $O(1)$ time independently of $σ$. These two predictors accept new symbols at any time, without knowing the alphabet in advance. For an alphabet known in advance, a different data structure gives $O(σ(1+\log z/\log\log n))$ time per symbol, which is constant when $σ=O(1)$ and $z=\mathrm{polylog}(n)$.

Tuesday, November 03 2026, 01:00

Monotone Local Search with Sample-Dependent Running Times

Authors: Serge Gaspers

Monotone Local Search turns an algorithm that extends a given partial solution into an algorithm that finds a solution from scratch: sample a few elements, hope that they lie in an unknown solution, and complete them. Its known analyses bound every call of the extension algorithm by a worst-case time that depends only on the number of sampled elements, and Esmer et al. (SODA 2024) showed that no algorithm that uses the extension algorithm as a black box with this running time beats the original form by more than a polynomial factor. We let this running time depend on the actual sample. If the extension algorithm decides in time $O^*(2^{g(I,X,k)})$, for a polynomial-time computable $g$, whether a sample $X$ can be completed to a solution with at most $k$ elements, our main theorem gives a randomized algorithm with one-sided error in time $O^*(2^{ψn})$ whenever, for some solution $S$ of size $k$ and some $t$, the exponential moment $\sum_{X\subseteq S,|X|=t}2^{-g(I,X,k)}$ is at least $2^{-ψn}\binom{n}{t}$. A corollary, multivariate Monotone Local Search, generalizes the theorems of Fomin et al. (STOC 2016, J. ACM 2019) and of Gaspers and Lee (ICALP 2017). For Max Leaf Spanning Tree and, equivalently, Connected Dominating Set, the previous best polynomial-space bound is $O^*(1.8619^n)$. Completing a sample is a Steiner tree problem, solved in time exponential in the number of terminals, which varies strongly with the sample. We obtain a randomized polynomial-space algorithm in time $O(1.6991^n)$ and a deterministic one in time $(\sqrt5-\frac12)^n 2^{o(n)}=O(1.7361^n)$. For connectivity problems in which the extension algorithm's running time doubles with every component of the sample, an analysis through a partition function gives randomized polynomial-space algorithms in time $O^*(1.8^n)$ for Connected Feedback Vertex Set and $O(1.6444^n)$ for Connected Vertex Cover.

Authors: Serge Gaspers

Monotone Local Search turns an algorithm that extends a given partial solution into an algorithm that finds a solution from scratch: sample a few elements, hope that they lie in an unknown solution, and complete them. Its known analyses bound every call of the extension algorithm by a worst-case time that depends only on the number of sampled elements, and Esmer et al. (SODA 2024) showed that no algorithm that uses the extension algorithm as a black box with this running time beats the original form by more than a polynomial factor. We let this running time depend on the actual sample. If the extension algorithm decides in time $O^*(2^{g(I,X,k)})$, for a polynomial-time computable $g$, whether a sample $X$ can be completed to a solution with at most $k$ elements, our main theorem gives a randomized algorithm with one-sided error in time $O^*(2^{ψn})$ whenever, for some solution $S$ of size $k$ and some $t$, the exponential moment $\sum_{X\subseteq S,|X|=t}2^{-g(I,X,k)}$ is at least $2^{-ψn}\binom{n}{t}$. A corollary, multivariate Monotone Local Search, generalizes the theorems of Fomin et al. (STOC 2016, J. ACM 2019) and of Gaspers and Lee (ICALP 2017). For Max Leaf Spanning Tree and, equivalently, Connected Dominating Set, the previous best polynomial-space bound is $O^*(1.8619^n)$. Completing a sample is a Steiner tree problem, solved in time exponential in the number of terminals, which varies strongly with the sample. We obtain a randomized polynomial-space algorithm in time $O(1.6991^n)$ and a deterministic one in time $(\sqrt5-\frac12)^n 2^{o(n)}=O(1.7361^n)$. For connectivity problems in which the extension algorithm's running time doubles with every component of the sample, an analysis through a partition function gives randomized polynomial-space algorithms in time $O^*(1.8^n)$ for Connected Feedback Vertex Set and $O(1.6444^n)$ for Connected Vertex Cover.

Tuesday, November 03 2026, 01:00

Fresh-Priority Eviction for Cuckoo Filters: Exploiting Latent Randomness Under XOR Hashing

Authors: Min Liu

Cuckoo filters are a widely adopted approximate membership query (AMQ) data structure that supports insertions, lookups, and deletions with low memory overhead. At high load factors, however, insertion performance degrades sharply due to the random-walk eviction process, which may require a large number of element relocations. Recent work on bucketized cuckoo hash tables has shown that \emph{fresh-priority eviction}, which preferentially evicts elements whose alternate bucket location has not yet been used, achieves $O(\ell m)$ total expected evictions for bucket size~$\ell$, an improvement over the standard random walk. In this paper, we show that this technique transfers to \emph{cuckoo filters} despite their use of XOR-based partial-key hashing. The key insight is that a fresh element's fingerprint, although stored in the table, is not examined by the eviction policy, so when the fingerprint length satisfies $k = Ω(\log m)$, the alternate bucket hash remains a near-uniform random variable, a property we call \emph{latent randomness}. We prove that under this condition the corruption graph remains subcritical and the total number of evictions is $O(\ell m \log m)$; we conjecture the tighter $O(\ell m)$ bound holds via the full amortized analysis of Bell and Kuszmaul. We further propose encoding the fresh/kicked state implicitly by the element's position within its bucket, requiring no additional per-element metadata. Experimental results on a filter with $2^{22}$ slots and $8$-bit fingerprints demonstrate $1.2\times$ to $2.4\times$ fewer evictions at load factors between $94\%$ and $99\%$ for bucket size $\ell = 8$, and a $1.73\times$ reduction in cumulative evictions to full capacity.

Authors: Min Liu

Cuckoo filters are a widely adopted approximate membership query (AMQ) data structure that supports insertions, lookups, and deletions with low memory overhead. At high load factors, however, insertion performance degrades sharply due to the random-walk eviction process, which may require a large number of element relocations. Recent work on bucketized cuckoo hash tables has shown that \emph{fresh-priority eviction}, which preferentially evicts elements whose alternate bucket location has not yet been used, achieves $O(\ell m)$ total expected evictions for bucket size~$\ell$, an improvement over the standard random walk. In this paper, we show that this technique transfers to \emph{cuckoo filters} despite their use of XOR-based partial-key hashing. The key insight is that a fresh element's fingerprint, although stored in the table, is not examined by the eviction policy, so when the fingerprint length satisfies $k = Ω(\log m)$, the alternate bucket hash remains a near-uniform random variable, a property we call \emph{latent randomness}. We prove that under this condition the corruption graph remains subcritical and the total number of evictions is $O(\ell m \log m)$; we conjecture the tighter $O(\ell m)$ bound holds via the full amortized analysis of Bell and Kuszmaul. We further propose encoding the fresh/kicked state implicitly by the element's position within its bucket, requiring no additional per-element metadata. Experimental results on a filter with $2^{22}$ slots and $8$-bit fingerprints demonstrate $1.2\times$ to $2.4\times$ fewer evictions at load factors between $94\%$ and $99\%$ for bucket size $\ell = 8$, and a $1.73\times$ reduction in cumulative evictions to full capacity.

Tuesday, November 03 2026, 01:00

Smoothed analysis of stability radii for the assignment problem

Authors: Michael P. Friedlander

Every linear program can tolerate small changes to its costs and still yield a solution of the original problem. The size of this tolerance measures how robust an optimal decision is to noise or numerical error, and how much regularization the problem can support. Classical sensitivity analysis can determine this tolerance only from an optimal solution. This paper provides a priori lower bounds on two measures of this tolerance -- the optimality gap and the $\infty$-norm stability radius -- for the $n$-by-$n$ linear assignment problem over the Birkhoff polytope. The bounds derive from a smoothed analysis, which adds Gaussian noise with standard deviation of at least $σ$ in every direction to an arbitrary base cost. At every fixed confidence level, uniformly over the base cost, both quantities are at least of order $σ/(n\sqrt{\log n})$. The radius bound is asymptotically sharp. Under isotropic noise, the worst case occurs when every assignment has equal cost, and in this case, a scaled stability radius converges in distribution to a standard exponential variable.

Authors: Michael P. Friedlander

Every linear program can tolerate small changes to its costs and still yield a solution of the original problem. The size of this tolerance measures how robust an optimal decision is to noise or numerical error, and how much regularization the problem can support. Classical sensitivity analysis can determine this tolerance only from an optimal solution. This paper provides a priori lower bounds on two measures of this tolerance -- the optimality gap and the $\infty$-norm stability radius -- for the $n$-by-$n$ linear assignment problem over the Birkhoff polytope. The bounds derive from a smoothed analysis, which adds Gaussian noise with standard deviation of at least $σ$ in every direction to an arbitrary base cost. At every fixed confidence level, uniformly over the base cost, both quantities are at least of order $σ/(n\sqrt{\log n})$. The radius bound is asymptotically sharp. Under isotropic noise, the worst case occurs when every assignment has equal cost, and in this case, a scaled stability radius converges in distribution to a standard exponential variable.

Tuesday, November 03 2026, 01:00

A logarithm-free subexponential algorithm for minimum quartet inconsistency

Authors: Chuang-Chieh Lin

We give a deterministic algorithm for complete, unweighted minimum quartet inconsistency (MQI) with time and space $2^{O(k^{1/4})}+n^{O(1)}$, where $n$ is the number of taxa and~$k$ is the allowed number of quartet disagreements. This removes the logarithmic factor from the exponent of the MQI bound announced in 2009. The algorithm combines low-error anchoring, chromatic coding, and dynamic programming that accounts for disagreements with all input quartets, with a refined count of colorings and merging states. We also record a generic ETH-based lower bound. The optimality of the fourth-root exponent remains open.

Authors: Chuang-Chieh Lin

We give a deterministic algorithm for complete, unweighted minimum quartet inconsistency (MQI) with time and space $2^{O(k^{1/4})}+n^{O(1)}$, where $n$ is the number of taxa and~$k$ is the allowed number of quartet disagreements. This removes the logarithmic factor from the exponent of the MQI bound announced in 2009. The algorithm combines low-error anchoring, chromatic coding, and dynamic programming that accounts for disagreements with all input quartets, with a refined count of colorings and merging states. We also record a generic ETH-based lower bound. The optimality of the fourth-root exponent remains open.

Tuesday, November 03 2026, 01:00

A Deterministic O(log m)-Competitive Algorithm for Online Minimum Cost Bipartite Matching with Delays

Authors: Tung-Wei Kuo

In the online Minimum cost Bipartite Perfect Matching with Delays (MBPMD) problem, $m$ requests and $m$ servers arrive over time in a metric space. An online algorithm has to match each request to a server and can postpone a matching decision by paying the delay cost. The objective is to minimize the total distance and delay cost. Azar, Chiplunkar, and Kaplan gave a randomized $O(\log n)$-competitive algorithm for metric spaces with $n$ points that are known in advance. For deterministic algorithms, however, the best known competitive ratio in general metric spaces was $O(m^{0.59})$, due to Azar and Jacob-Fanani. On the line, Kuo proposed the Virtual Robust Matching (VRM) algorithm and showed that it is $O(\sqrt{m}\log^2 m)$-competitive. We prove that VRM is $O(\log m)$-competitive for the MBPMD problem in any metric space. VRM is deterministic and does not need to know the metric space or $m$ in advance. It is the first deterministic polylogarithmic-competitive algorithm for the MBPMD problem in general metric spaces. Unlike the previous analysis of VRM, our analysis does not rely on a geometric decomposition of the metric space. Instead, it charges the waiting time of requests over time to the edges of a fixed optimal matching. We also extend VRM to the non-bipartite version and prove that the resulting deterministic algorithm is $O(\log m)$-competitive. This improves the $O(\log^5 m)$ bound of Dufay and Wattenhofer and matches a concurrent result of Ahmadian, Chawla, Kumar, Purohit, and Zhang. The proof that VRM is $O(\log m)$-competitive and its non-bipartite extension were found by GPT-6 Astra (OpenAI) with almost no human input. We used Claude Opus 5.5 (Anthropic) to simplify and present the results. See the AI Usage and Methodology section for details. We take full responsibility for this paper.

Authors: Tung-Wei Kuo

In the online Minimum cost Bipartite Perfect Matching with Delays (MBPMD) problem, $m$ requests and $m$ servers arrive over time in a metric space. An online algorithm has to match each request to a server and can postpone a matching decision by paying the delay cost. The objective is to minimize the total distance and delay cost. Azar, Chiplunkar, and Kaplan gave a randomized $O(\log n)$-competitive algorithm for metric spaces with $n$ points that are known in advance. For deterministic algorithms, however, the best known competitive ratio in general metric spaces was $O(m^{0.59})$, due to Azar and Jacob-Fanani. On the line, Kuo proposed the Virtual Robust Matching (VRM) algorithm and showed that it is $O(\sqrt{m}\log^2 m)$-competitive. We prove that VRM is $O(\log m)$-competitive for the MBPMD problem in any metric space. VRM is deterministic and does not need to know the metric space or $m$ in advance. It is the first deterministic polylogarithmic-competitive algorithm for the MBPMD problem in general metric spaces. Unlike the previous analysis of VRM, our analysis does not rely on a geometric decomposition of the metric space. Instead, it charges the waiting time of requests over time to the edges of a fixed optimal matching. We also extend VRM to the non-bipartite version and prove that the resulting deterministic algorithm is $O(\log m)$-competitive. This improves the $O(\log^5 m)$ bound of Dufay and Wattenhofer and matches a concurrent result of Ahmadian, Chawla, Kumar, Purohit, and Zhang. The proof that VRM is $O(\log m)$-competitive and its non-bipartite extension were found by GPT-6 Astra (OpenAI) with almost no human input. We used Claude Opus 5.5 (Anthropic) to simplify and present the results. See the AI Usage and Methodology section for details. We take full responsibility for this paper.

Tuesday, November 03 2026, 01:00

Deterministic $n^{O(\log n)}$ Graph Isomorphism Algorithm via Inherited Decompositions

Authors: MohammadTaghi Hajiaghayi, Mahdi JafariRaviz

We give a deterministic $n^{O(\log n)}$-time algorithm for graph isomorphism. To our knowledge, this is the first proof of such a worst-case bound for general graphs. The algorithm uses the local-certificate and color-refinement steps of Babai's quasipolynomial algorithm, but retains the decomposition of each recursive instance when it solves an auxiliary coloring problem. There are two reasons for retaining this information. First, a local test can make progress without removing many vertices: it can eliminate a layer of permutation symmetry. A logarithmic bound on the total weight of these layers allows tests of constant size. Second, the auxiliary bipartite graphs arising in the symmetry-breaking step can be represented inside the current instance, after retaining a row decomposition when necessary. These recursive calls inherit the progress already made. The analysis gives a recurrence with quadratic logarithmic cost and a fixed power decrease in every fresh recursive call, improving the explicit cubic logarithmic exponent in the analysis of the corrected Luks--Babai algorithm. We present the algorithm through colorings, partitions, and bipartite graphs, and collect the permutation-group ingredients in one appendix. The result extends to string isomorphism and returns one isomorphism and generators for the full automorphism group.

Authors: MohammadTaghi Hajiaghayi, Mahdi JafariRaviz

We give a deterministic $n^{O(\log n)}$-time algorithm for graph isomorphism. To our knowledge, this is the first proof of such a worst-case bound for general graphs. The algorithm uses the local-certificate and color-refinement steps of Babai's quasipolynomial algorithm, but retains the decomposition of each recursive instance when it solves an auxiliary coloring problem. There are two reasons for retaining this information. First, a local test can make progress without removing many vertices: it can eliminate a layer of permutation symmetry. A logarithmic bound on the total weight of these layers allows tests of constant size. Second, the auxiliary bipartite graphs arising in the symmetry-breaking step can be represented inside the current instance, after retaining a row decomposition when necessary. These recursive calls inherit the progress already made. The analysis gives a recurrence with quadratic logarithmic cost and a fixed power decrease in every fresh recursive call, improving the explicit cubic logarithmic exponent in the analysis of the corrected Luks--Babai algorithm. We present the algorithm through colorings, partitions, and bipartite graphs, and collect the permutation-group ingredients in one appendix. The result extends to string isomorphism and returns one isomorphism and generators for the full automorphism group.

Tuesday, November 03 2026, 01:00

Three-phase time minimization transportation problem with non-idle phases: a polynomial two-witness algorithm

Authors: Sailesh Kamalahasan, Raghav Paladugu, Archana Khurana

The three-phase time minimization transportation problem partitions the origin-terminal links into three sets used in consecutive phases and minimizes the sum of the phase shipment times. Optimal solutions may leave a phase idle, as in the main example of the original study. We require every phase to carry a positive integer quantity while leaving the split of the total requirement across phases free. Discarding idle solutions from the original algorithm does not, in general, solve this variant. We prove that the optimum is attained by fixing one witness link in each of two phases and deciding the third phase by a single network flow computation, with the optimal Phase-3 time of each witness pair located by binary search. The resulting exact algorithm is strongly polynomial. It extends to any fixed number of phases, whereas, when the number of phases is part of the input, even non-idle feasibility is strongly NP-complete; feasibility alone is fixed-parameter tractable in the number of phases via known rainbow-matching results. We also characterize the minimum time of each phase. On the main example, non-idleness raises the optimum from 38 to 41. On 3,650 random instances with up to $50\times50$ links the algorithm took at most 50.2 seconds, and non-idleness raised the optimal value in 41% of the 2,250 instances with equal phase sizes; on 90 sampled instances it was faster than a mixed integer programming solver on 80 with a direct formulation and on 65 with a tighter threshold formulation.

Authors: Sailesh Kamalahasan, Raghav Paladugu, Archana Khurana

The three-phase time minimization transportation problem partitions the origin-terminal links into three sets used in consecutive phases and minimizes the sum of the phase shipment times. Optimal solutions may leave a phase idle, as in the main example of the original study. We require every phase to carry a positive integer quantity while leaving the split of the total requirement across phases free. Discarding idle solutions from the original algorithm does not, in general, solve this variant. We prove that the optimum is attained by fixing one witness link in each of two phases and deciding the third phase by a single network flow computation, with the optimal Phase-3 time of each witness pair located by binary search. The resulting exact algorithm is strongly polynomial. It extends to any fixed number of phases, whereas, when the number of phases is part of the input, even non-idle feasibility is strongly NP-complete; feasibility alone is fixed-parameter tractable in the number of phases via known rainbow-matching results. We also characterize the minimum time of each phase. On the main example, non-idleness raises the optimum from 38 to 41. On 3,650 random instances with up to $50\times50$ links the algorithm took at most 50.2 seconds, and non-idleness raised the optimal value in 41% of the 2,250 instances with equal phase sizes; on 90 sampled instances it was faster than a mixed integer programming solver on 80 with a direct formulation and on 65 with a tighter threshold formulation.

Tuesday, November 03 2026, 01:00

Sharper Bounds for Graphical Two-Choice Allocation

Authors: Obinna Okechukwu

In graphical two-choice allocation, vertices are bins, and each arriving ball must be placed irrevocably at one endpoint of a uniformly random edge. The objective is to keep the maximum-minus-minimum load gap small at every time. On cycles, endpoint-local monotone rules incur polynomial gaps at long time horizons, whereas global rules can achieve polylogarithmic gaps. We prove a joint exponential-moment inequality for the orthogonal contrasts of a binary partition tree, controlling all contrasts along a root-to-leaf path simultaneously. Applied to the Bansal--Feldheim strategy, it improves the gap on every $κ$-edge-connected $d$-regular graph from $O((d/κ)\log^4 n\log\log n)$ to $O((d/κ)\log^3 n\log\log n)$. We give an explicit strategy on the $n$-cycle with expected gap $O(\log^2 n)$ uniformly over deterministic times and exponential tails at each such time. An Euler-tour projection yields $O(\bar d\log^2(2M))$ for every connected graph with $M$ edges and degree-proportional capacities. For lower bounds, we introduce a stochastic cut workload: the expected largest normalized overload forced by the requests, plus an indivisibility term. For every online integral strategy, the supremum over time of the expected gap is at least one sixth of this benchmark. Regular rings of cliques attain $Ω((d/κ)\log n)$ along an explicit family, improving the prior additive lower bound. With unequal capacities, the workload need not control the gap: one family has constant relative cut slack, bounded workload and gap $Ω(\log\log n)$; another has workload at most $20$ but forces gap $Ω(\log n)$ even for clairvoyant fractional allocations that remain irrevocable.

Authors: Obinna Okechukwu

In graphical two-choice allocation, vertices are bins, and each arriving ball must be placed irrevocably at one endpoint of a uniformly random edge. The objective is to keep the maximum-minus-minimum load gap small at every time. On cycles, endpoint-local monotone rules incur polynomial gaps at long time horizons, whereas global rules can achieve polylogarithmic gaps. We prove a joint exponential-moment inequality for the orthogonal contrasts of a binary partition tree, controlling all contrasts along a root-to-leaf path simultaneously. Applied to the Bansal--Feldheim strategy, it improves the gap on every $κ$-edge-connected $d$-regular graph from $O((d/κ)\log^4 n\log\log n)$ to $O((d/κ)\log^3 n\log\log n)$. We give an explicit strategy on the $n$-cycle with expected gap $O(\log^2 n)$ uniformly over deterministic times and exponential tails at each such time. An Euler-tour projection yields $O(\bar d\log^2(2M))$ for every connected graph with $M$ edges and degree-proportional capacities. For lower bounds, we introduce a stochastic cut workload: the expected largest normalized overload forced by the requests, plus an indivisibility term. For every online integral strategy, the supremum over time of the expected gap is at least one sixth of this benchmark. Regular rings of cliques attain $Ω((d/κ)\log n)$ along an explicit family, improving the prior additive lower bound. With unequal capacities, the workload need not control the gap: one family has constant relative cut slack, bounded workload and gap $Ω(\log\log n)$; another has workload at most $20$ but forces gap $Ω(\log n)$ even for clairvoyant fractional allocations that remain irrevocable.

Tuesday, November 03 2026, 01:00

Metric Testing: A Characterization of One-Sided Testability

Authors: Zihan Tan, Theo Teske

We give a unified framework for one-sided testability (constant-query testing) of metric families. For a broad class of families defined by fixed local predicates, we prove that testability is equivalent to two structural properties: \emph{packing amplification}, which turns many edge-disjoint local violations into a positive density of violations, and \emph{completion stability}, which allows locally consistent partial matrices with few missing entries to be cheaply completed. This characterization yields a common proof strategy and simpler proofs of classical testability results for ultrametrics, tree pseudometrics, Euclidean pseudometrics in fixed dimension, and several others. Our framework also gives new quantitative bounds for metric testing: a nonadaptive one-sided upper bound for matrices with entries in the bounded range $[1,C]$, and two-sided nonadaptive and adaptive lower bounds for general metrics.

Authors: Zihan Tan, Theo Teske

We give a unified framework for one-sided testability (constant-query testing) of metric families. For a broad class of families defined by fixed local predicates, we prove that testability is equivalent to two structural properties: \emph{packing amplification}, which turns many edge-disjoint local violations into a positive density of violations, and \emph{completion stability}, which allows locally consistent partial matrices with few missing entries to be cheaply completed. This characterization yields a common proof strategy and simpler proofs of classical testability results for ultrametrics, tree pseudometrics, Euclidean pseudometrics in fixed dimension, and several others. Our framework also gives new quantitative bounds for metric testing: a nonadaptive one-sided upper bound for matrices with entries in the bounded range $[1,C]$, and two-sided nonadaptive and adaptive lower bounds for general metrics.

Tuesday, November 03 2026, 01:00

Reading between the levels: new tools for the Thorup-Zwick hierarchy and their applications

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

In this paper, we develop three new analytical tools for the Thorup-Zwick hierarchy: the Mixed-Level Growth, Two-Level Growth, and Long-Short Growth lemmas. Using these tools, we obtain improved approximate distance oracles (ADOs), and algorithms for $n$-pairs shortest paths ($n$-PSP), multiple-source shortest paths (MSSP), and $ST$-shortest paths, as well as new hopsets. One of our main results is a $(\frac 65 k+O(1))$-stretch ADO with $\tilde O(mn^{1/k})$ preprocessing time and space (where $\tilde O$ hides polylogarithmic factors), and $\tilde O(n^{1/k})$ query time, breaking, for sparse graphs, the $(2k-O(1))$-stretch barrier for ADOs with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. In unweighted graphs, we also obtain a $(\frac 65 k+O(1),O(k))$-approximate ADO with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. This also yields a $(\frac 65 k+O(1))$-approximation algorithm for $m$-pairs shortest paths ($m$-PSP) in $\tilde O(mn^{1/k})$ time, answering the question of Chechik, Hoch, and Lifshitz [SODA'25] of whether the $2k-O(1)$ barrier for $n$-PSP in $\tilde O(mn^{1/k})$ time can be broken. For unweighted graphs, we give a new analysis of the existing $\tilde O(mn^{1/k}+n^{1+2/k})$-time $n$-PSP algorithm of [Chechik et al., SODA'25], and show that, without any modification, it is a $(k,k-1)$-approximation, i.e., $d(u,v)\le\hat d(u,v)\le k\cdot d(u,v)+k-1$ for every queried pair. This improves its multiplicative factor from $\frac 43 k+O(1)$ to $k$, and the leading coefficient $k$ is tight for this algorithm. Moreover, for graphs with $m=Ω(n^{1+1/k})$, this answers the question of Dalirrooyfard et al. [FOCS'22] up to the additive $k-1$.

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

In this paper, we develop three new analytical tools for the Thorup-Zwick hierarchy: the Mixed-Level Growth, Two-Level Growth, and Long-Short Growth lemmas. Using these tools, we obtain improved approximate distance oracles (ADOs), and algorithms for $n$-pairs shortest paths ($n$-PSP), multiple-source shortest paths (MSSP), and $ST$-shortest paths, as well as new hopsets. One of our main results is a $(\frac 65 k+O(1))$-stretch ADO with $\tilde O(mn^{1/k})$ preprocessing time and space (where $\tilde O$ hides polylogarithmic factors), and $\tilde O(n^{1/k})$ query time, breaking, for sparse graphs, the $(2k-O(1))$-stretch barrier for ADOs with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. In unweighted graphs, we also obtain a $(\frac 65 k+O(1),O(k))$-approximate ADO with $\tilde O(n^{1+1/k})$ space and $\tilde O(n^{1/k})$ query time. This also yields a $(\frac 65 k+O(1))$-approximation algorithm for $m$-pairs shortest paths ($m$-PSP) in $\tilde O(mn^{1/k})$ time, answering the question of Chechik, Hoch, and Lifshitz [SODA'25] of whether the $2k-O(1)$ barrier for $n$-PSP in $\tilde O(mn^{1/k})$ time can be broken. For unweighted graphs, we give a new analysis of the existing $\tilde O(mn^{1/k}+n^{1+2/k})$-time $n$-PSP algorithm of [Chechik et al., SODA'25], and show that, without any modification, it is a $(k,k-1)$-approximation, i.e., $d(u,v)\le\hat d(u,v)\le k\cdot d(u,v)+k-1$ for every queried pair. This improves its multiplicative factor from $\frac 43 k+O(1)$ to $k$, and the leading coefficient $k$ is tight for this algorithm. Moreover, for graphs with $m=Ω(n^{1+1/k})$, this answers the question of Dalirrooyfard et al. [FOCS'22] up to the additive $k-1$.

Tuesday, November 03 2026, 01:00

How Much Can Approximate Equilibria Change the Social Cost of Selfish Routing?

Authors: Yu Cheng, Tianle Jiang

We study approximate Nash equilibria in nonatomic congestion games, with the goal of understanding how much the social cost can change when the equilibrium condition is relaxed. We consider $ε$-approximate Nash equilibria ($ε$-NE), in which each player uses a strategy whose cost is at most $1 + ε$ times the minimum available cost. We compare the social costs of $ε$-NE and exact Nash equilibria. This lets us quantify, for example, how much a navigation app can reduce total travel time when drivers tolerate routes slightly slower than the fastest available route. For degree-$d$ polynomial latencies in both general nonatomic congestion games and single-commodity routing games on series-parallel networks, we give tight lower and upper bounds on the relative change in social cost from an exact Nash equilibrium to any $ε$-NE, with matching instances for every $ε$. For small $ε$, the largest possible relative reduction and increase in general congestion games are both $Θ(\sqrt{dε})$, independently of the number of resources and populations. In contrast, on series-parallel networks of depth $h$, the largest possible relative reduction is $Θ(hε)$, while the largest possible relative increase is known to be $ε$. For the classes of games we study, these results give a tight and quantitative characterization of the possible changes in social cost and establish a separation between general and series-parallel networks.

Authors: Yu Cheng, Tianle Jiang

We study approximate Nash equilibria in nonatomic congestion games, with the goal of understanding how much the social cost can change when the equilibrium condition is relaxed. We consider $ε$-approximate Nash equilibria ($ε$-NE), in which each player uses a strategy whose cost is at most $1 + ε$ times the minimum available cost. We compare the social costs of $ε$-NE and exact Nash equilibria. This lets us quantify, for example, how much a navigation app can reduce total travel time when drivers tolerate routes slightly slower than the fastest available route. For degree-$d$ polynomial latencies in both general nonatomic congestion games and single-commodity routing games on series-parallel networks, we give tight lower and upper bounds on the relative change in social cost from an exact Nash equilibrium to any $ε$-NE, with matching instances for every $ε$. For small $ε$, the largest possible relative reduction and increase in general congestion games are both $Θ(\sqrt{dε})$, independently of the number of resources and populations. In contrast, on series-parallel networks of depth $h$, the largest possible relative reduction is $Θ(hε)$, while the largest possible relative increase is known to be $ε$. For the classes of games we study, these results give a tight and quantitative characterization of the possible changes in social cost and establish a separation between general and series-parallel networks.

Tuesday, November 03 2026, 01:00

Fluid Limits and Upper Bounds for Randomized Greedy Matching

Authors: Sayan Mukherjee

The Modified Randomized Greedy (MRG) algorithm [Tinhofer (1984), Aronson-Dyer-Frieze-Suen (1995)] is a classical greedy algorithm for finding approximate maximum matchings. MRG repeatedly picks a uniformly random unmatched vertex and matches it to a uniformly random unmatched neighbor, if one exists. Since 1991, the best upper bound on its approximation ratio has been $2/3$, given by the Bomb graphs of [Dyer and Frieze (1991)]. We improve this bound to $0.6569$ on general graphs and $0.6607$ on bipartite graphs. Our constructions consist of copies of small gadgets, with specified vertex classes joined completely. We track each algorithm by counting gadget copies with each possible configuration of remaining vertices. Wormald's differential equation method [Wormald (1995)] shows that these counts, scaled by the size parameter of the construction, converge to a deterministic trajectory known as the \textit{fluid limit}, which is described by ordinary differential equations. This fluid limit approximation holds while the relevant vertex classes have positive density, and so we devise a separate counting argument determines the contribution from the end of the run. Furthermore, we apply our framework to Random Decision Order (RDO), in which vertices are processed in uniformly random order and choose unmatched neighbors according to fixed adversarial preferences. On bipartite graphs, we determine the exact limiting ratio on the Double-Bomb family of [Tang-Wu-Zhang (2020)]. Its minimum is $31/48$, confirming their numerical estimate of about $0.646$.On general graphs, we improve their upper bound of $5/8$, given by the paw graph, to $0.5946$ on a new family. Finally, we give one possible explanation for why $2/3$ seemed to be optimal for MRG in the concluding remarks.

Authors: Sayan Mukherjee

The Modified Randomized Greedy (MRG) algorithm [Tinhofer (1984), Aronson-Dyer-Frieze-Suen (1995)] is a classical greedy algorithm for finding approximate maximum matchings. MRG repeatedly picks a uniformly random unmatched vertex and matches it to a uniformly random unmatched neighbor, if one exists. Since 1991, the best upper bound on its approximation ratio has been $2/3$, given by the Bomb graphs of [Dyer and Frieze (1991)]. We improve this bound to $0.6569$ on general graphs and $0.6607$ on bipartite graphs. Our constructions consist of copies of small gadgets, with specified vertex classes joined completely. We track each algorithm by counting gadget copies with each possible configuration of remaining vertices. Wormald's differential equation method [Wormald (1995)] shows that these counts, scaled by the size parameter of the construction, converge to a deterministic trajectory known as the \textit{fluid limit}, which is described by ordinary differential equations. This fluid limit approximation holds while the relevant vertex classes have positive density, and so we devise a separate counting argument determines the contribution from the end of the run. Furthermore, we apply our framework to Random Decision Order (RDO), in which vertices are processed in uniformly random order and choose unmatched neighbors according to fixed adversarial preferences. On bipartite graphs, we determine the exact limiting ratio on the Double-Bomb family of [Tang-Wu-Zhang (2020)]. Its minimum is $31/48$, confirming their numerical estimate of about $0.646$.On general graphs, we improve their upper bound of $5/8$, given by the paw graph, to $0.5946$ on a new family. Finally, we give one possible explanation for why $2/3$ seemed to be optimal for MRG in the concluding remarks.

Tuesday, November 03 2026, 01:00

A smaller kernel for Ptolemaic Vertex Deletion

Authors: Dekel Tsur

In the Ptolemaic Vertex Deletion problem, the input consists of an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of at most $k$ vertices whose removal from $G$ results in a Ptolemaic graph. In this paper, we give a kernel for Ptolemaic Vertex Deletion with $O(k^5)$ vertices. Our result improves upon the $O(k^6)$ kernel of Agrawal et al. [IPEC 2021].

Authors: Dekel Tsur

In the Ptolemaic Vertex Deletion problem, the input consists of an undirected graph $G$ and an integer $k$. The goal is to decide whether there is a set of at most $k$ vertices whose removal from $G$ results in a Ptolemaic graph. In this paper, we give a kernel for Ptolemaic Vertex Deletion with $O(k^5)$ vertices. Our result improves upon the $O(k^6)$ kernel of Agrawal et al. [IPEC 2021].

Tuesday, November 03 2026, 01:00

Shared Randomness and Parallel Rounds in Permutation Sampling

Authors: Haoxiang Huang

We study the shared randomness needed to sample a uniform permutation in parallel output rounds. Coordinates committed in the same round are independent conditional on the complete history and the current shared choice. Previous outputs are public; local computation and private randomness are free. We charge each shared choice by the logarithm of its conditional support size and sum these costs along the worst execution path. At every fixed round density $ρ\in(0,1)$, the optimal limiting total-variation error under any $o(n)$ shared budget matches the conditional-marginal optimum, even with arbitrary local kernels and adaptive coordination. A fixed improvement requires linear cost. For exact sampling in $R=Θ(n)$ rounds, the optimum is $nΦ(R/n)+O(\log n)$, where $Φ$ linearly interpolates $Φ(1/q)=\log q$. A protocol attaining this rate uses an initial seed and at most one further binary shared choice. When $\lceil n/2\rceil\le R\le n$, the finite optimum is $\log\binom{2(n-R)}{n-R}$, attained by an initial seed alone. The approximate converse uses an entropy inequality for sparse injections to control overlap with the target. The exact construction reuses randomness retained in a shared label partition after the first-stage outputs are exposed.

Authors: Haoxiang Huang

We study the shared randomness needed to sample a uniform permutation in parallel output rounds. Coordinates committed in the same round are independent conditional on the complete history and the current shared choice. Previous outputs are public; local computation and private randomness are free. We charge each shared choice by the logarithm of its conditional support size and sum these costs along the worst execution path. At every fixed round density $ρ\in(0,1)$, the optimal limiting total-variation error under any $o(n)$ shared budget matches the conditional-marginal optimum, even with arbitrary local kernels and adaptive coordination. A fixed improvement requires linear cost. For exact sampling in $R=Θ(n)$ rounds, the optimum is $nΦ(R/n)+O(\log n)$, where $Φ$ linearly interpolates $Φ(1/q)=\log q$. A protocol attaining this rate uses an initial seed and at most one further binary shared choice. When $\lceil n/2\rceil\le R\le n$, the finite optimum is $\log\binom{2(n-R)}{n-R}$, attained by an initial seed alone. The approximate converse uses an entropy inequality for sparse injections to control overlap with the target. The exact construction reuses randomness retained in a shared label partition after the first-stage outputs are exposed.

Tuesday, November 03 2026, 01:00

Subexponential-Time Exact-One Hitting Set

Authors: Shiteng Chen, Tao Jiang, Yijie Zhu

Exact-One Hitting Set is one of Karp's original 21 NP-complete problems: given an explicitly listed family of subsets of an $r$-element ground set, decide whether a subset meets every input set in exactly one element. Combining our reduction with the fixed-width exact-counting theorem of OpenAI (2026), we obtain a deterministic polynomial-space algorithm that computes the exact number of feasible selections in time $2^{O(r\log\log(r+16)/\sqrt{\log(r+16)})}\operatorname{poly}(L+1)=2^{o(r)}\operatorname{poly}(L+1)$, where $L$ is the input length. The number and widths of the input sets are unrestricted. Known row selection retains at most $r+1$ input constraints, but their total width may remain quadratic. Our path representation of two-query counting cancellation constructs two ordinary 3-CNF formulas with at most $2r$ variables and $O(r^2)$ clauses each, whose model counts differ by the desired answer. The auxiliary-variable budget depends on the number of simultaneous conditions, not on their widths or on the numbers of witnesses. We extend this construction to signed equations with fixed bounded demands and positive integer coefficients. We also represent every explicitly listed Boolean relation of VC dimension at most one by binary clauses and signed exact-one equations on its original variables, obtaining the same two-query variable bound for unrestricted overlaps among such local constraints. A bounded-term version further handles fixed finite/cofinite local degree rules on arbitrary graphs. The path representation is related by variable elimination to existing marker--parity cancellation.

Authors: Shiteng Chen, Tao Jiang, Yijie Zhu

Exact-One Hitting Set is one of Karp's original 21 NP-complete problems: given an explicitly listed family of subsets of an $r$-element ground set, decide whether a subset meets every input set in exactly one element. Combining our reduction with the fixed-width exact-counting theorem of OpenAI (2026), we obtain a deterministic polynomial-space algorithm that computes the exact number of feasible selections in time $2^{O(r\log\log(r+16)/\sqrt{\log(r+16)})}\operatorname{poly}(L+1)=2^{o(r)}\operatorname{poly}(L+1)$, where $L$ is the input length. The number and widths of the input sets are unrestricted. Known row selection retains at most $r+1$ input constraints, but their total width may remain quadratic. Our path representation of two-query counting cancellation constructs two ordinary 3-CNF formulas with at most $2r$ variables and $O(r^2)$ clauses each, whose model counts differ by the desired answer. The auxiliary-variable budget depends on the number of simultaneous conditions, not on their widths or on the numbers of witnesses. We extend this construction to signed equations with fixed bounded demands and positive integer coefficients. We also represent every explicitly listed Boolean relation of VC dimension at most one by binary clauses and signed exact-one equations on its original variables, obtaining the same two-query variable bound for unrestricted overlaps among such local constraints. A bounded-term version further handles fixed finite/cofinite local degree rules on arbitrary graphs. The path representation is related by variable elimination to existing marker--parity cancellation.

Tuesday, November 03 2026, 01:00

Differential Privacy for Generalized Entropy Indices of Income Inequality

Authors: Jiahe Li, Jerome P. Reiter

The mean log deviation and the Theil index are two widely used measures of income inequality. The organization releasing these measures may be obligated to protect the confidentiality of the individual incomes used in the computations. In particular, the organization may wish to release versions of these measures with formal guarantees of privacy. The organization could do so by satisfying differential privacy; however, typical applications of differentially private algorithms are likely to add so much noise to the measures that they become unusable. We present differentially private algorithms for these indices with reasonable error properties. The algorithms rely on computing or approximating bounds on the smooth sensitivity, that is, the maximum absolute amount the inequality measures change when replacing any subset of the $n$ incomes in the reported data with arbitrarily chosen feasible values. For the mean log deviation, the algorithm runs in $O(n^2)$ time. For the Theil index, the algorithm runs in $O(n\log n)$ time. We illustrate the performance of each differentially private algorithm using individual-level and household-level income data from the Current Population Survey. We also examine repeated sampling properties of the algorithms using artificially generated data. The empirical results indicate when these differentially private algorithms can be expected to provide accurate estimates and when they are likely to introduce too much noise.

Authors: Jiahe Li, Jerome P. Reiter

The mean log deviation and the Theil index are two widely used measures of income inequality. The organization releasing these measures may be obligated to protect the confidentiality of the individual incomes used in the computations. In particular, the organization may wish to release versions of these measures with formal guarantees of privacy. The organization could do so by satisfying differential privacy; however, typical applications of differentially private algorithms are likely to add so much noise to the measures that they become unusable. We present differentially private algorithms for these indices with reasonable error properties. The algorithms rely on computing or approximating bounds on the smooth sensitivity, that is, the maximum absolute amount the inequality measures change when replacing any subset of the $n$ incomes in the reported data with arbitrarily chosen feasible values. For the mean log deviation, the algorithm runs in $O(n^2)$ time. For the Theil index, the algorithm runs in $O(n\log n)$ time. We illustrate the performance of each differentially private algorithm using individual-level and household-level income data from the Current Population Survey. We also examine repeated sampling properties of the algorithms using artificially generated data. The empirical results indicate when these differentially private algorithms can be expected to provide accurate estimates and when they are likely to introduce too much noise.

Tuesday, November 03 2026, 01:00

$\ell_{\infty}$ norm SVP in $\widetilde{O}(3^n)$

Authors: Yiming Gao

We give a classical randomized algorithm for the Shortest Vector Problem in the infinity norm in $\widetilde O(3^n)$ time and space, improving the previous bound of $2^{2.443n+o(n)}$. We adapt Gaussian difference dynamics to the infinity norm by repeatedly taking differences of vectors from two lists, producing lists in successively smaller cubes. The key is to restrict the midpoint of each pair to a fixed inner cube, so that every allowed difference has the same region of possible midpoints. Using random lattice translates, this rule preserves equal expected counts at each point of the smaller cube. We analyze the dependencies created by repeated differencing through the counting moments of Gaussian difference dynamics, together with a weighted packing bound for cubes. Finally, two searches based on coordinate comparisons generate the intermediate lists and recover a shortest lattice vector within the claimed time bound.

Authors: Yiming Gao

We give a classical randomized algorithm for the Shortest Vector Problem in the infinity norm in $\widetilde O(3^n)$ time and space, improving the previous bound of $2^{2.443n+o(n)}$. We adapt Gaussian difference dynamics to the infinity norm by repeatedly taking differences of vectors from two lists, producing lists in successively smaller cubes. The key is to restrict the midpoint of each pair to a fixed inner cube, so that every allowed difference has the same region of possible midpoints. Using random lattice translates, this rule preserves equal expected counts at each point of the smaller cube. We analyze the dependencies created by repeated differencing through the counting moments of Gaussian difference dynamics, together with a weighted packing bound for cubes. Finally, two searches based on coordinate comparisons generate the intermediate lists and recover a shortest lattice vector within the claimed time bound.

Tuesday, November 03 2026, 01:00

Facility Location with Customer Incompatibilities

Authors: Zachary Friggstad, Pengfei Wu

We study metric facility location with customer incompatibilities, where the clients assigned to each facility must satisfy prescribed independence constraints. We first consider the setting of facility-opening costs and where there is a single matroid $\mathcal M$ over the clients and the set of clients assigned to an open facility must be an independent set in $\mathcal M$. We develop a polynomial-time $(1+\sqrt{2}+ε)$-approximation algorithm for this setting. We further obtain a polynomial-time $(15+ε)$-approximation algorithm for the case of non-uniform facility-opening costs when $\mathcal M$ is a partition matroid that only permits $b_g$ clients to be assigned to a facility for each group $g$ in the partition. Both settings are substantial generalizations of Capacitated Facility Location with uniform capacities. Our algorithms are based on local search using add, delete, and single-swap operations. As with CFL, bounding the connection cost is reasonably straightforward using supermodularity of the minimum connection-cost function. However, bounding the facility opening costs requires novel arguments in both cases. We use a ``compression'' argument based on matroid exchange principles to identify a subset of the locally opened facilities whose size is no larger than that of an optimal facility set. The opening costs of facilities outside this subset can be bounded using supermodularity, while bounding the costs of the remaining facilities requires different arguments. It is natural to consider a variant where each possible facility $i$ has its own matroid $\mathcal M_i$ defining the subsets of clients it can feasibly serve. We also show problem is at least as hard to approximate as Set Cover even if all $\mathcal M_i$ are partition matroids that have the same partition of clients, just different group bounds $b_g$.

Authors: Zachary Friggstad, Pengfei Wu

We study metric facility location with customer incompatibilities, where the clients assigned to each facility must satisfy prescribed independence constraints. We first consider the setting of facility-opening costs and where there is a single matroid $\mathcal M$ over the clients and the set of clients assigned to an open facility must be an independent set in $\mathcal M$. We develop a polynomial-time $(1+\sqrt{2}+ε)$-approximation algorithm for this setting. We further obtain a polynomial-time $(15+ε)$-approximation algorithm for the case of non-uniform facility-opening costs when $\mathcal M$ is a partition matroid that only permits $b_g$ clients to be assigned to a facility for each group $g$ in the partition. Both settings are substantial generalizations of Capacitated Facility Location with uniform capacities. Our algorithms are based on local search using add, delete, and single-swap operations. As with CFL, bounding the connection cost is reasonably straightforward using supermodularity of the minimum connection-cost function. However, bounding the facility opening costs requires novel arguments in both cases. We use a ``compression'' argument based on matroid exchange principles to identify a subset of the locally opened facilities whose size is no larger than that of an optimal facility set. The opening costs of facilities outside this subset can be bounded using supermodularity, while bounding the costs of the remaining facilities requires different arguments. It is natural to consider a variant where each possible facility $i$ has its own matroid $\mathcal M_i$ defining the subsets of clients it can feasibly serve. We also show problem is at least as hard to approximate as Set Cover even if all $\mathcal M_i$ are partition matroids that have the same partition of clients, just different group bounds $b_g$.

Tuesday, November 03 2026, 01:00

$k$-Center under Weak and Strong Distance Oracles with an Application to Minimum Sum of Radii Clustering

Authors: Zachary Friggstad, Mahya Jamshidian, Ali Vakilian

In the weak-strong distance oracle model of Bateni, Dharangutte, Jayaram and Wang (COLT 2024), a metric $(X,d)$ on $n$ points is accessed through two oracles: a cheap weak oracle that returns each queried distance correctly with probability $1-δ$ and permits arbitrary values on corrupted pairs, chosen by an adversary who knows the entire metric and all corrupted pairs, and an expensive strong oracle that reveals a point exactly. The goal is to cluster well while making as few strong queries as possible. Our main result is a $(6+\varepsilon)$-approximation for $k$-center using $O(k\log n\log(\varepsilon^{-1}\log n))$ strong point queries, for any fixed $\varepsilon>0$. This improves both the $(14+\varepsilon)$ approximation factor and the $O(k\log^2 n\log(\varepsilon^{-1}\log n))$ query bound of Bateni et al. It also improves the query complexity of the recent $(6+\varepsilon)$-approximation of Pradhan, Bhattacharya and Jaiswal (AISTATS 2026), in the variant of the model they study. Our algorithm outputs an explicitly known sketch of an otherwise unknown metric: a concrete metric on at most $k$ points that a downstream algorithm can compute with directly, with no further queries and no noise. As an application, we give a simple FPT algorithm for Minimum Sum of Radii (MSR) that returns a constant-factor solution using $2^{O(k\log k)}\cdot O(\log n\log\log n)$ strong queries. We complement the upper bound with an $Ω(k\log(n/k))$ lower bound for $k$-center with assignment that holds for row-local algorithms. These algorithms assign two unqueried points to the same center whenever their weak distances to the strong-queried points are exactly the same. Our algorithm and those of Bateni et al. and Pradhan et al. are all row-local, so our query bound is optimal for this class up to the $O(\log\log n)$ factor.

Authors: Zachary Friggstad, Mahya Jamshidian, Ali Vakilian

In the weak-strong distance oracle model of Bateni, Dharangutte, Jayaram and Wang (COLT 2024), a metric $(X,d)$ on $n$ points is accessed through two oracles: a cheap weak oracle that returns each queried distance correctly with probability $1-δ$ and permits arbitrary values on corrupted pairs, chosen by an adversary who knows the entire metric and all corrupted pairs, and an expensive strong oracle that reveals a point exactly. The goal is to cluster well while making as few strong queries as possible. Our main result is a $(6+\varepsilon)$-approximation for $k$-center using $O(k\log n\log(\varepsilon^{-1}\log n))$ strong point queries, for any fixed $\varepsilon>0$. This improves both the $(14+\varepsilon)$ approximation factor and the $O(k\log^2 n\log(\varepsilon^{-1}\log n))$ query bound of Bateni et al. It also improves the query complexity of the recent $(6+\varepsilon)$-approximation of Pradhan, Bhattacharya and Jaiswal (AISTATS 2026), in the variant of the model they study. Our algorithm outputs an explicitly known sketch of an otherwise unknown metric: a concrete metric on at most $k$ points that a downstream algorithm can compute with directly, with no further queries and no noise. As an application, we give a simple FPT algorithm for Minimum Sum of Radii (MSR) that returns a constant-factor solution using $2^{O(k\log k)}\cdot O(\log n\log\log n)$ strong queries. We complement the upper bound with an $Ω(k\log(n/k))$ lower bound for $k$-center with assignment that holds for row-local algorithms. These algorithms assign two unqueried points to the same center whenever their weak distances to the strong-queried points are exactly the same. Our algorithm and those of Bateni et al. and Pradhan et al. are all row-local, so our query bound is optimal for this class up to the $O(\log\log n)$ factor.

Tuesday, November 03 2026, 01:00

Max Weight Independent Set in graphs with no long claws nor large induced bicliques

Authors: Maria Chudnovsky, Paweł Rzążewski

The complexity of Max Weight Independent Set (MWIS) in graphs excluding a fixed subdivided claw is one of the central open problems in algorithmic graph theory. We show that MWIS can be solved in polynomial time in graphs excluding a fixed subdivided claw and a fixed biclique as an induced subgraph. This strengthens a result of Abrishami, Chudnovsky, Dibek, Pilipczuk, and Rzążewski [ACM TALG 2026] who showed an analogous result, but under a stronger assumption that some biclique is excluded as a subgraph.

Authors: Maria Chudnovsky, Paweł Rzążewski

The complexity of Max Weight Independent Set (MWIS) in graphs excluding a fixed subdivided claw is one of the central open problems in algorithmic graph theory. We show that MWIS can be solved in polynomial time in graphs excluding a fixed subdivided claw and a fixed biclique as an induced subgraph. This strengthens a result of Abrishami, Chudnovsky, Dibek, Pilipczuk, and Rzążewski [ACM TALG 2026] who showed an analogous result, but under a stronger assumption that some biclique is excluded as a subgraph.

Tuesday, November 03 2026, 01:00

Online carpooling against an oblivious adversary

Here is the conference home page

Authors: Yuhao Lei

In the online carpooling problem, edges on $n$ vertices arrive one at a time and must be oriented immediately, so as to keep the difference between the indegree and the outdegree of every vertex small. We give a randomized online algorithm which bounds the largest absolute difference over the first $T$ edges by $O((\log T)^{1/3})$ with high probability on every input sequence fixed in advance. This proves a conjecture of Kulkarni, Reis and Rothvoss and matches the known worst case lower bound. The algorithm keeps the law of the discrepancy vector dominated in the convex order by a Gaussian measure, and forces the endpoint with the smaller discrepancy to receive the edge whenever the two discrepancies are far apart. In the analysis we show that a vertex of discrepancy $D$ is preceded by a set of $O(D)$ vertices with total discrepancy $Ω(D^2)$.

Authors: Yuhao Lei

In the online carpooling problem, edges on $n$ vertices arrive one at a time and must be oriented immediately, so as to keep the difference between the indegree and the outdegree of every vertex small. We give a randomized online algorithm which bounds the largest absolute difference over the first $T$ edges by $O((\log T)^{1/3})$ with high probability on every input sequence fixed in advance. This proves a conjecture of Kulkarni, Reis and Rothvoss and matches the known worst case lower bound. The algorithm keeps the law of the discrepancy vector dominated in the convex order by a Gaussian measure, and forces the endpoint with the smaller discrepancy to receive the edge whenever the two discrepancies are far apart. In the analysis we show that a vertex of discrepancy $D$ is preceded by a set of $O(D)$ vertices with total discrepancy $Ω(D^2)$.

Tuesday, November 03 2026, 01:00

Celebrating 100 Years: Avi 70 + CSDM 30

from Gil Kalai

Greetings everybody from Tel Aviv! Irit Dinur and Amir Shpilka asked me to spread the word through my blog about the celebration for Avi Wigderson and for computer science and discrete mathematics at IAS. Avi Fest @ IAS: Dates: June … Continue reading →

Greetings everybody from Tel Aviv!

Irit Dinur and Amir Shpilka asked me to spread the word through my blog about the celebration for Avi Wigderson and for computer science and discrete mathematics at IAS.

Avi Fest @ IAS: Dates: June 14-18, 2027

The Institute for Advanced Study’s School of Mathematics is pleased to announce a conference celebrating Avi Wigderson’s 70th birthday and retirement, together with 30 years of the Computer Science and Discrete Mathematics program (CSDM) at IAS. The conference will honor both Avi’s extraordinary scientific legacy and his transformative role in shaping the intellectual community around theoretical computer science and discrete mathematics at IAS.

Two weeks ago, Edna and Avi Wigderson hosted us in Princeton, where Edna prepared a wonderful dinner for me and my beloved friends—Avi, Edna, Noga, Nati, Elon, Abigail, Irit, and Hadar. The following week, just a day before the general election, we met again in Israel for the Rabin Memorial Conference and other activities. I plan to write here (mainly for myself) about my recent US trip and also to share my thoughts on a variety of uncertainties.

By Gil Kalai

Monday, November 02 2026, 20:48

TR26-291 | Reed–Muller Codes are Optimal Binary Linear 3-Query LCCs | Oliver Janzer, Peter Manohar

from ECCC Papers

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 $n \geq 2^{\Omega(\sqrt{k})}$. This improves on the prior best lower bound of $2^{\Omega(\sqrt{k/\log k})}$, 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.

Monday, November 02 2026, 16:04

Go for Two to Know Exactly What You Need

from Ben Recht

Checking in on Al Michaels and the go-for-two-down-eight strategy.

Hi there, argmin readers. Today I’m Monday Morning Quarterbacking and wrote one of my sporadic football analytics posts.

NFL Sunday was a wild and weird one. Rare events dominate the NFL and are what makes it so fun to watch. The Tampa Bay Bucs destroyed the Atlanta Falcons in a hurricane, rushing for 324 yards and passing for… zero yards. Zero! They didn’t even complete a pass. No team had won a game with zero completions in over fifty years. The Broncos, quarterbacked by Bo Nix, orchestrated his twelfth comeback win down double digits in the fourth quarter. He’s only started 42 games! And yet, everyone, including his coach, still thinks Bo Nix is a bad quarterback. And yesterday was a banner day for my favorite hobby horse, the stupidity of the go-for-2-down-8 (G42D8) gambit.

To the relief of most of my readership, I haven’t written about G42D8 in a while. So let me remind those who are fortunate enough not to share my dumb obsession. A team down 14 points in the fourth quarter starts a comeback by scoring a touchdown. They are now down by 8. They face the decision to kick an extra point or attempt a 2-point conversion. In 2015, analytic-brained weirdos convinced themselves that the laws of probability insisted your chances of winning were only 50% if you kicked the extra point, but were 62.5% if you attempted the 2-point conversion, even if you missed the first one. And for the next decade, they’d show you this stupid infographic and smugly yell about the probability differential to show you how smart they are.

Late in the third quarter, down 17-3 to the Cowboys, the Cardinals benched journeyman quarterback Jacoby Brissett for journeyman backup quarterback Gardner Minshew. The Cardinals immediately scored on a 63-yard run by Tyler Allgeier. Allgeier was apparently stoked about the QB change. Down 8, they kicked the extra point. Even though it was technically still 57 seconds until the fourth quarter, G42D8 zealot Seth Walder exclaimed they should have gone for two. Regardless, the Cardinals faced the down-eight test again after finding themselves again down 14 a few minutes into the fourth quarter. They scored a touchdown and kicked the extra point. The Cardinals failed the G42D8 test, twice in a single game. They weren’t having any of this shamalytics. Of course, they went on to win.

In 2024, I wrote about the full logic of the G42D8 gambit and why it was a terrible misapplication of decision theory.1 Since the start of the 2025 season, teams attempted the 2-point when down 8 13 times. Teams kicked extra points 11 times. The strategy is certainly mainstream enough now that teams apply it more than half the time. And announcers remain completely befuddled every time. The announcers are not wrong. The analytic explanation is not “simple math.” It’s tortured, mathematicized reasoning.

But since it’s becoming the dominant strategy for losing teams, it’s worth checking how those teams fare. The G42D8 gambit teams won 2 of the 13 games where they followed the analytics (and one win is a clear technicality, which I’ll address in a second). The stodgy, conservative, meathead teams won… twice as well. In fact, they additionally tied once. That is, team G42D8 is 2 and 11 (.154 pct). Team Meathead is 2, 8, and 1 (.227 pct).

Yes, these sample sizes are absurdly small, and it’s impossible to say one is better than the other. That is actually my point. We decide NFL seasons with 17 games, after all. There is no evidence that G42D8 leads to more wins than kicking extra points.

Since 2015, 45 teams have attempted the G42D8 gambit. In these games, 22 teams have missed the first two-point conversion. The G42D8 brain geniuses tell us this is okay, because when they score the second touchdown, they can go for two again and take the game to overtime. Except that in all 22 of these games, no team has ever taken the game to overtime. 19 of these teams never scored another touchdown! In one game, the Browns scored the second touchdown but missed the second 2-point conversion and lost. In another, the Jets made the 2nd 2-point conversion, but there was still 4:30 left in the game, and the Lions subsequently marched down the field to score a winning touchdown. The truly bizarre remaining game was a Thursday Night Football NFC South masterpiece between the aforementioned enemies, the Falcons and Bucs (IYKYK). The Falcons missed the 2nd 2-point conversion, but took advantage of miserable clock management by Bucs coach Todd Bowles and managed to get a third possession. They kicked a field goal to win by 1.2

A third possession in regulation isn’t even considered a possibility in the G42D8 decision tree, where you lose 100 percent of the time you miss both conversions. Except the Falcons miraculously won. I’ll allow this one to stay in the win column for team G42D8, but it’s certainly a technicality, and certainly one that shows football strategies are far more complex than what can be captured in an infographic. The models don’t describe the game of football as much as the statistical analysts think they do.

If you’re going to insist you’re smarter than everyone, at some point you need to show results. Otherwise, the people who don’t even understand what “percent” means are going to stuff you in a locker. Down 14 points in the fourth quarter means you’re probably going to lose. You can go for two. You can go for one. But trying to argue that one is obviously superior to the other is, and always will be, dumb. Just like fighting with people on sports shows and social media.

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1

Committed to my own dumb bits, I had a few follow-up posts in 2025 digging into the actual 2-point completion rates, the actual success rates of the strategy, and the absurdity of games where it “works.”

2

Bucs-Falcons games are always terrible and weird, and adding the sloppy, tired dynamics of Thursday night only made it weirder. Four plays into the drive after Atlanta’s botched 2-point attempt, Bucs QB Baker Mayfield threw an awful interception, giving the Falcons the ball back at their 33-yard line. They drove down the field and scored with 3:34 left. Even though the Falcons missed the second two-point conversion, 3:34 is an eternity in football, especially because the Falcons had all of their timeouts. They managed to stop the Bucs before the two-minute warning (there’s that clock management malpractice by Todd Bowles). With two minutes left and no timeouts, they drove 45 yards and kicked an easy 43-yard field goal. They won by 1, 29-28.

By Ben Recht

Monday, November 02 2026, 15:37

Postdoc at University of Colorado Boulder (apply by December 15, 2026)

from CCI: jobs

A postdoctoral position in theoretical computer science is available to work with Huck Bennett at the University of Colorado Boulder. The postdoc will perform research independently, with the PI, and with the PI’s group on topics including lattices, error-correcting codes, fine-grained complexity, algebraic and geometric algorithms, and more. Website: jobs.colorado.edu/jobs/JobDetail/PostDoctoral-Associate/75672 Email: huck.bennett@colorado.edu

A postdoctoral position in theoretical computer science is available to work with Huck Bennett at the University of Colorado Boulder. The postdoc will perform research independently, with the PI, and with the PI’s group on topics including lattices, error-correcting codes, fine-grained complexity, algebraic and geometric algorithms, and more.

Website: https://jobs.colorado.edu/jobs/JobDetail/PostDoctoral-Associate/75672
Email: huck.bennett@colorado.edu

By shacharlovett

Monday, November 02 2026, 06:02

Treedepth is NP-Hard on Planar Graphs of Maximum Degree 3

Authors: Kolja Kühn

Treedepth is an important parameter in algorithmic graph theory. It is known that the problem of computing the treedepth of a graph is NP-hard. It has been an open problem whether treedepth can be computed in polynomial time when the input graph is restricted to be planar, or of bounded degree. Assuming $P \neq NP$, we answer both questions in the negative: Under polynomial-time Turing reductions, it is NP-hard to compute the treedepth of planar graphs with maximum degree 3. This degree bound is best possible, since treedepth can be computed efficiently on graphs of maximum degree 2.

Authors: Kolja Kühn

Treedepth is an important parameter in algorithmic graph theory. It is known that the problem of computing the treedepth of a graph is NP-hard. It has been an open problem whether treedepth can be computed in polynomial time when the input graph is restricted to be planar, or of bounded degree. Assuming $P \neq NP$, we answer both questions in the negative: Under polynomial-time Turing reductions, it is NP-hard to compute the treedepth of planar graphs with maximum degree 3. This degree bound is best possible, since treedepth can be computed efficiently on graphs of maximum degree 2.

Monday, November 02 2026, 01:00

Exponential quantum-classical query separations for cyclic Forrelation

Authors: Yuxin Zhang, Changpeng Shao

We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of $k$-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given $k$ Boolean functions $f_1,\ldots,f_k:\mathbb{F}_2^n\to\{-1,1\}$, the goal is to estimate $T_k=N^{-1}\operatorname{Tr}(D_{f_1}H^{\otimes n}\cdots D_{f_k}H^{\otimes n}),$ where $D_{f_i}=\operatorname{diag}(f_i(x):x\in\mathbb{F}_2^n)$, $H$ is the Hadamard gate, and $N=2^n$. This normalized-trace problem can be solved with $k$ quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of $\widetildeΩ(N^{1-2/k})$. We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even $k\ge 4$, we prove a tight bound of $Θ_k(N^{1-2/k})$ for constant additive error. For odd $k\ge 5$, we obtain a lower bound of $Ω_k(N^{1-2/(k-1)})$ and an upper bound of $O_k(N^{1-2/k})$. Surprisingly, $k=3$ admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the $k$-cycle, leading to the exponent $1-1/\lfloor k/2\rfloor$, rather than the previously conjectured $1-2/k$, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of $Ω(N^{1/4})$ for every $k\ge4$. Although formally close to $k$-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.

Authors: Yuxin Zhang, Changpeng Shao

We study the randomized classical query complexity of cyclic Forrelation, a cyclic analogue of $k$-Forrelation [AA15, STOC'15; BS21, STOC'21]. Given $k$ Boolean functions $f_1,\ldots,f_k:\mathbb{F}_2^n\to\{-1,1\}$, the goal is to estimate $T_k=N^{-1}\operatorname{Tr}(D_{f_1}H^{\otimes n}\cdots D_{f_k}H^{\otimes n}),$ where $D_{f_i}=\operatorname{diag}(f_i(x):x\in\mathbb{F}_2^n)$, $H$ is the Hadamard gate, and $N=2^n$. This normalized-trace problem can be solved with $k$ quantum queries in the DQC1 model. Its classical query complexity was left open in [JLS+26, FOCS'26], which conjectured a lower bound of $\widetildeΩ(N^{1-2/k})$. We establish a nearly complete characterization of the nonadaptive classical query complexity of cyclic Forrelation. For even $k\ge 4$, we prove a tight bound of $Θ_k(N^{1-2/k})$ for constant additive error. For odd $k\ge 5$, we obtain a lower bound of $Ω_k(N^{1-2/(k-1)})$ and an upper bound of $O_k(N^{1-2/k})$. Surprisingly, $k=3$ admits a polylogarithmic-query nonadaptive classical algorithm, disproving the conjecture in this case. These results suggest an unexpected parity-dependent structure: the nonadaptive complexity may be governed by the independence number of the $k$-cycle, leading to the exponent $1-1/\lfloor k/2\rfloor$, rather than the previously conjectured $1-2/k$, which was motivated by a Fourier-growth heuristic. Finally, allowing adaptivity, we prove a lower bound of $Ω(N^{1/4})$ for every $k\ge4$. Although formally close to $k$-Forrelation, the cyclic structure leads to rather different classical behavior and requires different techniques. Our lower bounds are based on hidden-shift collision constructions, and our upper bounds follow from randomized sampling together with cycle-specific conditional-moment estimates. Our results provide a systematic understanding of its query advantages over classical computation.

Monday, November 02 2026, 01:00

Quantum query complexity of Hurwitz stability testing

Authors: Maxwell Fitzsimmons, Mohamed Serry, Meenu Kumari, Jun Liu

Determining whether every trajectory of the linear system $\dot{x}=Ax$ converges to zero (or, equivalently, whether $A$ is Hurwitz stable) is a fundamental problem in control theory. It arises, for example, in verifying the stability of feedback controllers and state observers, as well as of linearized dynamical models. We study the quantum query complexity of deciding whether a matrix $A\in\mathbb{C}^{n\times n}$ is Hurwitz stable, given an exact normalized block-encoding of $A$. The query complexity is characterized in terms of the system dimension $n$ and a supplied lower bound $0<ν\leq ω_A$ on the imaginary-axis distance $ω_A:=\min_{t\in\mathbb{R}}σ_{\min}(A-\mathrm{i}tI)$, where $σ_{\min}(A)$ is the smallest singular value of $A$. We prove a query-complexity lower bound of $Ω(\sqrt{n}/ν)$, which holds even when $A$ is restricted to be real and diagonal. For general matrices with nonzero $ω_A$, we encode an approximation of the right-half-plane spectral projector $\operatorname{sign}^+(A):=(I+\operatorname{sign}(A))/2$ as a block of a matrix inverse, where $A$ is Hurwitz if and only if the projector is zero. To test this, we produce a quantum algorithm which determines if a $(β,b,1/(4\sqrt{n}))$-block-encoded projector matrix (like $\operatorname{sign}^+(A)$) is zero or not; for constant success probability, it has complexity $Θ(β\sqrt{n})$. When the zero test is applied to specific approximations of $\operatorname{sign}^+(A)$, this yields an upper bound of $O(\sqrt{n}ν^{-2}\log(\sqrt{n}/ν))$ queries in the general case, and $O(\sqrt{n}ν^{-1}\log n)$ when $A$ is Hermitian or normal. The latter upper bound matches the lower bound up to a logarithmic factor. Thus the dimension dependence of quantum Hurwitz testing is nearly optimal, while the remaining factor in the non-normal case isolates an open conditioning gap.

Authors: Maxwell Fitzsimmons, Mohamed Serry, Meenu Kumari, Jun Liu

Determining whether every trajectory of the linear system $\dot{x}=Ax$ converges to zero (or, equivalently, whether $A$ is Hurwitz stable) is a fundamental problem in control theory. It arises, for example, in verifying the stability of feedback controllers and state observers, as well as of linearized dynamical models. We study the quantum query complexity of deciding whether a matrix $A\in\mathbb{C}^{n\times n}$ is Hurwitz stable, given an exact normalized block-encoding of $A$. The query complexity is characterized in terms of the system dimension $n$ and a supplied lower bound $0<ν\leq ω_A$ on the imaginary-axis distance $ω_A:=\min_{t\in\mathbb{R}}σ_{\min}(A-\mathrm{i}tI)$, where $σ_{\min}(A)$ is the smallest singular value of $A$. We prove a query-complexity lower bound of $Ω(\sqrt{n}/ν)$, which holds even when $A$ is restricted to be real and diagonal. For general matrices with nonzero $ω_A$, we encode an approximation of the right-half-plane spectral projector $\operatorname{sign}^+(A):=(I+\operatorname{sign}(A))/2$ as a block of a matrix inverse, where $A$ is Hurwitz if and only if the projector is zero. To test this, we produce a quantum algorithm which determines if a $(β,b,1/(4\sqrt{n}))$-block-encoded projector matrix (like $\operatorname{sign}^+(A)$) is zero or not; for constant success probability, it has complexity $Θ(β\sqrt{n})$. When the zero test is applied to specific approximations of $\operatorname{sign}^+(A)$, this yields an upper bound of $O(\sqrt{n}ν^{-2}\log(\sqrt{n}/ν))$ queries in the general case, and $O(\sqrt{n}ν^{-1}\log n)$ when $A$ is Hermitian or normal. The latter upper bound matches the lower bound up to a logarithmic factor. Thus the dimension dependence of quantum Hurwitz testing is nearly optimal, while the remaining factor in the non-normal case isolates an open conditioning gap.

Monday, November 02 2026, 01:00

Stochastic and Noisy Oracles in Smooth Continuous Optimization

Authors: Albert S. Berahas, Liyuan Cao, Katya Scheinberg

We review models of inexact computational oracles in smooth continuous optimization and their implications for the convergence, computational complexity, and attainable accuracy of optimization algorithms. We adopt a framework in which zeroth-,first-, and second-order oracles are characterized by quantitative error bounds and may accept additional control parameters that influence both their accuracy and computational cost. We introduce deterministic and stochastic oracle models and illustrate their properties through a variety of examples, including finite-difference approximations, stochastic gradients, and randomized derivative estimators. Particular attention is given to the distinction between absolute and relative errors, bias and variance, and stochastic function evaluations with independent or coupled randomness. A central theme of the review is the use of common analytical tools to recover and compare convergence and complexity guarantees under different oracle models. For deterministic and stochastic first-order methods, we show how a general descent inequality leads to complexity bounds under various assumptions on the objective function and all proposed oracle models. In the stochastic setting, we develop bounds on expected stopping times and the accuracy that can be guaranteed under different oracle conditions. We also examine the total computational complexity of algorithms relying on specific oracle constructions and briefly review models of inexact second- order information and their use in optimization algorithms.

Authors: Albert S. Berahas, Liyuan Cao, Katya Scheinberg

We review models of inexact computational oracles in smooth continuous optimization and their implications for the convergence, computational complexity, and attainable accuracy of optimization algorithms. We adopt a framework in which zeroth-,first-, and second-order oracles are characterized by quantitative error bounds and may accept additional control parameters that influence both their accuracy and computational cost. We introduce deterministic and stochastic oracle models and illustrate their properties through a variety of examples, including finite-difference approximations, stochastic gradients, and randomized derivative estimators. Particular attention is given to the distinction between absolute and relative errors, bias and variance, and stochastic function evaluations with independent or coupled randomness. A central theme of the review is the use of common analytical tools to recover and compare convergence and complexity guarantees under different oracle models. For deterministic and stochastic first-order methods, we show how a general descent inequality leads to complexity bounds under various assumptions on the objective function and all proposed oracle models. In the stochastic setting, we develop bounds on expected stopping times and the accuracy that can be guaranteed under different oracle conditions. We also examine the total computational complexity of algorithms relying on specific oracle constructions and briefly review models of inexact second- order information and their use in optimization algorithms.

Monday, November 02 2026, 01:00

Clifford synthesis is NP-complete