Theory of Computing Report

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

Representation-Aware Closed-Surface Parameterization: Exact Sensitivity Analysis and Pole-Aware Planar Preconditioning

Authors: Zixi Yi, Gary P. T. Choi

Surface parameterization is a fundamental task in geometry processing and shape analysis. For topologically closed surfaces, it is common to parameterize them onto a standardized closed domain in $\mathbb{R}^3$, with a prime example being a unit sphere for genus-0 closed surfaces. Many mapping methods use planar projections and perform computations on the plane rather than directly on the 3D surface. However, perturbations introduced before projection or geometric evolution propagate through the computation and may affect map quality, convergence, stopping behavior, and geometric validity. In this work, we formulate representation sensitivity as a branchwise input--output problem, separating smooth propagation through active Jacobians from changes in anchors, stopping rules, and fallback events. Under consistent piecewise-linear assembly, the unmodified south-pole Beltrami correction in the FLASH method for spherical conformal parameterization reduces exactly to source-mesh harmonic extension. On a fixed-anchor branch, the corrected map factors into chart transition, harmonic propagation, and inverse stereographic projection, yielding an explicit sensitivity Jacobian and a finite-triangle conditioning bound. Separately, we introduce a bounded five-parameter affine preconditioner over planar scale, translation, and anisotropy and evaluate candidates on the corrected spherical map. Across 30 genus-0 surfaces, it reduces median polar upper-tail log-condition and chord-angle distortion by $10.17\%$ and $10.97\%$, respectively, with both measures improving on every mesh. We also apply the framework to other spherical mapping methods including the conformalized mean-curvature flow and spherical density-equalizing maps. Altogether, our work highlights the importance of representation selection as a structural and practical component of spherical parameterization and downstream computation.

Authors: Zixi Yi, Gary P. T. Choi

Surface parameterization is a fundamental task in geometry processing and shape analysis. For topologically closed surfaces, it is common to parameterize them onto a standardized closed domain in $\mathbb{R}^3$, with a prime example being a unit sphere for genus-0 closed surfaces. Many mapping methods use planar projections and perform computations on the plane rather than directly on the 3D surface. However, perturbations introduced before projection or geometric evolution propagate through the computation and may affect map quality, convergence, stopping behavior, and geometric validity. In this work, we formulate representation sensitivity as a branchwise input--output problem, separating smooth propagation through active Jacobians from changes in anchors, stopping rules, and fallback events. Under consistent piecewise-linear assembly, the unmodified south-pole Beltrami correction in the FLASH method for spherical conformal parameterization reduces exactly to source-mesh harmonic extension. On a fixed-anchor branch, the corrected map factors into chart transition, harmonic propagation, and inverse stereographic projection, yielding an explicit sensitivity Jacobian and a finite-triangle conditioning bound. Separately, we introduce a bounded five-parameter affine preconditioner over planar scale, translation, and anisotropy and evaluate candidates on the corrected spherical map. Across 30 genus-0 surfaces, it reduces median polar upper-tail log-condition and chord-angle distortion by $10.17\%$ and $10.97\%$, respectively, with both measures improving on every mesh. We also apply the framework to other spherical mapping methods including the conformalized mean-curvature flow and spherical density-equalizing maps. Altogether, our work highlights the importance of representation selection as a structural and practical component of spherical parameterization and downstream computation.

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

Constant Approximation of Fréchet Distance in Near-Linear Time

Authors: Haoqiang Huang, Zihe Wang, Shuo Zhang

We present a deterministic algorithm that approximates the Fréchet distance between two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices within a constant factor in $O((m+n)^{1+\varepsilon})$ time, for every fixed $\varepsilon>0$ and fixed dimension $d$. The approximation factor depends only on $\varepsilon$ and $d$. No assumption on the input curves is needed, and the algorithm also reports a matching certifying the approximation. The same result holds for the discrete Fréchet distance. Our result extends the line of work initiated by Cheng, Huang, and Zhang (2025), who gave the first constant-factor approximation in strongly subquadratic time, to near-linear time. For two $n$-vertex curves, the best previous running time was $\widetilde{O}(n^{5/3})$, due to van der Horst and Ophelders (2026). Our algorithm bootstraps between two primitives: a data structure that preprocesses a single curve for comparing any two of its subcurves, and a query procedure that compares subcurves of two different curves. Given the data structure for one curve, we compare it with another curve using a surrogate assembled from subcurves of the former. The surrogate is shared by all possible starting points of a matching. Conversely, faster queries make the next data structure cheaper to build. After a number of rounds that depends only on $\varepsilon$, both preprocessing and query time are $O((m+n)^{1+\varepsilon})$. To make each round's output reusable by the next, we compactly summarize how matchings can enter and leave a pair of subcurves. These summaries can be applied to different starting points and retain the information needed to report the matchings.

Authors: Haoqiang Huang, Zihe Wang, Shuo Zhang

We present a deterministic algorithm that approximates the Fréchet distance between two polygonal curves in $\mathbb{R}^d$ with $n$ and $m$ vertices within a constant factor in $O((m+n)^{1+\varepsilon})$ time, for every fixed $\varepsilon>0$ and fixed dimension $d$. The approximation factor depends only on $\varepsilon$ and $d$. No assumption on the input curves is needed, and the algorithm also reports a matching certifying the approximation. The same result holds for the discrete Fréchet distance. Our result extends the line of work initiated by Cheng, Huang, and Zhang (2025), who gave the first constant-factor approximation in strongly subquadratic time, to near-linear time. For two $n$-vertex curves, the best previous running time was $\widetilde{O}(n^{5/3})$, due to van der Horst and Ophelders (2026). Our algorithm bootstraps between two primitives: a data structure that preprocesses a single curve for comparing any two of its subcurves, and a query procedure that compares subcurves of two different curves. Given the data structure for one curve, we compare it with another curve using a surrogate assembled from subcurves of the former. The surrogate is shared by all possible starting points of a matching. Conversely, faster queries make the next data structure cheaper to build. After a number of rounds that depends only on $\varepsilon$, both preprocessing and query time are $O((m+n)^{1+\varepsilon})$. To make each round's output reusable by the next, we compactly summarize how matchings can enter and leave a pair of subcurves. These summaries can be applied to different starting points and retain the information needed to report the matchings.

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

Finding and Certifying Weighted Minimum Cuts with Cut Queries

Authors: Troy Lee

We study the deterministic and nondeterministic cut query complexity of minimum cut in undirected graphs with arbitrary nonnegative edge weights. For the nondeterministic case, we show that every $n$-vertex graph has an $O(n\log^2(2n))$-query certificate of its exact global minimum-cut value, such that any other nonnegatively weighted graph on the same vertex set that agrees on these queries has the same minimum cut value. A key ingredient is a way to certify a maximum $s$--$t$ flow value with only $O(n)$ cut queries. For the deterministic case we give an algorithm that finds an exact global minimum cut using $O(n^{3/2}\log^3(2n)(\log\log(2n))^2)$ cut queries. Previously no $O(n^{2-\varepsilon})$ deterministic bound was known for arbitrary nonnegatively weighted graphs. The algorithm follows Karger's two-respecting tree approach. The key new ingredient is a deterministic construction of a weak skeleton of the input graph $G$: an explicit weighted graph $H$ in which every minimum cut of $G$ is a $5/4$-approximate minimum cut.

Authors: Troy Lee

We study the deterministic and nondeterministic cut query complexity of minimum cut in undirected graphs with arbitrary nonnegative edge weights. For the nondeterministic case, we show that every $n$-vertex graph has an $O(n\log^2(2n))$-query certificate of its exact global minimum-cut value, such that any other nonnegatively weighted graph on the same vertex set that agrees on these queries has the same minimum cut value. A key ingredient is a way to certify a maximum $s$--$t$ flow value with only $O(n)$ cut queries. For the deterministic case we give an algorithm that finds an exact global minimum cut using $O(n^{3/2}\log^3(2n)(\log\log(2n))^2)$ cut queries. Previously no $O(n^{2-\varepsilon})$ deterministic bound was known for arbitrary nonnegatively weighted graphs. The algorithm follows Karger's two-respecting tree approach. The key new ingredient is a deterministic construction of a weak skeleton of the input graph $G$: an explicit weighted graph $H$ in which every minimum cut of $G$ is a $5/4$-approximate minimum cut.

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

Exact Tie-Breaking and Tight Degree Bounds for Minimum-Degree Greedy on Interval Graphs

Authors: Ruize Zhao

Minimum-degree greedy repeatedly selects a vertex of minimum current degree and deletes its closed neighborhood; let $B(G)$ and $W(G)$ be its largest and smallest output sizes over complete legal executions. Given a closed interval representation of an $n$-vertex graph, we compute both values and construct attaining executions in $O(n^3)$ time and $O(n^2)$ space using endpoint-window dynamic programming and legal interleaving. For interval graphs with $Δ(G)\le4$, we prove the tight bound $B(G)\ge3α(G)/4$ by reduction to unit interval graphs and induction with virtual boundary vertices. A connected unit interval family $J_t$ attains equality, with $B(J_t)=W(J_t)=3t$ and $α(J_t)=4t$. For each degree bound $d\ge5$, a degree-five family $H_t$ approaches the known $2/3$ lower bound, giving infimum $2/3$. Together with known optimal favorable tie-breaking for $d\le3$, these results determine the favorable guarantee for every degree bound.

Authors: Ruize Zhao

Minimum-degree greedy repeatedly selects a vertex of minimum current degree and deletes its closed neighborhood; let $B(G)$ and $W(G)$ be its largest and smallest output sizes over complete legal executions. Given a closed interval representation of an $n$-vertex graph, we compute both values and construct attaining executions in $O(n^3)$ time and $O(n^2)$ space using endpoint-window dynamic programming and legal interleaving. For interval graphs with $Δ(G)\le4$, we prove the tight bound $B(G)\ge3α(G)/4$ by reduction to unit interval graphs and induction with virtual boundary vertices. A connected unit interval family $J_t$ attains equality, with $B(J_t)=W(J_t)=3t$ and $α(J_t)=4t$. For each degree bound $d\ge5$, a degree-five family $H_t$ approaches the known $2/3$ lower bound, giving infimum $2/3$. Together with known optimal favorable tie-breaking for $d\le3$, these results determine the favorable guarantee for every degree bound.

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

Testing MAX-CUT in the Adjacency Matrix Model: Tight Non-Adaptive Bounds and a Cubic Adaptive Lower Bound

Authors: Songhua He

We study the query complexity of approximating the MAX-CUT value to additive error $\varepsilon n^2$ in the adjacency matrix model. This was one of the first problems studied in the area of graph property testing [GGR, JACM'1998]. Previously, the best known upper bound for both adaptive and non-adaptive algorithms was $O(1/\varepsilon^6)$ [MS, SODA'2008; GMRS, ICALP'2022]. The only known lower bound was a trivial $Ω(1/\varepsilon^2)$ bound. We close this gap from both sides for non-adaptive algorithms, proving that its query complexity is exactly $Θ(1/\varepsilon^4)$. We also prove an adaptive $Ω(1/\varepsilon^3)$ lower bound. Our upper bound is achieved by a canonical tester that samples $O(1/\varepsilon^2)$ vertices in expectation and outputs the rescaled MAX-CUT value of the induced subgraph. We prove its correctness by applying spectral truncation to the adjacency matrix, using a geometric argument to control the sampling error of the retained low-rank part and a cut-norm restriction estimate [RV, JACM'2007] for the remainder. As an application, an analogous proof also yields a tight $O(1/\varepsilon^4)$ non-adaptive query upper bound for MAX-2CSP. For our lower bounds, we show indistinguishability between a uniform random graph and a distribution that plants a hidden partition. For the non-adaptive lower bound, we observe that one plus the $χ^2$-divergence between the query transcript distributions is exactly a normalized Ising partition function, which can be bounded using Boolean hypercontractivity. For the adaptive lower bound, we view each query through the algorithm's ability to predict whether its endpoints lie in the same part of the hidden partition. We bound the combined advantage of these predictions using the completed transcript, which lets us analyze the adaptively chosen pairs together despite their dependence on earlier answers.

Authors: Songhua He

We study the query complexity of approximating the MAX-CUT value to additive error $\varepsilon n^2$ in the adjacency matrix model. This was one of the first problems studied in the area of graph property testing [GGR, JACM'1998]. Previously, the best known upper bound for both adaptive and non-adaptive algorithms was $O(1/\varepsilon^6)$ [MS, SODA'2008; GMRS, ICALP'2022]. The only known lower bound was a trivial $Ω(1/\varepsilon^2)$ bound. We close this gap from both sides for non-adaptive algorithms, proving that its query complexity is exactly $Θ(1/\varepsilon^4)$. We also prove an adaptive $Ω(1/\varepsilon^3)$ lower bound. Our upper bound is achieved by a canonical tester that samples $O(1/\varepsilon^2)$ vertices in expectation and outputs the rescaled MAX-CUT value of the induced subgraph. We prove its correctness by applying spectral truncation to the adjacency matrix, using a geometric argument to control the sampling error of the retained low-rank part and a cut-norm restriction estimate [RV, JACM'2007] for the remainder. As an application, an analogous proof also yields a tight $O(1/\varepsilon^4)$ non-adaptive query upper bound for MAX-2CSP. For our lower bounds, we show indistinguishability between a uniform random graph and a distribution that plants a hidden partition. For the non-adaptive lower bound, we observe that one plus the $χ^2$-divergence between the query transcript distributions is exactly a normalized Ising partition function, which can be bounded using Boolean hypercontractivity. For the adaptive lower bound, we view each query through the algorithm's ability to predict whether its endpoints lie in the same part of the hidden partition. We bound the combined advantage of these predictions using the completed transcript, which lets us analyze the adaptively chosen pairs together despite their dependence on earlier answers.

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

Authors: Søren Fuglede Jørgensen

We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.

Authors: Søren Fuglede Jørgensen

We show that the problem of synthesizing a quantum circuit for a given Clifford operator using at most a specified amount of resources is NP-complete for several resource models of practical interest. Specifically, we study synthesis of a given Clifford operator using single-qubit Clifford gates and two-qubit Clifford gates, with full qubit connectivity and no additional ancilla qubits. With single-qubit gates free and CNOT as the only two-qubit gate, we show that deciding whether a given target admits an implementation using at most a specified number of CNOT gates is NP-complete, and we show that the problem remains NP-complete when any two-qubit Clifford gate is allowed and the budget counts all such gates. Moreover, we show that both results continue to hold when SWAP gates may be inserted for free, and when single-qubit gates are also considered part of the budget. In each case, we build on recent work by Li and Guan and show that there are polynomial-time reductions from Hamiltonian path in grid graphs to these decision problems, using a number of qubits quadratic in the number of grid vertices.

Monday, November 02 2026, 01:00

On the NP-Hardness of Rank-Metric Minimum Distance and Syndrome Decoding, at Every Rate

Authors: Venkata Gandikota, Vatsal Jha, S. Venkitesh

In this work, we investigate the hardness of computing minimum distance, as well as the hardness of syndrome decoding, for rank-metric codes. It is known (Gaborit and Zémor, IEEE TIT 2016) that for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$, minimum distance when $m=O(N)$ and syndrome decoding when $m=O(N^2)$ are NP-hard under randomized reductions. We obtain a derandomization of these results and a bit more. We show that for any fixed small $\varepsilon>0$, for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$ with $m=O(N^{1+\varepsilon})$, the problems of minimum distance, nearest codeword, and syndrome decoding are NP-hard under deterministic reductions. Further, we can prescribe the rate; that is, if we restrict to codes having any fixed rational rate $R\in(0,1)$, then the NP-hardness for all three problems persists. Our main technical contribution is simple, and possibly of independent interest. The reduction of Gaborit and Zémor (IEEE TIT 2016) relies on a distance-preserving embedding of a Hamming-metric code into a rank-metric code, obtained by multiplying its coordinates by a uniformly random vector. We show that we can choose an explicit vector from a slightly larger extension field.

Authors: Venkata Gandikota, Vatsal Jha, S. Venkitesh

In this work, we investigate the hardness of computing minimum distance, as well as the hardness of syndrome decoding, for rank-metric codes. It is known (Gaborit and Zémor, IEEE TIT 2016) that for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$, minimum distance when $m=O(N)$ and syndrome decoding when $m=O(N^2)$ are NP-hard under randomized reductions. We obtain a derandomization of these results and a bit more. We show that for any fixed small $\varepsilon>0$, for $\mathbb{F}_{q^m}$-linear rank-metric codes of length $N$ with $m=O(N^{1+\varepsilon})$, the problems of minimum distance, nearest codeword, and syndrome decoding are NP-hard under deterministic reductions. Further, we can prescribe the rate; that is, if we restrict to codes having any fixed rational rate $R\in(0,1)$, then the NP-hardness for all three problems persists. Our main technical contribution is simple, and possibly of independent interest. The reduction of Gaborit and Zémor (IEEE TIT 2016) relies on a distance-preserving embedding of a Hamming-metric code into a rank-metric code, obtained by multiplying its coordinates by a uniformly random vector. We show that we can choose an explicit vector from a slightly larger extension field.

Monday, November 02 2026, 01:00

A General Theoretical Framework for Transformed Geometric Pattern Matching in Music

Authors: David Meredith

In this paper, we present a framework of concepts, terminology and notation for geometric pattern matching, centred around the idea of a {\em transformation class}. After formally defining the concept of a transformation class, we provide detailed definitions of a number of specific transformation classes that are particularly relevant to pattern matching and pattern discovery in music, which is the primary application domain within which the framework has been developed. We then provide formal definitions of the notions of {\em match}, {\em occurrence}, {\em maximal match\/} and {\em maximal occurrence\/} and prove some results relating to these concepts. We then define and explore in detail the idea of an {\em occurrence set\/} and the more general notion of a {\em heterogeneous occurrence set}. Finally, we explore in detail the concept of {\em transformational equivalence}, including a consideration of {\em transformation groups}, along with the group-theoretic and transformational equivalence properties of selected transformation classes.

Authors: David Meredith

In this paper, we present a framework of concepts, terminology and notation for geometric pattern matching, centred around the idea of a {\em transformation class}. After formally defining the concept of a transformation class, we provide detailed definitions of a number of specific transformation classes that are particularly relevant to pattern matching and pattern discovery in music, which is the primary application domain within which the framework has been developed. We then provide formal definitions of the notions of {\em match}, {\em occurrence}, {\em maximal match\/} and {\em maximal occurrence\/} and prove some results relating to these concepts. We then define and explore in detail the idea of an {\em occurrence set\/} and the more general notion of a {\em heterogeneous occurrence set}. Finally, we explore in detail the concept of {\em transformational equivalence}, including a consideration of {\em transformation groups}, along with the group-theoretic and transformational equivalence properties of selected transformation classes.

Monday, November 02 2026, 01:00

Catalytic space-efficient algorithms for Distance and Determinant

Authors: Srijan Chakraborty, Samir Datta, Kushagra Shandilya

Catalytic computation reuses potentially large but in-use space (or catalytic space) which must be restored to its initial contents while the clean space is small. This allows all problems in $\mathsf{NL}$ and even larger classes to be in catalytic logspace [Buhrman et al, STOC2014]. In [Chmel et al, CCC2026] the catalytic space for reachability was restricted to sublinear space (specifically, $n \big/ 2^{θ(\sqrt{\log n})}$ space) while remaining in polynomial time. This matches the non-catalytic bounds from [Barnes et al, SICOMP1998]. We extend the reachability bound to shortest distance, proving a simultaneous sublinear catalytic space and polynomial time bound (henceforth called, sublinear $\mathsf{CLP}$). This is the first simultaneous sublinear space and polynomial time bound for distance even without the catalytic discipline. Moreover, our bound works even for polynomially weighted graphs. Along the way we prove simultaneous near linear catalytic space ($\mathcal{O}(n\log^{O(1)}{n})$) and polynomial time bounds for the determinant. We also show some applications to planar matching problems. Our starting point was catalytic computation of reachability in directed graphs by [Cook-Pyne, ITCS2026]. [Edenhofer, arXiv:2602.21088] used it to give a tradeoff between catalytic and non-catalytic space, while [Chmel et al, CCC2026] compress the graph to a smaller graph preserving $s,t$-reachability to achieve a smaller space bound. All three can be modified to yield an algorithm for unweighted distance. Our contribution is to define an approximation to the walk polynomial in weighted graphs that allows us to infer weighted $s,t$-distance in sublinear $\mathsf{CLP}$. We do so by non-trivially combining elements from each of the above three papers.

Authors: Srijan Chakraborty, Samir Datta, Kushagra Shandilya

Catalytic computation reuses potentially large but in-use space (or catalytic space) which must be restored to its initial contents while the clean space is small. This allows all problems in $\mathsf{NL}$ and even larger classes to be in catalytic logspace [Buhrman et al, STOC2014]. In [Chmel et al, CCC2026] the catalytic space for reachability was restricted to sublinear space (specifically, $n \big/ 2^{θ(\sqrt{\log n})}$ space) while remaining in polynomial time. This matches the non-catalytic bounds from [Barnes et al, SICOMP1998]. We extend the reachability bound to shortest distance, proving a simultaneous sublinear catalytic space and polynomial time bound (henceforth called, sublinear $\mathsf{CLP}$). This is the first simultaneous sublinear space and polynomial time bound for distance even without the catalytic discipline. Moreover, our bound works even for polynomially weighted graphs. Along the way we prove simultaneous near linear catalytic space ($\mathcal{O}(n\log^{O(1)}{n})$) and polynomial time bounds for the determinant. We also show some applications to planar matching problems. Our starting point was catalytic computation of reachability in directed graphs by [Cook-Pyne, ITCS2026]. [Edenhofer, arXiv:2602.21088] used it to give a tradeoff between catalytic and non-catalytic space, while [Chmel et al, CCC2026] compress the graph to a smaller graph preserving $s,t$-reachability to achieve a smaller space bound. All three can be modified to yield an algorithm for unweighted distance. Our contribution is to define an approximation to the walk polynomial in weighted graphs that allows us to infer weighted $s,t$-distance in sublinear $\mathsf{CLP}$. We do so by non-trivially combining elements from each of the above three papers.

Monday, November 02 2026, 01:00

Lowered crossover points of Anthropic's subexponential 3-SAT algorithm

Authors: Ruhollah Majdoddin

The October 2026 refutations of the Exponential Time Hypothesis count satisfying assignments in time $2^{o(n)}$, but an asymptotic bound does not say from which input size, the crossover point, they beat the best known algorithms. We make the #3-SAT recursion of Anthropic's algorithm explicit and lower its crossover points, computed from the main term of the proved bound, by proved changes. For the bound in the number $m$ of clauses, at clause density one, the crossover point drops to about $2^{420}$ clauses from $2^{33858}$; for general 3-SAT, in the number $n$ of variables, our estimate drops to $2^{8.6\cdot10^9}$ variables from $2^{2.2\cdot10^{12}}$. With integer tables the recursion counts, for every $j$, the assignments satisfying exactly $j$ clauses in time $2^{O(m/\sqrt{\log m})}$. We observe that this refutes the Sparse Max-3-SAT Hypothesis of Alman and Vassilevska Williams, which also follows from either refutation by the reduction of Impagliazzo, Paturi and Zane, and we lower the crossover point of this computation to about $2^{1016}$ clauses from $2^{68364}$. The distance is structural: we define a rewrite-and-merge class that contains the recursion and all our constructions, and prove that in the main-term model no grid algorithm in it can gain for general 3-SAT below about $2^{6000}$ variables. For two balanced homogeneous merges we prove with quantum functionals an exponent of at least $(\log_2K)^{-3}$ per edge bit, which excludes running times $2^{n^{1-c}}$. Independently of SAT, we formalise the algorithm as a model of computation, Signed Network Sums, which contains rank-one methods and contraction trees. On the complete graph with four nodes, one-level constructions contract the network in $N^{23/6}$ if $ω=2$ and in about $N^{3.93}$ on the current (preprint) matrix-multiplication curve, below the bound $N^4$ of every rank-one method or contraction tree.

Authors: Ruhollah Majdoddin

The October 2026 refutations of the Exponential Time Hypothesis count satisfying assignments in time $2^{o(n)}$, but an asymptotic bound does not say from which input size, the crossover point, they beat the best known algorithms. We make the #3-SAT recursion of Anthropic's algorithm explicit and lower its crossover points, computed from the main term of the proved bound, by proved changes. For the bound in the number $m$ of clauses, at clause density one, the crossover point drops to about $2^{420}$ clauses from $2^{33858}$; for general 3-SAT, in the number $n$ of variables, our estimate drops to $2^{8.6\cdot10^9}$ variables from $2^{2.2\cdot10^{12}}$. With integer tables the recursion counts, for every $j$, the assignments satisfying exactly $j$ clauses in time $2^{O(m/\sqrt{\log m})}$. We observe that this refutes the Sparse Max-3-SAT Hypothesis of Alman and Vassilevska Williams, which also follows from either refutation by the reduction of Impagliazzo, Paturi and Zane, and we lower the crossover point of this computation to about $2^{1016}$ clauses from $2^{68364}$. The distance is structural: we define a rewrite-and-merge class that contains the recursion and all our constructions, and prove that in the main-term model no grid algorithm in it can gain for general 3-SAT below about $2^{6000}$ variables. For two balanced homogeneous merges we prove with quantum functionals an exponent of at least $(\log_2K)^{-3}$ per edge bit, which excludes running times $2^{n^{1-c}}$. Independently of SAT, we formalise the algorithm as a model of computation, Signed Network Sums, which contains rank-one methods and contraction trees. On the complete graph with four nodes, one-level constructions contract the network in $N^{23/6}$ if $ω=2$ and in about $N^{3.93}$ on the current (preprint) matrix-multiplication curve, below the bound $N^4$ of every rank-one method or contraction tree.

Monday, November 02 2026, 01:00

Faster #PPM from faster #3SAT

Authors: László Kozma

The counting variant (#PPM) of the permutation pattern matching (PPM) problem asks for the number of occurrences of a permutation pattern in a host permutation of length $n$. The problem admits faster algorithms than the trivial $O^*(2^n)$ time enumeration; the best known running time, for both PPM and #PPM, is $O(1.415^n)$ (Gawrychowski and Rzepecki). It is also known that PPM is ETH-hard: an algorithm with running time $2^{o(n/\log{n})}$ would imply a subexponential algorithm for 3SAT (Jelínek, Opler, and Pekárek). We give a reduction in the opposite direction, showing that an algorithm for counting satisfying assignments of a 3-CNF formula (#3SAT) in time $2^{O({n}/{f(n)})}$, where $n$ is the number of variables, yields an algorithm for #PPM with running time $2^{O({n \sqrt{\log{f(n)}/{f(n)}}})}$. In particular, the recently claimed refutation of ETH by OpenAI implies a #3SAT algorithm with an improvement factor $f(n) \in Ω(\sqrt{\log{n}}/\log\log{n})$ in the exponent; this yields, through our reduction, a subexponential algorithm for #PPM.

Authors: László Kozma

The counting variant (#PPM) of the permutation pattern matching (PPM) problem asks for the number of occurrences of a permutation pattern in a host permutation of length $n$. The problem admits faster algorithms than the trivial $O^*(2^n)$ time enumeration; the best known running time, for both PPM and #PPM, is $O(1.415^n)$ (Gawrychowski and Rzepecki). It is also known that PPM is ETH-hard: an algorithm with running time $2^{o(n/\log{n})}$ would imply a subexponential algorithm for 3SAT (Jelínek, Opler, and Pekárek). We give a reduction in the opposite direction, showing that an algorithm for counting satisfying assignments of a 3-CNF formula (#3SAT) in time $2^{O({n}/{f(n)})}$, where $n$ is the number of variables, yields an algorithm for #PPM with running time $2^{O({n \sqrt{\log{f(n)}/{f(n)}}})}$. In particular, the recently claimed refutation of ETH by OpenAI implies a #3SAT algorithm with an improvement factor $f(n) \in Ω(\sqrt{\log{n}}/\log\log{n})$ in the exponent; this yields, through our reduction, a subexponential algorithm for #PPM.

Monday, November 02 2026, 01:00

Implicit Additively Weighted Geodesic Voronoi Diagrams and Applications

Authors: Bruce W. Brewer, Haitao Wang

Let $P$ be a simple polygon with $m$ vertices. After $O(m)$-time preprocessing of $P$, for any set $S$ of $n$ additively weighted point sites, we compute an implicit additively weighted geodesic Voronoi diagram in $O(n\log n+n\log^2 m)$ time, supporting nearest-neighbor queries in $O(\log(n+m))$ time. In contrast, explicitly constructing the diagram requires $Ω(m)$ time. Our algorithm introduces a new sweep-line paradigm that avoids explicitly constructing the diagram, together with a novel technique for point location in implicit subdivisions. As applications, we obtain new dynamic data structures for additively weighted geodesic nearest-neighbor searching and the first efficient algorithms for shortest paths in geodesic disk graphs. After the same $O(m)$-time preprocessing, single-source shortest paths in a geodesic disk graph of $n$ sites can be computed in $O(n\log n+n\log^2 m)$ time in the unweighted case and $O(n^{3/2}(\log n+\log^2 m))$ time in the weighted case. We also extend our techniques to implicit farthest geodesic Voronoi diagrams for unweighted sites, obtaining analogous bounds and dynamic farthest-neighbor data structures.

Authors: Bruce W. Brewer, Haitao Wang

Let $P$ be a simple polygon with $m$ vertices. After $O(m)$-time preprocessing of $P$, for any set $S$ of $n$ additively weighted point sites, we compute an implicit additively weighted geodesic Voronoi diagram in $O(n\log n+n\log^2 m)$ time, supporting nearest-neighbor queries in $O(\log(n+m))$ time. In contrast, explicitly constructing the diagram requires $Ω(m)$ time. Our algorithm introduces a new sweep-line paradigm that avoids explicitly constructing the diagram, together with a novel technique for point location in implicit subdivisions. As applications, we obtain new dynamic data structures for additively weighted geodesic nearest-neighbor searching and the first efficient algorithms for shortest paths in geodesic disk graphs. After the same $O(m)$-time preprocessing, single-source shortest paths in a geodesic disk graph of $n$ sites can be computed in $O(n\log n+n\log^2 m)$ time in the unweighted case and $O(n^{3/2}(\log n+\log^2 m))$ time in the weighted case. We also extend our techniques to implicit farthest geodesic Voronoi diagrams for unweighted sites, obtaining analogous bounds and dynamic farthest-neighbor data structures.

Monday, November 02 2026, 01:00

Optimal Average Distortion for Non-Expansive Embeddings of Euclidean Metrics into the Line: With applications to cut-matching games and vertex expansion

Authors: Jack Spalding-Jamieson

We show that any $n$-point Euclidean metric admits a non-expansive embedding into the line with average distortion at most $O\left(\sqrt{\frac{\log n}{\log\log n}}\right)$, matching a known lower bound. We also present two applications of this result. First, we give a cut-player strategy with an optimal spectral guarantee for the undirected cut-matching game. Second, we give an improved spectral approximation for vertex expansion in high-degree graphs.

Authors: Jack Spalding-Jamieson

We show that any $n$-point Euclidean metric admits a non-expansive embedding into the line with average distortion at most $O\left(\sqrt{\frac{\log n}{\log\log n}}\right)$, matching a known lower bound. We also present two applications of this result. First, we give a cut-player strategy with an optimal spectral guarantee for the undirected cut-matching game. Second, we give an improved spectral approximation for vertex expansion in high-degree graphs.

Monday, November 02 2026, 01:00

The Pareto Secretary Problem

Authors: Andrés Cristi, Yilin Kong, Jiechen Zhang

The Pareto secretary problem asks for online selection of a Pareto-maximal observation from an i.i.d. sample: no other observation is at least as good in every coordinate and strictly better in one. For every fixed dimension $d\ge2$ and independent continuous coordinates, the optimal failure probability is of order $1/[(\log n)^{d-1}\log\log n]$, both with full information and with coordinate ranks alone. A deterministic rank rule attains the optimal leading term uniformly over the marginals. For every fixed positive definite Gaussian covariance, the same failure order holds, with a leading constant depending on the correlation matrix, and a rank-based rule attains the optimal leading failure probability without knowing the Gaussian parameters. In both models, we determine the optimal fixed cutoff and its leading failure constant; its failure probability exceeds the unrestricted optimum by a factor $Θ_d((\log\log n)^2)$. Under arbitrary dependence, a fixed joint density with uniform marginals can instead have optimal full-information success tending to approximately $0.58$. The proof uses the scarcity of observations with very small domination probability to bound every stopping rule. A matching rule selects a currently maximal point when this probability is below a threshold that increases as time runs out. Coordinate ranks from an initial sample provide sufficiently accurate estimates to preserve the optimal leading failure probability. For singular Gaussian distributions with known correlations, we characterize the optimal failure asymptotics attainable using ranks. Singular dependence can make every observation acceptable or recover the difficulty of selecting a single best point.

Authors: Andrés Cristi, Yilin Kong, Jiechen Zhang

The Pareto secretary problem asks for online selection of a Pareto-maximal observation from an i.i.d. sample: no other observation is at least as good in every coordinate and strictly better in one. For every fixed dimension $d\ge2$ and independent continuous coordinates, the optimal failure probability is of order $1/[(\log n)^{d-1}\log\log n]$, both with full information and with coordinate ranks alone. A deterministic rank rule attains the optimal leading term uniformly over the marginals. For every fixed positive definite Gaussian covariance, the same failure order holds, with a leading constant depending on the correlation matrix, and a rank-based rule attains the optimal leading failure probability without knowing the Gaussian parameters. In both models, we determine the optimal fixed cutoff and its leading failure constant; its failure probability exceeds the unrestricted optimum by a factor $Θ_d((\log\log n)^2)$. Under arbitrary dependence, a fixed joint density with uniform marginals can instead have optimal full-information success tending to approximately $0.58$. The proof uses the scarcity of observations with very small domination probability to bound every stopping rule. A matching rule selects a currently maximal point when this probability is below a threshold that increases as time runs out. Coordinate ranks from an initial sample provide sufficiently accurate estimates to preserve the optimal leading failure probability. For singular Gaussian distributions with known correlations, we characterize the optimal failure asymptotics attainable using ranks. Singular dependence can make every observation acceptable or recover the difficulty of selecting a single best point.

Monday, November 02 2026, 01:00

Differentially Private Synthetic Data from Sliced Chebyshev Moments

Authors: Lucas Rosenblatt, R. Teal Witter, Christopher Musco

Research on practical differentially private (DP) synthetic data has largely focused on methods that approximately preserve low-dimensional marginals of a target distribution, and can thus only hope to preserve higher-dimensional, global structure as a lucky by-product. At the same time, existing methods that target global accuracy guarantees pay a price exponential in the data dimension, $d$. In this work, we address this ``curse of dimensionality'' by presenting a new DP synthetic data algorithm, SPLICE, that comes with a compelling global distributional guarantee, even in high-dimensions, and moreover, is practically state-of-the-art. Our algorithm is based on measuring noisy sliced Chebyshev polynomial moments of the target distribution and fitting a distribution to match, a twist on recent polynomial methods for 1-D synthetic data. We prove that, given a dataset of $n$ examples, SPLICE returns a private distribution with Sliced Wasserstein-1 distance $\tilde{O}(1/\sqrt{n})$ from the target, for constant privacy parameters. Notably, this bound is independent of the data dimension, $d$. Along the way, we prove a moment matching result of independent interest: for a distribution on the unit ball, matching the first $O(1/α)$ Chebyshev moments along $\tilde{O}(d/α^2)$ random directions guarantees $α$ Sliced Wasserstein error. Finally, we provide a practical implementation of the algorithm that borrows tools from the tensor decomposition literature and achieves state-of-the-art performance compared to prior DP synthetic data methods on an extensive benchmark.

Authors: Lucas Rosenblatt, R. Teal Witter, Christopher Musco

Research on practical differentially private (DP) synthetic data has largely focused on methods that approximately preserve low-dimensional marginals of a target distribution, and can thus only hope to preserve higher-dimensional, global structure as a lucky by-product. At the same time, existing methods that target global accuracy guarantees pay a price exponential in the data dimension, $d$. In this work, we address this ``curse of dimensionality'' by presenting a new DP synthetic data algorithm, SPLICE, that comes with a compelling global distributional guarantee, even in high-dimensions, and moreover, is practically state-of-the-art. Our algorithm is based on measuring noisy sliced Chebyshev polynomial moments of the target distribution and fitting a distribution to match, a twist on recent polynomial methods for 1-D synthetic data. We prove that, given a dataset of $n$ examples, SPLICE returns a private distribution with Sliced Wasserstein-1 distance $\tilde{O}(1/\sqrt{n})$ from the target, for constant privacy parameters. Notably, this bound is independent of the data dimension, $d$. Along the way, we prove a moment matching result of independent interest: for a distribution on the unit ball, matching the first $O(1/α)$ Chebyshev moments along $\tilde{O}(d/α^2)$ random directions guarantees $α$ Sliced Wasserstein error. Finally, we provide a practical implementation of the algorithm that borrows tools from the tensor decomposition literature and achieves state-of-the-art performance compared to prior DP synthetic data methods on an extensive benchmark.

Monday, November 02 2026, 01:00

The Quartic Gap Between Quantum and Dequantized Algorithms for Linear Algebra Is Tight

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

Quantum algorithms for linear algebra that access a matrix $A$ through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to $A$, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm $ρ=\|A\|_F/\|A\|$ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use $\widetilde O(ρ)$ queries, whereas the best dequantized algorithms use $\widetilde O(ρ^4)$. We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant $κ_0>1$, we construct dense linear systems with condition number at most $κ_0$ for which sampling from the solution within a sufficiently small constant total variation distance requires $Ω(ρ^4/\logρ)$ classical queries but only $Θ(ρ)$ quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.

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

Quantum algorithms for linear algebra that access a matrix $A$ through quantum random access memory (QRAM) prepare quantum states proportional to the solutions of linear systems and related problems. Dequantized classical algorithms, when given sampling and query access to $A$, return sampling and query access to an approximate solution. Both kinds of algorithms use a number of queries that is polynomial in the normalized Frobenius norm $ρ=\|A\|_F/\|A\|$ and does not directly depend on the dimension. For linear systems at fixed condition number and accuracy, quantum algorithms use $\widetilde O(ρ)$ queries, whereas the best dequantized algorithms use $\widetilde O(ρ^4)$. We prove that this quartic gap is tight in the worst case, up to a logarithmic factor. For every constant $κ_0>1$, we construct dense linear systems with condition number at most $κ_0$ for which sampling from the solution within a sufficiently small constant total variation distance requires $Ω(ρ^4/\logρ)$ classical queries but only $Θ(ρ)$ quantum queries. The same separation holds, for errors below smaller thresholds, for positive-definite systems, least squares, ridge regression, matrix powers and square roots, imaginary-time and real-time Hamiltonian evolution, and sampling from the top principal component. Our results therefore show that many quantum algorithms for linear algebra, achieve a near-quartic speedup in the worst case, over classical algorithms with sampling and query access. All of these classical lower bounds follow from the same framework, which is based on a lower bound for sampling from distributions that depend on a fixed set of Fourier coefficients of an unknown Boolean function.

Monday, November 02 2026, 01:00

Two choices suffice: $k$-out sampling leaves $O(n/k)$ intercomponent edges for $k\geq 2$

Authors: Yanyu Chen, Yu Chen

Given a simple undirected graph on $n$ vertices, random $k$-out sampling forms a spanning subgraph by having each vertex independently choose $k$ incident edges uniformly with replacement. We prove that, for every integer $k\geq2$, the expected number of original edges joining distinct sampled components is $O(n/k)$. This resolves the conjecture of Holm, King, Thorup, Zamir, and Zwick in its full range, improving their logarithmic sampling requirement to the optimal threshold of two choices per vertex. The bound fails for $k=1$, and the dependence $O(n/k)$ is tight up to constant factors. Our result holds with replacement and consequently also without replacement. In particular, this implies that contracting just two independent local choices per vertex suffices to reduce an arbitrary simple graph to a quotient with $O(n)$ edges in expectation, which is crucial in reducing the instance size for many contraction-based algorithms. Our proof uses switching arguments to relate intercomponent edges to bridges and a degree-weighted component potential, whose expectation we also bound with a switching argument.

Authors: Yanyu Chen, Yu Chen

Given a simple undirected graph on $n$ vertices, random $k$-out sampling forms a spanning subgraph by having each vertex independently choose $k$ incident edges uniformly with replacement. We prove that, for every integer $k\geq2$, the expected number of original edges joining distinct sampled components is $O(n/k)$. This resolves the conjecture of Holm, King, Thorup, Zamir, and Zwick in its full range, improving their logarithmic sampling requirement to the optimal threshold of two choices per vertex. The bound fails for $k=1$, and the dependence $O(n/k)$ is tight up to constant factors. Our result holds with replacement and consequently also without replacement. In particular, this implies that contracting just two independent local choices per vertex suffices to reduce an arbitrary simple graph to a quotient with $O(n)$ edges in expectation, which is crucial in reducing the instance size for many contraction-based algorithms. Our proof uses switching arguments to relate intercomponent edges to bridges and a degree-weighted component potential, whose expectation we also bound with a switching argument.

Monday, November 02 2026, 01:00

Optimal Online Weighted Connectivity Augmentation

Authors: Mohit Garg, Felix Hommelsheim, Alexander Lindermayr

In the online Weighted Connectivity Augmentation Problem (WCAP), we are given a $k$-edge-connected graph on $n$ vertices and a set of additional edges, called links, with nonnegative costs. Pairs of vertices arrive one by one, and an online algorithm must immediately and irrevocably buy links so that each pair becomes $(k+1)$-edge-connected. The goal is to minimize the total cost of the purchased links. We obtain a deterministic polynomial-time $O(\log n)$-competitive algorithm for WCAP, improving the previous $O(\log^2 n)$ bound (SODA 2026). Our bound is asymptotically optimal, even for randomized algorithms in the random-order model. The general problem reduces to the case where the underlying graph is a cycle, for which link costs in $\{0,1\}$ already yield an $Ω(\log n)$ lower bound. Surprisingly, for unit costs on a cycle, we obtain a deterministic polynomial-time $4$-competitive algorithm, improving the previous $O(\log n)$ bound (SODA 2026), and prove that this ratio is optimal among deterministic algorithms. Thus, with unit costs, cycles are easier than trees, where the optimal competitive ratio is $Θ(\log n)$.

Authors: Mohit Garg, Felix Hommelsheim, Alexander Lindermayr

In the online Weighted Connectivity Augmentation Problem (WCAP), we are given a $k$-edge-connected graph on $n$ vertices and a set of additional edges, called links, with nonnegative costs. Pairs of vertices arrive one by one, and an online algorithm must immediately and irrevocably buy links so that each pair becomes $(k+1)$-edge-connected. The goal is to minimize the total cost of the purchased links. We obtain a deterministic polynomial-time $O(\log n)$-competitive algorithm for WCAP, improving the previous $O(\log^2 n)$ bound (SODA 2026). Our bound is asymptotically optimal, even for randomized algorithms in the random-order model. The general problem reduces to the case where the underlying graph is a cycle, for which link costs in $\{0,1\}$ already yield an $Ω(\log n)$ lower bound. Surprisingly, for unit costs on a cycle, we obtain a deterministic polynomial-time $4$-competitive algorithm, improving the previous $O(\log n)$ bound (SODA 2026), and prove that this ratio is optimal among deterministic algorithms. Thus, with unit costs, cycles are easier than trees, where the optimal competitive ratio is $Θ(\log n)$.

Monday, November 02 2026, 01:00

3SUM Made Simpler (Without Matrix Multiplication), $k$-SUM Hypothesis Refuted for Odd $k$, and Real 3SUM/APSP Derandomized

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

Recently, a truly subquadratic algorithm for the 3SUM problem and a truly subcubic algorithm for the all-pairs shortest paths (APSP) problem were found by Anthropic, as reported by Alman and Vassilevska Williams (2026). These algorithms relied on a new subroutine for rectangular matrix multiplication with selected output entries, which was obtained by modifying known matrix multiplication techniques of Schönhage (1981) and Coppersmith (1982). 1. We present a different truly subquadratic randomized algorithm for 3SUM for integer inputs. It completely avoids the use of matrix multiplication but instead modifies known dynamic programming algorithms for subset convolution and the zeta transform, which rely only on simple subtraction tricks via inclusion--exclusion (and does not even require fast Fourier transform), and so is conceptually easier to understand (the whole solution is about 5 pages long). 2. Following a similar approach, we obtain a randomized algorithm for $k$-SUM for integer inputs for odd constant $k$ in $O(n^{\lceil k/2\rceil-\varepsilon_k})$ time for some constant $\varepsilon_k>0$, disproving the $k$-SUM Hypothesis for all odd $k$. 3. Existing reductions from 3SUM and APSP to selected-entries rectangular matrix multiplication were randomized in the case of real-valued input. We present new deterministic reductions, yielding a truly subquadratic deterministic algorithm for Real 3SUM and a truly subcubic deterministic algorithm for Real APSP.

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

Recently, a truly subquadratic algorithm for the 3SUM problem and a truly subcubic algorithm for the all-pairs shortest paths (APSP) problem were found by Anthropic, as reported by Alman and Vassilevska Williams (2026). These algorithms relied on a new subroutine for rectangular matrix multiplication with selected output entries, which was obtained by modifying known matrix multiplication techniques of Schönhage (1981) and Coppersmith (1982). 1. We present a different truly subquadratic randomized algorithm for 3SUM for integer inputs. It completely avoids the use of matrix multiplication but instead modifies known dynamic programming algorithms for subset convolution and the zeta transform, which rely only on simple subtraction tricks via inclusion--exclusion (and does not even require fast Fourier transform), and so is conceptually easier to understand (the whole solution is about 5 pages long). 2. Following a similar approach, we obtain a randomized algorithm for $k$-SUM for integer inputs for odd constant $k$ in $O(n^{\lceil k/2\rceil-\varepsilon_k})$ time for some constant $\varepsilon_k>0$, disproving the $k$-SUM Hypothesis for all odd $k$. 3. Existing reductions from 3SUM and APSP to selected-entries rectangular matrix multiplication were randomized in the case of real-valued input. We present new deterministic reductions, yielding a truly subquadratic deterministic algorithm for Real 3SUM and a truly subcubic deterministic algorithm for Real APSP.

Monday, November 02 2026, 01:00

The C-3PO Framework: Algorithms for $\#\mathrm{BIS}$-Hard Problems on Graphs with Low Threshold Rank or High Girth

Authors: Ijay Narang

We give approximate counting and sampling algorithms for spin models on graphs of low threshold rank or high girth. Given any $n$-vertex graph $G$, let $k_η= k_η(G)$ count normalized-Laplacian eigenvalues strictly below $η$. For $\varepsilon > 0$, an $\varepsilon$-approximate counting algorithm outputs an estimate $\widehat Z$ of the partition function $Z$ satisfying $(1-\varepsilon)Z\le\widehat Z\le(1+\varepsilon)Z$ with probability at least $3/4$. An $\varepsilon$-approximate sampling algorithm outputs a configuration whose distribution is within total variation distance $\varepsilon$ of the target distribution. We give the following results: (a) Ferromagnetic Potts. Fix $q\ge2$ and $η\in(0,1]$. There exists $β_0(q,η)$ such that, for every inverse temperature $β\geβ_0$, the zero-field $q$-state Potts model admits approximate counting and sampling algorithms: (i) On arbitrary graphs in time $\exp(O(k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) Fix $Δ\ge 3$. On $Δ$-regular graphs of girth $g$ in time $\exp(O(n e^{-g})) \operatorname{poly}(n,1/\varepsilon).$ (b) Bipartite hard-core. Fix $Δ\ge3$ and $η\in(0,1]$. There exists $λ_0(Δ,η)$ such that, for every fugacity $λ\geλ_0$, the hard-core model admits approximate counting and sampling: (i) On $Δ$-regular vertex-transitive bipartite graphs in time $\exp(O (k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) On $Δ$-regular bipartite graphs of girth $g$ in $\exp(O(n e^{- g}))\operatorname{poly}(n,1/\varepsilon).$ All four results follow from verifying the hypotheses of a new framework we call C-3PO. At a high level, the C-3PO algorithm proceeds by enumerating small cuts and reconstructing the original measure through cluster expansions. Underlying the algorithm is an improvement of subspace enumeration methods via active regression.

Authors: Ijay Narang

We give approximate counting and sampling algorithms for spin models on graphs of low threshold rank or high girth. Given any $n$-vertex graph $G$, let $k_η= k_η(G)$ count normalized-Laplacian eigenvalues strictly below $η$. For $\varepsilon > 0$, an $\varepsilon$-approximate counting algorithm outputs an estimate $\widehat Z$ of the partition function $Z$ satisfying $(1-\varepsilon)Z\le\widehat Z\le(1+\varepsilon)Z$ with probability at least $3/4$. An $\varepsilon$-approximate sampling algorithm outputs a configuration whose distribution is within total variation distance $\varepsilon$ of the target distribution. We give the following results: (a) Ferromagnetic Potts. Fix $q\ge2$ and $η\in(0,1]$. There exists $β_0(q,η)$ such that, for every inverse temperature $β\geβ_0$, the zero-field $q$-state Potts model admits approximate counting and sampling algorithms: (i) On arbitrary graphs in time $\exp(O(k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) Fix $Δ\ge 3$. On $Δ$-regular graphs of girth $g$ in time $\exp(O(n e^{-g})) \operatorname{poly}(n,1/\varepsilon).$ (b) Bipartite hard-core. Fix $Δ\ge3$ and $η\in(0,1]$. There exists $λ_0(Δ,η)$ such that, for every fugacity $λ\geλ_0$, the hard-core model admits approximate counting and sampling: (i) On $Δ$-regular vertex-transitive bipartite graphs in time $\exp(O (k_η))\operatorname{poly}(n,1/\varepsilon).$ (ii) On $Δ$-regular bipartite graphs of girth $g$ in $\exp(O(n e^{- g}))\operatorname{poly}(n,1/\varepsilon).$ All four results follow from verifying the hypotheses of a new framework we call C-3PO. At a high level, the C-3PO algorithm proceeds by enumerating small cuts and reconstructing the original measure through cluster expansions. Underlying the algorithm is an improvement of subspace enumeration methods via active regression.

Monday, November 02 2026, 01:00

Revisiting the directed pathwidth of h-semicomplete digraphs

Authors: Guilherme C. M. Gomes, Raul Lopes, Matthias Mnich

A digraph is $h$-semicomplete if every vertex has at most $h$ non-neighbors. In semicomplete digraphs, $k$-triples are obstructions to bounded directed pathwidth with many algorithmic applications: Fradkin and Seymour [JCTB 2013] showed that a large triple contains a subdivision of every small digraph, and triples were used for irrelevant vertex arguments in routing problems by Fomin and Pilipczuk [JCTB 2019] and Gomes, Lopes, and Sau [ICALP 2025]. Fradkin and Seymour also showed that there is a function $f(k)$ such that every semicomplete digraph of directed pathwidth at least $f(k)$ contains a $k$-triple, and their proof gives $f(k)$ triply exponential in $k$. For $h$-semicomplete digraphs, a result of Kitsunai et al. [ESA 2015] also yields a $k$-triple, but only existentially, and finding one in fixed-parameter time remained open. We improve the bound on $f(k)$ to a single exponential and extend the result to $h$-semicomplete digraphs: we show that every $h$-semicomplete digraph of directed pathwidth greater than $(h+1) \cdot k \cdot 2^{27k+9}$ contains a $k$-triple, which we find in polynomial time, and no bound subexponential in $k$ or sublinear in $h$ is possible. We also extend the approximation algorithm for directed pathwidth of Fomin and Pilipczuk to $h$-semicomplete digraphs, so its certificates of large directed pathwidth remain valid. Our proof is much shorter than the analysis of Kitsunai et al., and while their algorithm is exact, ours runs in polynomial time. As an application, we show that Arc-Disjoint Paths on $h$-semicomplete digraphs is fixed-parameter tractable parameterized by the number of paths and $h$, generalizing the semicomplete case. Finally, unlike in semicomplete digraphs, directed pathwidth does not bound directed cliquewidth: for every $k$, some acyclic $1$-semicomplete digraph, so of directed pathwidth zero, has directed cliquewidth at least $k$.

Authors: Guilherme C. M. Gomes, Raul Lopes, Matthias Mnich

A digraph is $h$-semicomplete if every vertex has at most $h$ non-neighbors. In semicomplete digraphs, $k$-triples are obstructions to bounded directed pathwidth with many algorithmic applications: Fradkin and Seymour [JCTB 2013] showed that a large triple contains a subdivision of every small digraph, and triples were used for irrelevant vertex arguments in routing problems by Fomin and Pilipczuk [JCTB 2019] and Gomes, Lopes, and Sau [ICALP 2025]. Fradkin and Seymour also showed that there is a function $f(k)$ such that every semicomplete digraph of directed pathwidth at least $f(k)$ contains a $k$-triple, and their proof gives $f(k)$ triply exponential in $k$. For $h$-semicomplete digraphs, a result of Kitsunai et al. [ESA 2015] also yields a $k$-triple, but only existentially, and finding one in fixed-parameter time remained open. We improve the bound on $f(k)$ to a single exponential and extend the result to $h$-semicomplete digraphs: we show that every $h$-semicomplete digraph of directed pathwidth greater than $(h+1) \cdot k \cdot 2^{27k+9}$ contains a $k$-triple, which we find in polynomial time, and no bound subexponential in $k$ or sublinear in $h$ is possible. We also extend the approximation algorithm for directed pathwidth of Fomin and Pilipczuk to $h$-semicomplete digraphs, so its certificates of large directed pathwidth remain valid. Our proof is much shorter than the analysis of Kitsunai et al., and while their algorithm is exact, ours runs in polynomial time. As an application, we show that Arc-Disjoint Paths on $h$-semicomplete digraphs is fixed-parameter tractable parameterized by the number of paths and $h$, generalizing the semicomplete case. Finally, unlike in semicomplete digraphs, directed pathwidth does not bound directed cliquewidth: for every $k$, some acyclic $1$-semicomplete digraph, so of directed pathwidth zero, has directed cliquewidth at least $k$.

Monday, November 02 2026, 01:00

Compressing the Lawler-Moore Dynamic Program for Weighted Tardy Jobs

Authors: Klaus Jansen, Björn Schumacher, Corinna Wambsganz

The classic scheduling problem of minimizing the total weight of tardy jobs ($P_m||\sum_jw_jU_j$) asks for a schedule of jobs each with a processing time, weight, and due date on a constant number of parallel identical machines. A job is late in a schedule if it completes after its due date, and an optimal schedule is one where the total weight of all late jobs is minimized. The classic dynamic programming algorithm of Lawler and Moore [Manag. Sci., 1969] solves this problem in $O(nd_{\max}^m)$ where $n$ is the number of jobs, $d_{\max}$ is the maximum due date of any job, and $m$ is the number of identical machines. We show that it is sufficient to only compute part of the dynamic program and obtain the running time $\tilde{O}(n + d_{\max}^{m + 1})$ where $\tilde{O}$ suppresses polylogarithmic factors. This affirmatively answers the open question posed by Hermelin, Molter, and Shabtay [INFORMS J. Comput., 2024], whether a running time of $\tilde{O}(n + d_{\max}^2)$ is possible on a single machine.

Authors: Klaus Jansen, Björn Schumacher, Corinna Wambsganz

The classic scheduling problem of minimizing the total weight of tardy jobs ($P_m||\sum_jw_jU_j$) asks for a schedule of jobs each with a processing time, weight, and due date on a constant number of parallel identical machines. A job is late in a schedule if it completes after its due date, and an optimal schedule is one where the total weight of all late jobs is minimized. The classic dynamic programming algorithm of Lawler and Moore [Manag. Sci., 1969] solves this problem in $O(nd_{\max}^m)$ where $n$ is the number of jobs, $d_{\max}$ is the maximum due date of any job, and $m$ is the number of identical machines. We show that it is sufficient to only compute part of the dynamic program and obtain the running time $\tilde{O}(n + d_{\max}^{m + 1})$ where $\tilde{O}$ suppresses polylogarithmic factors. This affirmatively answers the open question posed by Hermelin, Molter, and Shabtay [INFORMS J. Comput., 2024], whether a running time of $\tilde{O}(n + d_{\max}^2)$ is possible on a single machine.

Monday, November 02 2026, 01:00

Degree Realization with Maximum Matching

Authors: Amotz Bar-Noy, Igor Kalinichev, David Peleg, Dror Rawitz

The Maximum Matching Degree Realization problem requires, for a sequence $d$ of $n$ positive integers, to compute a realizing graph(whose degree sequence corresponds to $d$) with the maximum possible matching. We present a polynomial-time algorithm for this problem. The decision version of this problem requires to determine whether a sequence $d$ admits a realization with a matching of size $ν$ (without computing such a realization). Recently, it was shown that it can be done by verifying a series of inequalities. We improve the existing characterization by reducing the number of inequalities.

Authors: Amotz Bar-Noy, Igor Kalinichev, David Peleg, Dror Rawitz

The Maximum Matching Degree Realization problem requires, for a sequence $d$ of $n$ positive integers, to compute a realizing graph(whose degree sequence corresponds to $d$) with the maximum possible matching. We present a polynomial-time algorithm for this problem. The decision version of this problem requires to determine whether a sequence $d$ admits a realization with a matching of size $ν$ (without computing such a realization). Recently, it was shown that it can be done by verifying a series of inequalities. We improve the existing characterization by reducing the number of inequalities.

Monday, November 02 2026, 01:00

A Fully Polynomial Randomized Approximation Scheme for Hypergraph Unreliability

Authors: Zhijie Zheng

We give a fully polynomial randomized approximation scheme for all-terminal unreliability in explicitly represented undirected hypergraphs of arbitrary rank. The algorithm estimates the probability that independent hyperedge failures disconnect the hypergraph, allowing a different rational failure probability for every edge. For a common failure probability, this improves the quasi-polynomial approximation schemes of Cen, Li, and Panigrahi to polynomial time and answers the polynomial-time approximation question reiterated in their subsequent work on graphs. The construction turns their representative-survival bound into an importance sampler whose relative second moment is bounded solely in terms of the number of vertices. We protect one randomly chosen representative pair during contraction, correct the path likelihood, and divide by the number of pairs surviving at the end. The resulting unbiased estimator has relative second moment at most $\binom n2$, even when unreliability is exponentially small. A minimum-cut normalization makes this estimator implementable with polynomially bounded precision: the residual likelihood becomes a survival probability in an auxiliary absorbing process, which admits rational discretization with a fixed random-bit budget. Logarithmic rate rounding and an exact weighted minimum cut extend the construction to unequal rational failure probabilities. The algorithm's worst-case running time and random-bit usage are polynomial in the input length, the inverse relative error, and $\log(1/δ)$ for confidence $1-δ$.

Authors: Zhijie Zheng

We give a fully polynomial randomized approximation scheme for all-terminal unreliability in explicitly represented undirected hypergraphs of arbitrary rank. The algorithm estimates the probability that independent hyperedge failures disconnect the hypergraph, allowing a different rational failure probability for every edge. For a common failure probability, this improves the quasi-polynomial approximation schemes of Cen, Li, and Panigrahi to polynomial time and answers the polynomial-time approximation question reiterated in their subsequent work on graphs. The construction turns their representative-survival bound into an importance sampler whose relative second moment is bounded solely in terms of the number of vertices. We protect one randomly chosen representative pair during contraction, correct the path likelihood, and divide by the number of pairs surviving at the end. The resulting unbiased estimator has relative second moment at most $\binom n2$, even when unreliability is exponentially small. A minimum-cut normalization makes this estimator implementable with polynomially bounded precision: the residual likelihood becomes a survival probability in an auxiliary absorbing process, which admits rational discretization with a fixed random-bit budget. Logarithmic rate rounding and an exact weighted minimum cut extend the construction to unequal rational failure probabilities. The algorithm's worst-case running time and random-bit usage are polynomial in the input length, the inverse relative error, and $\log(1/δ)$ for confidence $1-δ$.

Monday, November 02 2026, 01:00

A Better Lower Bound for Sublinear Metric Steiner Tree

Authors: Sepideh Mahabadi, Jakub Tarnawski, Ali Vakilian

In the metric Steiner tree problem, we are given an $n$-point metric space $(V,w)$ and a set of terminals $T\subseteq V$, and we are interested in the smallest total weight $\mathsf{st}(V,T,w)$ of a tree that contains every terminal and may also use points of $V\setminus T$. We work in the distance-oracle model: the algorithm knows $V$ and $T$ in advance and learns $w$ only by querying a single entry $w(u,v)$ at a time. The goal is to estimate $\mathsf{st}(V,T,w)$ with the minimum number of queries. A factor of $2$ is achievable with nearly linear number of queries, so the interesting question is the query complexity of algorithms with an approximation ratio strictly below $2$. We prove that for every fixed $η\in(0,1)$, every randomized $(2-η)$-approximation algorithm must make more than $\widetildeΩ(n^{7/5})$ queries, improving upon the previously known bound of $\widetildeΩ(n^{6/5})$. Given that the best known upper bound shows that $\widetilde O(n^{5/3})$ queries suffice, the exponent of this problem now lies between $7/5$ and $5/3$, up to logarithmic factors.

Authors: Sepideh Mahabadi, Jakub Tarnawski, Ali Vakilian

In the metric Steiner tree problem, we are given an $n$-point metric space $(V,w)$ and a set of terminals $T\subseteq V$, and we are interested in the smallest total weight $\mathsf{st}(V,T,w)$ of a tree that contains every terminal and may also use points of $V\setminus T$. We work in the distance-oracle model: the algorithm knows $V$ and $T$ in advance and learns $w$ only by querying a single entry $w(u,v)$ at a time. The goal is to estimate $\mathsf{st}(V,T,w)$ with the minimum number of queries. A factor of $2$ is achievable with nearly linear number of queries, so the interesting question is the query complexity of algorithms with an approximation ratio strictly below $2$. We prove that for every fixed $η\in(0,1)$, every randomized $(2-η)$-approximation algorithm must make more than $\widetildeΩ(n^{7/5})$ queries, improving upon the previously known bound of $\widetildeΩ(n^{6/5})$. Given that the best known upper bound shows that $\widetilde O(n^{5/3})$ queries suffice, the exponent of this problem now lies between $7/5$ and $5/3$, up to logarithmic factors.

Monday, November 02 2026, 01:00

List 3-Coloring in tame classes of graphs in quasipolynomial time

Authors: Paweł Rzążewski

A class of graphs $\mathcal{G}$ is tame if there is a constant c such that every $n$-vertex graph in $\mathcal{G}$ has at most $n^c$ minimal separators. Many important graph classes are tame, including chordal graphs, permutation graphs, and circle graphs. Several fundamental problems, most notably Max Weight Independent Set, can be solved in polynomial time on every tame class, which explains in a unified way their tractability on all these classes. We show that List 3-Coloring can be solved in quasipolynomial time on every tame class of graphs.

Authors: Paweł Rzążewski

A class of graphs $\mathcal{G}$ is tame if there is a constant c such that every $n$-vertex graph in $\mathcal{G}$ has at most $n^c$ minimal separators. Many important graph classes are tame, including chordal graphs, permutation graphs, and circle graphs. Several fundamental problems, most notably Max Weight Independent Set, can be solved in polynomial time on every tame class, which explains in a unified way their tractability on all these classes. We show that List 3-Coloring can be solved in quasipolynomial time on every tame class of graphs.

Monday, November 02 2026, 01:00

Steiner Tree Parameterized by Multiway Cut: Single-Exponential Time and Polynomial Space

Authors: Lukas Van de Voorde, Brecht Verbeken, Vincent Ginis

In the Steiner Tree problem, the goal is to find a minimum-weight tree containing a given set of terminals in an edge-weighted graph. Given a multiway cut of size $s$, we propose a deterministic algorithm running in $2^{O(s)}\operatorname{poly}(n,b)$ time and polynomial space ($n$ vertices, non-negative $b$-bit integer weights). A multiway cut for the terminals is a set of vertices whose deletion leaves at most one terminal in each connected component. This result answers a question of Jansen and Swennenhuis (ESA 2024). Building on their component decomposition, we show that it suffices to connect the selected cut vertices together with $O(s)$ additional terminals and attach the remaining terminals by shortest paths. We use colour coding to optimize over the choice of these additional terminals.

Authors: Lukas Van de Voorde, Brecht Verbeken, Vincent Ginis

In the Steiner Tree problem, the goal is to find a minimum-weight tree containing a given set of terminals in an edge-weighted graph. Given a multiway cut of size $s$, we propose a deterministic algorithm running in $2^{O(s)}\operatorname{poly}(n,b)$ time and polynomial space ($n$ vertices, non-negative $b$-bit integer weights). A multiway cut for the terminals is a set of vertices whose deletion leaves at most one terminal in each connected component. This result answers a question of Jansen and Swennenhuis (ESA 2024). Building on their component decomposition, we show that it suffices to connect the selected cut vertices together with $O(s)$ additional terminals and attach the remaining terminals by shortest paths. We use colour coding to optimize over the choice of these additional terminals.

Monday, November 02 2026, 01:00

New bounds for online coloring three-colorable graphs

Authors: Yaqiao Li

We prove new lower and upper bounds for online coloring of three-colorable graphs, improving classical bounds of Vishwanathan [FOCS'1990] that have stood for more than three decades. For the lower bound against deterministic algorithms, we improve Vishwanathan's $Ω(\log^2n)$ to $\exp\!\bigl(Ω((\log n)^{1/3}(\log\log n)^{2/3})\bigr)$. For the upper bound of randomized algorithms, we give a randomized algorithm that achieves $O(\sqrt{n})$ improving Vishwanathan's $O(\sqrt{n \log n})$. Technically, for the lower bound construction, we introduce the language of $t$-fold coloring for simpler description of recursive construction, and the notion of post-degree for future control, and use the flexibility of two-sided colors of bipartite graph, constructed using the independent set amplification method of Vishwanathan, as building blocks to ensure three-colorability. The randomized algorithm integrates, in a new way, some of the ideas in offline coloring of Wigderson and Blum with ideas of Vishwanathan in the online setting, such as the degree control from Wigderson, the offline-monochromatic class from Blum, and the anchor-sampling from Vishwanathan and the CBIP algorithm by Lov{á}sz, Saks and Trotter.

Authors: Yaqiao Li

We prove new lower and upper bounds for online coloring of three-colorable graphs, improving classical bounds of Vishwanathan [FOCS'1990] that have stood for more than three decades. For the lower bound against deterministic algorithms, we improve Vishwanathan's $Ω(\log^2n)$ to $\exp\!\bigl(Ω((\log n)^{1/3}(\log\log n)^{2/3})\bigr)$. For the upper bound of randomized algorithms, we give a randomized algorithm that achieves $O(\sqrt{n})$ improving Vishwanathan's $O(\sqrt{n \log n})$. Technically, for the lower bound construction, we introduce the language of $t$-fold coloring for simpler description of recursive construction, and the notion of post-degree for future control, and use the flexibility of two-sided colors of bipartite graph, constructed using the independent set amplification method of Vishwanathan, as building blocks to ensure three-colorability. The randomized algorithm integrates, in a new way, some of the ideas in offline coloring of Wigderson and Blum with ideas of Vishwanathan in the online setting, such as the degree control from Wigderson, the offline-monochromatic class from Blum, and the anchor-sampling from Vishwanathan and the CBIP algorithm by Lov{á}sz, Saks and Trotter.

Monday, November 02 2026, 01:00

Improved Approximation Guarantees for Periodic-Review Stochastic Inventory Control

Authors: Danny Segev

In periodic-review stochastic inventory control, replenishment policies must balance holding and backlogging costs while responding to uncertain demands and evolving forecasts that may be arbitrarily correlated and nonstationary. In this general information setting, Levi, P{á}l, Roundy, and Shmoys (Mathematics of Operations Research 2007) established that the fully adaptive optimum can be approximated within factor $2$, which had remained the best-known performance guarantee since then. Their marginal cost accounting and dual-balancing framework made this comparison possible without computing optimal decisions across future trajectories, opening a broad research program on approximation algorithms for stochastic inventory systems. We break this long-standing barrier for periodic-review stochastic inventory control, showing how to compete against the fully adaptive benchmark within factor $\frac{ e }{ e-1 } \approx 1.582$ while retaining general demand correlations and evolving forecasts. We attain these advances by discovering a structural connection between marginal inventory accounting and weighted stopping problems, where each unit's conditional holding charge forms a supermartingale, while postponing its order incurs a history-dependent backlog charge. To leverage the random thresholding approach of Ma and Tzamos (ICML 2023) in this context, we develop a potential-function analysis, leading to an $\frac{ e }{ e-1 }$-approximation with respect to any unit's stopping problem. Subsequently, our common-threshold policy coordinates these marginal decisions into a global replenishment policy without additional loss.

Authors: Danny Segev

In periodic-review stochastic inventory control, replenishment policies must balance holding and backlogging costs while responding to uncertain demands and evolving forecasts that may be arbitrarily correlated and nonstationary. In this general information setting, Levi, P{á}l, Roundy, and Shmoys (Mathematics of Operations Research 2007) established that the fully adaptive optimum can be approximated within factor $2$, which had remained the best-known performance guarantee since then. Their marginal cost accounting and dual-balancing framework made this comparison possible without computing optimal decisions across future trajectories, opening a broad research program on approximation algorithms for stochastic inventory systems. We break this long-standing barrier for periodic-review stochastic inventory control, showing how to compete against the fully adaptive benchmark within factor $\frac{ e }{ e-1 } \approx 1.582$ while retaining general demand correlations and evolving forecasts. We attain these advances by discovering a structural connection between marginal inventory accounting and weighted stopping problems, where each unit's conditional holding charge forms a supermartingale, while postponing its order incurs a history-dependent backlog charge. To leverage the random thresholding approach of Ma and Tzamos (ICML 2023) in this context, we develop a potential-function analysis, leading to an $\frac{ e }{ e-1 }$-approximation with respect to any unit's stopping problem. Subsequently, our common-threshold policy coordinates these marginal decisions into a global replenishment policy without additional loss.

Monday, November 02 2026, 01:00

Random-Sign IBLTs: Calibration-Free Self-Sizing After Decoding Failure

Authors: Min Wu

We propose the random-sign Invertible Bloom Lookup Table (IBLT) for calibration-free self-sizing after decoding failure. In a two-round set reconciliation protocol, the receiver uses the failed first-round table to estimate an upper bound on the unknown difference size $d$ and size the second-round table. In standard IBLTs, failure selection couples the overlap bias with the sign composition of the set difference, making practical sizing depend on offline simulation calibration. A shared pseudorandom sign independent of the position mapping removes this dependence while preserving cell fields, field widths, and ideal peeling recovery. We prove exact unbiasedness on every fixed mapping, a Gaussian limit after failure for $d\gg\sqrt M$, and a discrete limit dominated by duplicate edge pairs at smaller scales, where $M$ is the number of cells. These results yield a Skellam upper-bound rule whose conditional miss rate is asymptotically at most the target risk, uniformly over all $d\ge2$, without offline calibration. More than 500 million paired decoding trials and end-to-end protocol experiments support the design. For small differences, the Gaussian rule's conditional miss rate exceeds the target by more than a factor of 8. The Skellam rule meets the target within sampling error across all 189 tested load configurations. Total two-round communication approaches that of an oracle using the true $d$ in the same protocol. With fixed deployment parameters, precomputed table lookup reduces sizing to less than one microsecond, with total protocol time within 0.5 percent of the Gaussian rule.

Authors: Min Wu

We propose the random-sign Invertible Bloom Lookup Table (IBLT) for calibration-free self-sizing after decoding failure. In a two-round set reconciliation protocol, the receiver uses the failed first-round table to estimate an upper bound on the unknown difference size $d$ and size the second-round table. In standard IBLTs, failure selection couples the overlap bias with the sign composition of the set difference, making practical sizing depend on offline simulation calibration. A shared pseudorandom sign independent of the position mapping removes this dependence while preserving cell fields, field widths, and ideal peeling recovery. We prove exact unbiasedness on every fixed mapping, a Gaussian limit after failure for $d\gg\sqrt M$, and a discrete limit dominated by duplicate edge pairs at smaller scales, where $M$ is the number of cells. These results yield a Skellam upper-bound rule whose conditional miss rate is asymptotically at most the target risk, uniformly over all $d\ge2$, without offline calibration. More than 500 million paired decoding trials and end-to-end protocol experiments support the design. For small differences, the Gaussian rule's conditional miss rate exceeds the target by more than a factor of 8. The Skellam rule meets the target within sampling error across all 189 tested load configurations. Total two-round communication approaches that of an oracle using the true $d$ in the same protocol. With fixed deployment parameters, precomputed table lookup reduces sizing to less than one microsecond, with total protocol time within 0.5 percent of the Gaussian rule.

Monday, November 02 2026, 01:00

Traveling Salesman Problem on Graphs with Proportional Independent Set

Authors: Ethan X. Fang, Shengtao Guo, Junwei Lu, Yuan Yao

We give a deterministic algorithm for minimum-cost Hamiltonian cycle in undirected graphs with arbitrary real edge costs and an independent set of size at least $αn$, for any fixed $3/7<α\le1/2$. It runs in $O^*(γ(α)^n)$ time for an explicit constant $γ(α)<2$; for bipartite graphs, the bound is $O^*((1+2\sqrt2)^{n/2})\approx O^*(1.957^n)$. The algorithm is based on a restricted Held--Karp dynamic program. Its state enumeration uses structural properties of minimum-cost assignments arising from a regular triangulation of a product of simplices.

Authors: Ethan X. Fang, Shengtao Guo, Junwei Lu, Yuan Yao

We give a deterministic algorithm for minimum-cost Hamiltonian cycle in undirected graphs with arbitrary real edge costs and an independent set of size at least $αn$, for any fixed $3/7<α\le1/2$. It runs in $O^*(γ(α)^n)$ time for an explicit constant $γ(α)<2$; for bipartite graphs, the bound is $O^*((1+2\sqrt2)^{n/2})\approx O^*(1.957^n)$. The algorithm is based on a restricted Held--Karp dynamic program. Its state enumeration uses structural properties of minimum-cost assignments arising from a regular triangulation of a product of simplices.

Monday, November 02 2026, 01:00

The Competitive Ratio of Online Budgeted Allocation is Strictly Smaller than $1-1/e$

Authors: Debmalya Panigrahi, Ian Zhang

A well-known conjecture in online algorithms is that the (randomized) competitive ratio of online budgeted allocation is $1-1/e$. We refute this conjecture and show that every randomized online algorithm has competitive ratio at most $1 - 1/e - \varepsilon$ for a (small) absolute constant $\varepsilon > 0$.

Authors: Debmalya Panigrahi, Ian Zhang

A well-known conjecture in online algorithms is that the (randomized) competitive ratio of online budgeted allocation is $1-1/e$. We refute this conjecture and show that every randomized online algorithm has competitive ratio at most $1 - 1/e - \varepsilon$ for a (small) absolute constant $\varepsilon > 0$.

Monday, November 02 2026, 01:00

Omniprediction for All Mixable Proper Losses and Beyond

Authors: Junyan Liu, Haipeng Luo, Lillian J. Ratliff

Omniprediction concerns the task of making predictions that are simultaneously accurate with respect to different losses. Earlier results by [CV09] show that in an online learning setting, $O(\log(|\mathcal{F}||\mathcal{L}|))$ regret is achievable simultaneously against every predictor in a finite predictor class $\mathcal{F}$ and for every loss in a finite class $\mathcal{L}$ of mixable proper losses, while recent advances [OKK25, FP26] show that $\widetilde{O}(\sqrt{T\log(|\mathcal{F}||\mathcal{L}|)})$ regret is achievable simultaneously for every loss in a finite class $\mathcal{L}$ of bounded losses. In this work, we ask whether sublinear regret remains possible when $\mathcal{L}$ is an infinite class of proper losses. Our first result gives a negative answer in general: even when $\mathcal{F}$ contains only two linear predictors, it is impossible to achieve sublinear regret simultaneously for all bounded proper losses. We then show that $\widetilde{O}(T^{1/3}(\log|\mathcal{F}|)^{2/3})$ regret is achievable simultaneously for \textit{all} mixable proper losses and that this bound is optimal up to logarithmic factors. This optimal rate stands in stark contrast to the constant (in $T$) regret achievable for a finite class of mixable proper losses. Finally, we extend our algorithm to obtain a best-of-all-worlds result: in addition to the new regret guarantee for all mixable proper losses, the resulting algorithm simultaneously achieves the existing guarantees for finite loss classes. With appropriate choices of these finite classes, our algorithm simultaneously achieves constant regret for squared loss and log loss, $\widetilde{O}(T^{1/3}(\log|\mathcal{F}|)^{2/3})$ regret for all mixable proper losses, and $\widetilde{O}(\sqrt{T\log|\mathcal{F}|})$ regret for $0$-$1$ loss, all Lipschitz losses, all GLM losses, and other structured families.

Authors: Junyan Liu, Haipeng Luo, Lillian J. Ratliff

Omniprediction concerns the task of making predictions that are simultaneously accurate with respect to different losses. Earlier results by [CV09] show that in an online learning setting, $O(\log(|\mathcal{F}||\mathcal{L}|))$ regret is achievable simultaneously against every predictor in a finite predictor class $\mathcal{F}$ and for every loss in a finite class $\mathcal{L}$ of mixable proper losses, while recent advances [OKK25, FP26] show that $\widetilde{O}(\sqrt{T\log(|\mathcal{F}||\mathcal{L}|)})$ regret is achievable simultaneously for every loss in a finite class $\mathcal{L}$ of bounded losses. In this work, we ask whether sublinear regret remains possible when $\mathcal{L}$ is an infinite class of proper losses. Our first result gives a negative answer in general: even when $\mathcal{F}$ contains only two linear predictors, it is impossible to achieve sublinear regret simultaneously for all bounded proper losses. We then show that $\widetilde{O}(T^{1/3}(\log|\mathcal{F}|)^{2/3})$ regret is achievable simultaneously for \textit{all} mixable proper losses and that this bound is optimal up to logarithmic factors. This optimal rate stands in stark contrast to the constant (in $T$) regret achievable for a finite class of mixable proper losses. Finally, we extend our algorithm to obtain a best-of-all-worlds result: in addition to the new regret guarantee for all mixable proper losses, the resulting algorithm simultaneously achieves the existing guarantees for finite loss classes. With appropriate choices of these finite classes, our algorithm simultaneously achieves constant regret for squared loss and log loss, $\widetilde{O}(T^{1/3}(\log|\mathcal{F}|)^{2/3})$ regret for all mixable proper losses, and $\widetilde{O}(\sqrt{T\log|\mathcal{F}|})$ regret for $0$-$1$ loss, all Lipschitz losses, all GLM losses, and other structured families.

Monday, November 02 2026, 01:00

Hypergraph cycle enumeration for efficient inference of planted k-XOR with smooth trade-off

Authors: Arman Babakhani, Enrico Fontana, Junhyung Lyle Kim, Shouvanik Chakrabarti, Ruslan Shaydulin

We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.

Authors: Arman Babakhani, Enrico Fontana, Junhyung Lyle Kim, Shouvanik Chakrabarti, Ruslan Shaydulin

We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.

Monday, November 02 2026, 01:00

Stochastic Minimum Spanning Tree Identification with Binary Weights

Authors: Haya Diwan, Lisa Hellerstein, Sanjeev Khanna, Junkai Song

We introduce the Stochastic MST Identification Problem, with binary weights. In this problem, we need to identify a minimum spanning tree (MST) of a graph, each of whose edge weights is either 0 or 1. The weight of each edge $e$ is a random variable which is 1 with probability $p_e$ and 0 with probability $1-p_e$. The probability $p_e$ is given for each edge $e$, but the realized weight of an edge can only be determined by querying the edge. The problem is then to determine a strategy for querying the edges of the graph that minimizes the expected number of queries that need to be performed until we can identify a subset of edges that form an MST of the graph. We address both adaptive and non-adaptive versions of this problem. For the non-adaptive version of our Stochastic MST Identification problem, we give an $O(\log E)$-approximation algorithm via a reduction to submodular ranking. For the adaptive version, we show that solving the problem is trivial when all $p_e$ are equal to 1/2, because of a relation to the Tutte polynomial. However, computing the expected cost of an optimal querying strategy is #P-hard, even when all $p_e$ equal $\frac{1}{2}$. For arbitrary $p_e$ values, we present a constant-factor approximation algorithm for simple planar graphs. We investigate the question of whether the submodular goal value approach of Deshpande et al. could be used to achieve a $\log$-factor approximation algorithm that would work for arbitrary $p_e$ values and general graphs. In this context, we address the structural question of determining optimal submodular goal values for MST identification in graphs.

Authors: Haya Diwan, Lisa Hellerstein, Sanjeev Khanna, Junkai Song

We introduce the Stochastic MST Identification Problem, with binary weights. In this problem, we need to identify a minimum spanning tree (MST) of a graph, each of whose edge weights is either 0 or 1. The weight of each edge $e$ is a random variable which is 1 with probability $p_e$ and 0 with probability $1-p_e$. The probability $p_e$ is given for each edge $e$, but the realized weight of an edge can only be determined by querying the edge. The problem is then to determine a strategy for querying the edges of the graph that minimizes the expected number of queries that need to be performed until we can identify a subset of edges that form an MST of the graph. We address both adaptive and non-adaptive versions of this problem. For the non-adaptive version of our Stochastic MST Identification problem, we give an $O(\log E)$-approximation algorithm via a reduction to submodular ranking. For the adaptive version, we show that solving the problem is trivial when all $p_e$ are equal to 1/2, because of a relation to the Tutte polynomial. However, computing the expected cost of an optimal querying strategy is #P-hard, even when all $p_e$ equal $\frac{1}{2}$. For arbitrary $p_e$ values, we present a constant-factor approximation algorithm for simple planar graphs. We investigate the question of whether the submodular goal value approach of Deshpande et al. could be used to achieve a $\log$-factor approximation algorithm that would work for arbitrary $p_e$ values and general graphs. In this context, we address the structural question of determining optimal submodular goal values for MST identification in graphs.

Monday, November 02 2026, 01:00

Tight Lower Bounds for Distributional Paging

Authors: Ali Vakilian

Distributional paging seeks a polynomial-time eviction rule that competes with the optimal \emph{online} policy $\mathrm{OPT}$ when requests come from a known stochastic source. Two algorithms by Lund, Phillips, and Reingold [FOCS, 1994] are canonical. Their randomized \emph{dominating distribution} algorithm, for Markov sources, evicts a page drawn from a distribution over the cache satisfying pairwise linear inequalities; their deterministic \emph{median} algorithm, for sources with independent inter-arrival times across pages, evicts the cached page with the largest lower median of its conditional next-request time. Their original guarantees of $4$ and $5$ were recently improved to $2$ and $4$ by Pabbaraju and Vakilian [ICALP 2025]. We show that neither improved constant can be reduced in its respective source model. For Markov paging, we give two complementary factor-$2$ lower-bound constructions, each with an explicit polynomial-time dominating selector. A directed-cycle family has ratio $2k/(k+1)$, while a recurrent four-page family with cache size three has ratio $2-4\varepsilon/3$. For the independent inter-arrival model, we show that the median algorithm is not $c$-competitive for any $c<4$. These are separate results for distinct stochastic models. Matching the known upper bounds, they establish tight competitive ratios of $2$ and $4$, respectively, improving on previous lower bounds of $1.5907$ and $1.511$.

Authors: Ali Vakilian

Distributional paging seeks a polynomial-time eviction rule that competes with the optimal \emph{online} policy $\mathrm{OPT}$ when requests come from a known stochastic source. Two algorithms by Lund, Phillips, and Reingold [FOCS, 1994] are canonical. Their randomized \emph{dominating distribution} algorithm, for Markov sources, evicts a page drawn from a distribution over the cache satisfying pairwise linear inequalities; their deterministic \emph{median} algorithm, for sources with independent inter-arrival times across pages, evicts the cached page with the largest lower median of its conditional next-request time. Their original guarantees of $4$ and $5$ were recently improved to $2$ and $4$ by Pabbaraju and Vakilian [ICALP 2025]. We show that neither improved constant can be reduced in its respective source model. For Markov paging, we give two complementary factor-$2$ lower-bound constructions, each with an explicit polynomial-time dominating selector. A directed-cycle family has ratio $2k/(k+1)$, while a recurrent four-page family with cache size three has ratio $2-4\varepsilon/3$. For the independent inter-arrival model, we show that the median algorithm is not $c$-competitive for any $c<4$. These are separate results for distinct stochastic models. Matching the known upper bounds, they establish tight competitive ratios of $2$ and $4$, respectively, improving on previous lower bounds of $1.5907$ and $1.511$.

Monday, November 02 2026, 01:00

FastHeapDict: Permanent Slot Indexing for Efficient Decrease-Key Operations in Indexed Priority Queues

Authors: Hojjat T. Goudarzi, Arash Torabi Goodarzi, Mohammad Amir Eshraghi

Efficient decrease-key support remains an important practical concern in priority queues used by shortest-path and graph-processing algorithms. Although duplicate-entry heaps avoid explicit decrease-key operations and are simple to implement, they may accumulate many stale entries when tentative priorities are repeatedly improved. This paper presents FastHeapDict, an indexed binary min-heap based on permanent slot indexing. Instead of repeatedly locating keys through hash-table lookups during heap repair, FastHeapDict assigns each key a stable integer slot and stores that slot directly in heap entries. As a result, position updates during sift-up and sift-down operations are performed through integer indexing while preserving the usual indexed-heap interface. We evaluate FastHeapDict in both Python and C++20 using Dijkstra's algorithm as the primary experimental workload. The benchmarks include constructed dense stress graphs, randomized dense graph variants, and comparisons against duplicate-entry heaps and existing priority-queue alternatives. Across the tested Python workloads, FastHeapDict consistently outperforms the duplicate-entry heapq baseline and the tested heapdict and pqdict implementations. In the C++20 implementation, FastHeapDict also achieves strong constant-factor improvements over std::priority_queue duplicates, std::set-based decrease-key, and std::multiset duplicates. These results indicate that permanent slot indexing can provide a compact and effective implementation strategy for indexed priority queues in workloads where priority updates are frequent.

Authors: Hojjat T. Goudarzi, Arash Torabi Goodarzi, Mohammad Amir Eshraghi

Efficient decrease-key support remains an important practical concern in priority queues used by shortest-path and graph-processing algorithms. Although duplicate-entry heaps avoid explicit decrease-key operations and are simple to implement, they may accumulate many stale entries when tentative priorities are repeatedly improved. This paper presents FastHeapDict, an indexed binary min-heap based on permanent slot indexing. Instead of repeatedly locating keys through hash-table lookups during heap repair, FastHeapDict assigns each key a stable integer slot and stores that slot directly in heap entries. As a result, position updates during sift-up and sift-down operations are performed through integer indexing while preserving the usual indexed-heap interface. We evaluate FastHeapDict in both Python and C++20 using Dijkstra's algorithm as the primary experimental workload. The benchmarks include constructed dense stress graphs, randomized dense graph variants, and comparisons against duplicate-entry heaps and existing priority-queue alternatives. Across the tested Python workloads, FastHeapDict consistently outperforms the duplicate-entry heapq baseline and the tested heapdict and pqdict implementations. In the C++20 implementation, FastHeapDict also achieves strong constant-factor improvements over std::priority_queue duplicates, std::set-based decrease-key, and std::multiset duplicates. These results indicate that permanent slot indexing can provide a compact and effective implementation strategy for indexed priority queues in workloads where priority updates are frequent.

Monday, November 02 2026, 01:00

The STOC Video

from Sophie Huiberts

new videooo

Since 2020, STOC has been asking all authors to contribute a video presentation of their paper. This was originally a response to COVID, but it has generally been fun to get to see talks without having to travel to the conference. The ability to experience part of the conference without the need for travel is great for many reasons, including the climate impact of heavier-than-air flight and the participatory impacts of organizing community events in far-away countries. Even as an in-person participant I have found videos nice, either to watch back later or to watch in advance before attending the IRL talk.

Those who know my work probably know that I also enjoy making these videos. I put in half a week's worth of time to make a nice video, and the extra effort is appreciated by the people. STOC 2026 was no exception on my part. I think this is my best video yet. I am learning how to make pretty animations and graphics, how to use the grammar of YouTube thumbnails, and I added collaborator cameos in there.

Like always: if a paper is worth the effort of reading, then it is a good look to make the video be worth watching. Our paper this year is by far the best paper that I have ever worked on, and likely will hold that title for years to come. It is very good work.

What is different this year is that the STOC organization didn't follow through on tradition. For 2026 they initially told authors that there would be pre-recorded videos, but at the last moment they backed out and promised to record all talks at the conference instead. As of today (4 months after the conference), it appears that videos from Day 1 of the event have been posted. The remainder is nowhere to be seen.

Unwilling to wait any longer, last Monday I posted our paper's video talk to my own YouTube channel. Go watch it, it is good fun. In this post, I will say a couple more things about the video.

Numbers

The nice thing about this video being on my own channel, is that I can elucidate what numbers you get on a video like this. YT's view count is very opaque to the outside observer, especially now that they inflated all measurements earlier this year. So even though it says 2.9K views, we should probably not think of this as an auditorium of 2900 people. What counts as a view requires much lower participation than sitting in a lecture room for 21 minutes.

To better estimate the effective 'audience size' compared to an in-person talk, we can use the watch time statistic instead. At 41 hours watched and a 21:34 video runtime, that makes an 'average audience size' of 41 hours/21:34 = 114 playbacks. In that light, we might interpret this as effectively a talk given to a 114-person audience.

Besides the total number of watch hours, YT also provides a retention that looks like this:

This indicates, for each moment in the video, something like how many viewers watched that bit of the video. But if people slow down the video, or play back a section a second time, then it counts for more time. I think that is why the graph isn't exactly monotonically decreasing.

If we want to count the interested audience members, we may want to count only the people who played back the entire video. Using the retention graph, we can pretend that this represents a drop-out model and do some isotonic regression to figure out the total watchtime accounted for by people who watched the whole video. With the diet problem video as a second comparison, this is the result:

I computed this graph with the retention data of the by-the-book video at 4 days since posting. But by then it already had 38/41 watch hours, so I think the picture wouldn't change much. Anyway, so we can see in the graph that playbacks of 90% or more of the video accounted for 54% of the total watch time. With the 38 watch hours from a few days ago, that makes for a round 57.06 complete playbacks. If we count each complete playback as an audience member, then that is still quite a favorable number.

For comparison we can do a similar calculation on the diet problem video. This video has 137.5 logged watch hours as I write this. With 64% of that being accounted for by people who watch 90% or more, that makes there are around 105 people who watched the entire video. Or at least, 105 full playbacks of the video.

Reach

This is good reach for the first few days. But I think more important is the value of videos like these as a backlog. Almost every paper I wrote has a video link attached on my website, and I believe it is links like these that drive most traffic in the long haul.

Of the 137.5 watch hours on the diet problem video, 31 were attributed to 'thumbnail impressions'. That is, people who see the thumbnail through YouTube recommendations or search. I think that means the remaining 106.5 hours come from direct traffic to the applicable link. If so, that would include social media shares and links on my own website. As we can see from the watch time graph below, there is a large early spike from social media shares. But the vast majority of watchtime lies in the long tail that starts 7 days post-publishing and continues for years after. See this graph with days-since-publication on the horizontal axis and total watch hours on the vertical axis:

Total video watch time in hours, over the course of 690 days. 35.4 hours after 7 days, 105.3 hours after 365 days, 137.5 hours after 690 days.

Interaction

These are good-sized crowds, reached from the comfort of my own home. The total time investment is comparable to traveling for an invited seminar talk. The main thing that lacks somewhat, is after-talk social interaction. I do get emails and messages on social media from viewers sometimes, but not as many as from a normal in-person talk. YouTube comments are mostly rude and unnecessary. But what I do always get, is people I meet in person who tell me they watched my videos. So the videos really are reaching the right people, and I get a chance to have more in-depth conversations in those moments.

Conferences

This is all why I think it is so important for our community to have a tradition of video recordings. It broadens participation, and the benefits of a video backlog are hard to overstate.

I hope that the STOC organization will post their remaining videos sometime soon, and that future conferences (all of which are in MAGA USA next year) will commit to doing videos for every paper presented.

In the meantime, if you have a nice paper but your publication venue doesn't do video: please record a video anyway and submit it elsewhere! In pure math there has been buzz about some new initiative, but they do not consider TCS to be math. Instead, I welcome you to submit to the EATCS channel. We are open to all contributions. Shoot me or Ian Mertz a message for more information.

Monday, November 02 2026, 00:00

TR26-290 | Asymmetry and Complexity of Nondeterministic Computations | Jinqiao Hu, Halley Goldberg, Zhenjian Lu, Jingyi Lyu, Igor Oliveira

from ECCC Papers

Symmetry of Information (SoI) [ZL70] is a fundamental theorem of classical Kolmogorov complexity whose status for time-bounded computation is closely related to central open problems in complexity theory and cryptography. In a recent development, [HMO26] showed that SoI fails for $rKt$ complexity, namely in the setting of randomized computation. By contrast, many basic properties of Kolmogorov complexity are known to remain valid in the presence of nondeterminism [HLO26], suggesting that SoI may provide a fruitful structural principle for exploring the relative power of randomness and nondeterminism in computation. Our results reveal that the validity of SoI for nondeterministic computation, namely for $nKt$ complexity, is a subtle question connected to several frontiers. We show that symmetry of information holds on average for randomized extensions of $nKt$, while its worst case validity would imply breakthrough complexity separations. Although the standard cryptographic argument for asymmetry does not extend to the nondeterministic setting, we provide evidence that information remains asymmetric in nondeterministic computations. Among other results, we show that SoI would imply a collapse of exponential time classes to low levels of the polynomial hierarchy. Then, building on the work of [Hir22b, GK22, KK26, HMO26], we develop a general framework connecting symmetry of information, meta-complexity, and explicit constructions in the nondeterministic setting. More precisely, we prove that suitable formulations of these three notions are quantitatively equivalent. We apply different aspects of this correspondence to make progress on complexity lower bounds and on the failure of SoI for nondeterministic computation. These consequences are unconditional and provide some of the strongest known lower bounds in meta-complexity.

Sunday, November 01 2026, 22:25

3SUM and APSP and ETH, Oh My!

from Computational Complexity

(Throughout this post I will assume that the recent results by OpenAI and Anthropic contain correct proofs of what they claim.)

I wrote the following about 2 years ago but did not post it. It's dated, but that's the point.

BEGIN DOCUMENT FROM TWO YEARS AGO

Why do we think $P\ne NP$?

a) Lots of people have worked on SAT and other NP-complete problems, without getting them in P. This is not mathematical evidence but it is worth noting.

b) If SAT is in P then all problems in NP are in P. This implication is one reason I think $P\ne NP$ is true. I find this more compelling than (a), though it's still not a proof.

c) Just my opinion: I think SAT is not in P.

-------------------------

3SUM is the following problem: Given integers $a_1,\ldots,a_n$ is there a triple of indices $i< j < k $ such that $a_i + a_j + a_k = 0$? There is a clever algorithm for this problem that takes $O(n^2)$ time. It is conjectured that 3SUM has no subquadratic algorithm (that is, for every fixed $\epsilon>0$ there is no $O(n^{2-\epsilon})$ algorithm). (My spellcheck doesn't think subquadratic is a word. Are they right?)

This conjecture is widely believed. There is a notion of 3SUM-hardness and 3SUM-completeness. $ A \le B $ if there is a linear time function $f$ such that $x\in A$ iff $f(x)\in B$. A problem $A$ is 3SUM-complete if both $A\le 3SUM$ and $3SUM \le A$.

There are around 15-20 3SUM-complete problems. Either they are all subquadratic (which includes 3SUM being subquadratic) or none are subquadratic. Given the conjecture, none of them are subquadratic. That is the common belief.

Why do we think 3SUM has no subquadratic algorithm?

a) Lots of people have worked on 3SUM without getting it into subquadratic time. This is not mathematical evidence but it is worth noting.

b) Darn! There is no Cook-Levin-type theorem for 3SUM.

c) Just my opinion: I think 3SUM is not subquadratic. I only wish there were a Cook-Levin-type theorem for this belief. I doubt this is possible.

--------------------------

APSP (All Pairs Shortest Paths) is the following problem: Given a weighted graph $G$, find, for all pairs, the cost of the shortest path. There is an $O(n^3$) algorithm for APSP. It is widely believed that there is no subcubic algorithm for APSP (that is, for every fixed $\epsilon>0$ there is no $O(n^{3-\epsilon})$ algorithm). This conjecture is widely believed. There is a notion of APSP-hardness and APSP-completeness. $ A \le B $ if there is a linear time function $f$ such that $x\in A$ iff $f(x) \in B$. A problem $A$ is APSP-complete if both $A\le APSP$ and $APSP \le A$. (My spellcheck doesn't think subcubic is a word. Are they right?)

There are around 15-20 problems that are APSP-complete. Either they are all subcubic (which includes APSP being subcubic) or none are subcubic. Given the conjecture, none of them are subcubic. That is the common belief.

Why do we think APSP has no subcubic algorithm?

(a) and (b) similar to that for 3SUM. I state (c)

c) Just my opinion: I think APSP is not subcubic. I only wish there were a Cook-Levin-type theorem for this belief. I doubt this is possible.

---------------------------

ETH (Exponential Time Hypothesis) Not only is 3SAT not in P but 3SAT on $n$ variables requires $2^{\Omega(n)}$ time. ETH is widely believed.

Informally, if ETH is true, and $A$ is NP-complete by a linear reduction, then $A$ requires exponential time. There are many such NP-complete problems.

(a) and (b) similar to that for 3SUM. I state (c)

c) Just my opinion: I think 3SAT requires $2^{cn}$ time for some $c$. I only wish there was a Cook-Levin-type theorem for this belief. I doubt this is possible.

END DOCUMENT FROM TWO YEARS AGO

Three of my favorite hardness assumptions had a bad month. I'm glad the Unique Game Conjecture was proven true so I am 1-for-4 rather than 0-for-3 (if it were still open) or 0-for-4 (if it was refuted).

1) Anthropic has obtained an $O(n^{1.9992})$ algorithm for 3SUM. Darling thinks this result is hilarious.

A writeup of this by J. Alman and V.V. Williams is here.

2) Anthropic has obtained an $O(n^{2.9995})$ algorithm for APSP with integer weights (with no negative cycles). Darling, thinks this result is hilarious. (Spell check thinks Antrhopic is a word. Are they right?)

The same writeup in item (1) also contains this result.

3) OpenAI has obtained a $2^{O(n\log\log n/\sqrt{\log n})}$ algorithm for 3SAT Darling thinks this result is hilarious. I agree with her. (My spellcheck things OpenAI is not a word. Are they right?)

OpenAI's announcement is here.

Lance blogged about it here.

So now what?

0) All of these results are being looked at by experts.

1) 1.9992 is close to 2. And we still think 3SUM can't get that much better. could conjecture that, for every fixed $\epsilon$, there is no $O(n^{1.5-\epsilon})$ algorithm for $3SUM$.

Assuming the conjecture, for every fixed $\epsilon$, for every 3SUM-complete problem $A$ there is no $O(n^{1.5-\epsilon})$ algorithm for $A$.

At least until AI gives an $O(n^{1.499999})$ algorithm for 3SUM.

2) Similar for APSP and, say, $O(n^{2.5})$.

3) ETH. Replace with the following: there exists $0<c<1$ such that there is no $O(2^{n^c})$ algorithm for $3SAT$. Or something else like it.

4) We can Regard this as good news. All of those problems that are 3SUM-complete are now known to have subquadratic complexity. Similar for APSP and ETH. Yeah(?)

By gasarch

(Throughout this post I will assume that the recent results by OpenAI and Anthropic contain correct proofs of what they claim.)

I wrote the following about 2 years ago but did not post it. It's dated, but that's the point.

BEGIN DOCUMENT FROM TWO YEARS AGO

Why do we think $P\ne NP$?

a) Lots of people have worked on SAT and other NP-complete problems, without getting them in P. This is not mathematical evidence but it is worth noting.

b) If SAT is in P then all problems in NP are in P. This implication is one reason I think $P\ne NP$ is true. I find this more compelling than (a), though it's still not a proof.

c) Just my opinion: I think SAT is not in P.

-------------------------

3SUM is the following problem: Given integers $a_1,\ldots,a_n$ is there a triple of indices $i< j < k $ such that $a_i + a_j + a_k = 0$? There is a clever algorithm for this problem that takes $O(n^2)$ time. It is conjectured that 3SUM has no subquadratic algorithm (that is, for every fixed $\epsilon>0$ there is no $O(n^{2-\epsilon})$ algorithm). (My spellcheck doesn't think subquadratic is a word. Are they right?)

This conjecture is widely believed. There is a notion of 3SUM-hardness and 3SUM-completeness. $ A \le B $ if there is a linear time function $f$ such that $x\in A$ iff $f(x)\in B$. A problem $A$ is 3SUM-complete if both $A\le 3SUM$ and $3SUM \le A$.

There are around 15-20 3SUM-complete problems. Either they are all subquadratic (which includes 3SUM being subquadratic) or none are subquadratic. Given the conjecture, none of them are subquadratic. That is the common belief.

Why do we think 3SUM has no subquadratic algorithm?

a) Lots of people have worked on 3SUM without getting it into subquadratic time. This is not mathematical evidence but it is worth noting.

b) Darn! There is no Cook-Levin-type theorem for 3SUM.

c) Just my opinion: I think 3SUM is not subquadratic. I only wish there were a Cook-Levin-type theorem for this belief. I doubt this is possible.

--------------------------

APSP (All Pairs Shortest Paths) is the following problem: Given a weighted graph $G$, find, for all pairs, the cost of the shortest path. There is an $O(n^3$) algorithm for APSP. It is widely believed that there is no subcubic algorithm for APSP (that is, for every fixed $\epsilon>0$ there is no $O(n^{3-\epsilon})$ algorithm). This conjecture is widely believed. There is a notion of APSP-hardness and APSP-completeness. $ A \le B $ if there is a linear time function $f$ such that $x\in A$ iff $f(x) \in B$. A problem $A$ is APSP-complete if both $A\le APSP$ and $APSP \le A$. (My spellcheck doesn't think subcubic is a word. Are they right?)

There are around 15-20 problems that are APSP-complete. Either they are all subcubic (which includes APSP being subcubic) or none are subcubic. Given the conjecture, none of them are subcubic. That is the common belief.

Why do we think APSP has no subcubic algorithm?

(a) and (b) similar to that for 3SUM. I state (c)

c) Just my opinion: I think APSP is not subcubic. I only wish there were a Cook-Levin-type theorem for this belief. I doubt this is possible.

---------------------------

ETH (Exponential Time Hypothesis) Not only is 3SAT not in P but 3SAT on $n$ variables requires $2^{\Omega(n)}$ time. ETH is widely believed.

Informally, if ETH is true, and $A$ is NP-complete by a linear reduction, then $A$ requires exponential time. There are many such NP-complete problems.

(a) and (b) similar to that for 3SUM. I state (c)

c) Just my opinion: I think 3SAT requires $2^{cn}$ time for some $c$. I only wish there was a Cook-Levin-type theorem for this belief. I doubt this is possible.

END DOCUMENT FROM TWO YEARS AGO

Three of my favorite hardness assumptions had a bad month. I'm glad the Unique Game Conjecture was proven true so I am 1-for-4 rather than 0-for-3 (if it were still open) or 0-for-4 (if it was refuted).

1) Anthropic has obtained an $O(n^{1.9992})$ algorithm for 3SUM. Darling thinks this result is hilarious.

A writeup of this by J. Alman and V.V. Williams is here.

2) Anthropic has obtained an $O(n^{2.9995})$ algorithm for APSP with integer weights (with no negative cycles). Darling, thinks this result is hilarious. (Spell check thinks Antrhopic is a word. Are they right?)

The same writeup in item (1) also contains this result.

3) OpenAI has obtained a $2^{O(n\log\log n/\sqrt{\log n})}$ algorithm for 3SAT Darling thinks this result is hilarious. I agree with her. (My spellcheck things OpenAI is not a word. Are they right?)

OpenAI's announcement is here.

Lance blogged about it here.

So now what?

0) All of these results are being looked at by experts.

1) 1.9992 is close to 2. And we still think 3SUM can't get that much better. could conjecture that, for every fixed $\epsilon$, there is no $O(n^{1.5-\epsilon})$ algorithm for $3SUM$.

Assuming the conjecture, for every fixed $\epsilon$, for every 3SUM-complete problem $A$ there is no $O(n^{1.5-\epsilon})$ algorithm for $A$.

At least until AI gives an $O(n^{1.499999})$ algorithm for 3SUM.

2) Similar for APSP and, say, $O(n^{2.5})$.

3) ETH. Replace with the following: there exists $0<c<1$ such that there is no $O(2^{n^c})$ algorithm for $3SAT$. Or something else like it.

4) We can Regard this as good news. All of those problems that are 3SUM-complete are now known to have subquadratic complexity. Similar for APSP and ETH. Yeah(?)

By gasarch

Sunday, November 01 2026, 20:23

A Subversive Message of Hope

from Theory Dish: Stanford Blog

In my previous post, I argued that the mathematical community needs to respond actively, continuously and bravely to the changes brought by AI. Here, I would like to take a step back and say a little more about the spirit in which I wrote those words. It is hard to share a sense of optimism without sounding detached from the very real fears people are experiencing, especially when writing from the relative comfort and security of my own position. But I want to do exactly that, even though I have no clear picture of where we are heading and I deeply share many of the fears that others do. I cannot predict how far machine capabilities will go or how quickly they will develop. I do not have a secret list of things that humans, and theoretical computer scientists among them, will always do better. But I find hope in the people around me and the incredibly creative community I am part of. Theoretical computer science has shown remarkable adaptability and relevance through the decades. I believe that we can and will bring that same creativity to shaping our own future. And this belief extends well beyond our field, and [...]

In my previous post, I argued that the mathematical community needs to respond actively, continuously and bravely to the changes brought by AI. Here, I would like to take a step back and say a little more about the spirit in which I wrote those words.

It is hard to share a sense of optimism without sounding detached from the very real fears people are experiencing, especially when writing from the relative comfort and security of my own position. But I want to do exactly that, even though I have no clear picture of where we are heading and I deeply share many of the fears that others do.

I cannot predict how far machine capabilities will go or how quickly they will develop. I do not have a secret list of things that humans, and theoretical computer scientists among them, will always do better. But I find hope in the people around me and the incredibly creative community I am part of. Theoretical computer science has shown remarkable adaptability and relevance through the decades. I believe that we can and will bring that same creativity to shaping our own future. And this belief extends well beyond our field, and well beyond scientific research.

I worry that, out of fear of losing our agency, we are already giving some of it away. When we describe the fate of our field, or of humanity, as a predictable consequence of the next technological breakthrough, we leave our own choices out of the story. We imagine machines changing radically while the social arrangements around them remain fixed and then treat the resulting picture as inevitable.

Our political, social, and economic institutions have changed throughout history, through human choices and struggles. They will change again. We can find new ways to distribute resources and make room for meaningful human existence. Technology changes the circumstances in which we make choices, sometimes dramatically. It does not relieve us of making them.

Part of our commitment should be to ensure that the power to shape our future is widely shared, including with those who currently have the least say in it. At the same time, our ability to find meaning does not depend entirely on how much control we have over our circumstances. The capacity to adapt also belongs to individuals.

In recent months, I have been reminding myself of Viktor Frankl’s Man’s Search for Meaning, and of his account of finding purpose in some of the least humane circumstances imaginable. What stays with me is the resilience of the human capacity to find meaning: in caring for another person, in taking responsibility, in deciding that something remains worth doing even when so much lies beyond our control.

I do not know how we will find our way through the changes ahead. But there may be choices and wonderful possibilities ahead that we have not yet imagined. I find reasons to believe in our ability, as individuals and communities, to discover meaningful ways forward, together.

And this is something we need to keep doing now. We cannot postpone the search for purpose until the future becomes clear. What is worth understanding, creating, or protecting today? What can you find purpose in, even while so much remains uncertain? And how can you support a fellow human in their own struggles?

So, in the spirit of living and struggling in the moment, let me make a sharp transition from Frankl to the film version of Tolkien, and offer Aragorn’s words at the Black Gate of Mordor, with a couple of small adjustments:

My brothers and sisters,

I see in your eyes the same fear that would take the heart of me.

A day may come when our courage fails, when we forsake our friends and break all bonds of fellowship, but it is not this day.

An hour of wolves and shattered shields when the age of humans comes crashing down, but it is not this day!

This day we fight!

By all that you hold dear on this good earth, I bid you stand.

By Omer Reingold

Sunday, November 01 2026, 19:39

TR26-289 | UPP$^{\text{cc}}$ is Not Closed Under Intersection | Mandar Juvekar, Mark Bun

from ECCC Papers

The sign rank of a ${\pm 1}$-valued matrix $A$ is the minimum rank of a real matrix that agrees with $A$ entrywise in sign. For every sufficiently large $d$, we give an explicit matrix $A$ with $2^{\Theta(d^2)}$ rows and columns and sign rank $\Theta(d)$ such that $A \land A$ has sign rank $2^{\Theta(d)}$. Here, $A \land A$ is the communication matrix of the problem obtained by evaluating $A$ on disjoint inputs and taking the conjunction of the results. As the logarithm of sign rank characterizes unbounded-error communication complexity, this implies that the communication class $\mathrm{UPP}^{\mathrm{cc}}$ is not closed under intersection. The starting point for our lower bound is Forster's method, which relates the sign rank of a matrix to the average margin of representations of its rows and columns by points and halfspaces. We study a continuous analog for conjunctions of inner-product sign tests on the unit sphere. Rotational symmetry connects this problem to polynomial sign representations, yielding an exponentially small upper bound on the average margin. This bound holds even when sign constraints are relaxed near the sign tests' decision boundary, allowing us to transfer it to explicit finite sign matrices arising from intersections of integer halfspaces.

Sunday, November 01 2026, 13:24

Podcast about the Topic: Part III, with Brian Greene

from Scott Aaronson

Boooooo! What spooooky times we now live in, with AI rapidly advancing in capabilities and solving more and more major open math problems! So it seems like a perfect occasion to share yet another podcast. I know I’ve been dropping them like every day, but this is a special one, with Brian Greene for his […]

Boooooo! What spooooky times we now live in, with AI rapidly advancing in capabilities and solving more and more major open math problems!

So it seems like a perfect occasion to share yet another podcast. I know I’ve been dropping them like every day, but this is a special one, with Brian Greene for his World Science Festival, recorded in-person when I visited Columbia University a week ago. Note that we don’t talk about OpenAI’s refutation of the Exponential Time Hypothesis because the announcement of that was still two days in the future!

By Scott

Saturday, October 31 2026, 21:40

Linkage for a rainy Halloween

from David Eppstein

Elliot Kienzle illustrates the AI proof of the Mahler conjecture ($\mathbb{M}$) on symplectic embeddings of a ball into the Cartesian product of a convex body and its dual, showing that the product’s volume is always at least as large as the ball.

Elliot Kienzle illustrates the AI proof of the Mahler conjecture ($\mathbb{M}$) on symplectic embeddings of a ball into the Cartesian product of a convex body and its dual, showing that the product’s volume is always at least as large as the ball.
On the generation of names of weevils and popes by L-systems ($\mathbb{M}$).
Marginalia search ($\mathbb{M}$), a web search engine that prioritizes the stuff the big ones don’t: text-heavy, non-commercial pages, personal blogs, old university pages, and the like.
Sudoku but you have to prove your answer before it’s allowed. Which raises the question: prove under what assumptions? Are you allowed to assume uniqueness in your proofs?
Borromean hexagonal tilings, embedded orthogonally in $\mathbb{R}^3$ in such a way that removing one of them allows the other two to be pulled apart.
Lance Fortnow on reading messy proofs ($\mathbb{M}$). He quotes Noam Nisan as saying that “reading a badly written proof leads to better understanding since you really need to figure out the details”.
Mathematics Genealogy Project temporarily offline ($\mathbb{M}$) because of the recently-discovered security vulnerabilities in PHP. Unfortunately their software was incompatible with recent versions so it wasn’t a matter of a simple upgrade. They’re asking for donations to cover both a backend overhaul (estimated to cost more than half their typical annual budget) and catching up on their severe backlog of updates (almost a year deep).
The Perles configuration of nine points and nine lines ($\mathbb{M}$) is based on a regular pentagon/pentagram; it cannot be realized by points with rational-number Cartesian coordinates, even in less-symmetrical arrangements of its points and lines. But there are stronger ways to be difficult to realize than irrationality. We can’t hope for transcendentality: when a configuration can be realized it can always be realized using algebraic number coordinates. An intermediate step between these two is unconstructibility: can it be drawn with compass and straightedge? Because the Perles configuration is based on a regular pentagon, it is obviously constructible.

The Grünbaum–Rigby configuration is based in a similar way on regular heptagons/heptagrams. Because these cannot be constructed with compass and straightedge, neither can the symmetric form of the Grünbaum–Rigby configuration. I suspect the same is true for any asymmetric forms it might have but I have no proof. (I did try asking a couple of advanced-model LLMs; one told me it had no idea and another told me it was unconstructible but retreated when pressed for details.) One property of the Perles configuration that helps prove its irrationality is projective uniqueness: any two realizations are related by a projective transformation, together with a permutation of the points and lines of the configuration. If this were true of the Grünbaum–Rigby configuration it would prove unconstructibility but I don’t know whether it’s true.
3d-printed tiles from the aperiodic chair tiling snap together like Lego ($\mathbb{M}$) in this short (2:18) video by Bob Mackay explaining its recursive structure.
LLMs still can’t count. A story about trying to count the teeth on a bicycle sprocket wheel, getting an answer that isn’t a valid tooth count, asking the LLM to label the teeth on a photo, and getting back a replacement photo with more teeth that still aren’t even counted correctly.
OEIS A400K contest winners ($\mathbb{M}$). The top entry: A400000, the “city-building sequence” in which $a(n)$ is the height $h$ of the shortest building which can see exactly $h$ earlier buildings of heights $a(i)$, in an unobstructed line of site from its roof.
I updated the talk slides for my talk “Computational complexities of folding” ($\mathbb{M}$), on connections between computational complexity theory and origami, for a presentation to a local student group. The updates included a last-minute change to a slide on the exponential time hypothesis to take into account the OpenAI claimed disproof, but their result doesn’t really change things for what I was using it for: if the exponential part of a time bound has an exponent that is linear in a certain parameter (as pre-disproof ETH would imply), or near-linear in it (consistent with the disproof), it’s still a big time bound. Since I originally presented this material in 2024 I’ve published a journal version detailing much of the talk; see also my publication metadata page.
The lattice of stable matchings ($\mathbb{M}$) organizes the set of all solutions of a stable matching problem into a structure in which each two matchings have a least upper bound (giving all the participants on one side their preference among the two partners from the two matchings) and a greatest lower bound (the same for the participants on the other side). Its uses involve algorithms for finding weight-maximizing or regret-minimizing stable matchings. A little technical still, but now a Good Article on Wikipedia.
Why Richard Elwes loves Stella ($\mathbb{M}$), software for constructing and visualizing polyhedra.
How inappropriate it is for referees/editors to use AI to help review any original work. A computer can never be held accountable for breaching the confidentiality of submissions; therefore, a computer must never be given access to confidential submissions.

By David Eppstein

Saturday, October 31 2026, 18:15

Talk by Dana Moshkovitz: The Unique Games Conjecture – Proved (!!?)

from Scott Aaronson

In my “Mathocalypse” post, I mentioned that my wife Dana—who’s been working on Subhash Khot’s Unique Games Conjecture since ~2009, for the entire time I’ve known her—has for the past few weeks (while on sabbatical at the Simons Institute in Berkeley) been spending essentially all of her time laboring to understand OpenAI’s recent proof of […]

In my “Mathocalypse” post, I mentioned that my wife Dana—who’s been working on Subhash Khot’s Unique Games Conjecture since ~2009, for the entire time I’ve known her—has for the past few weeks (while on sabbatical at the Simons Institute in Berkeley) been spending essentially all of her time laboring to understand OpenAI’s recent proof of the conjecture. Having now studied the proof closely in collaboration with Subhash Khot himself, Johan Håstad, Dor Minzer, and other colleagues, and (naturally) GPT-6 Astra, Dana probably now understands this crown jewel of OpenAI’s release as well as anyone does. This week, she gave two well-received whiteboard talks on the proof at Simons Institute. So, I and others encouraged her to give the same talks next week at UT Austin when she’s back here, and Dana agreed! Because of the large timely interest, we’re going to do a live Zoom option, and will also record the talk for later sharing on YouTube. Please note: While all are welcome, this is a 2-hour technical talk. The first hour will review key ideas from hardness of approximation, while the second hour will describe the new construction. —SA

Title: The Unique Games Conjecture – Proved (!!?)

Speaker: Dana Moshkovitz (UT Austin)

When: Wednesday Nov. 4, 3:30-5:30pm US Central (with a brief break in the middle)

Where: UT Austin, Gates Dell Complex 6.302. Live Zoom will be available here, and a recording will be available later.

Abstract: The Unique Games Conjecture of Khot says that it is NP-hard, given a system of linear equations of the form x_i-x_j=b_ij (mod p) over variables x₁…x_n, where 99% fraction of the equations can be satisfied, to find an assignment satisfying 1% of the equations (here p is a sufficiently large constant).

This conjecture implies optimal inapproximability results for many optimization problems.

A few weeks ago OpenAI released a writeup with a proof of the Unique Games Conjecture, as well as a Lean formalization of the proof. The writeup, like others released that day, is not written in a way that’s especially readable to human mathematicians.

My goal in this talk is to offer my mental map of the proof, in a way that’s helpful to theoretical computer scientists who are familiar with the area. I think of the innovation as a new code construction, “Hadamard with a noise test.” I’ll sketch how it fits into the NP-hardness reduction. Then I’ll describe the code construction fully and sketch its soundness analysis.

I’ll also point out a couple of ingredients of the proof that are of independent interest, like a simple quadratic that behaves “like a linear,” and a probability bound using a trick with harmonic numbers.

By Scott

Friday, October 30 2026, 23:27

TR26-288 | Polynomial Interpolation for Sherali-Adams Lower Bounds | Nathan Acheampong, Ben Davis, Robert Robere

from ECCC Papers

We resolve two open problems regarding the Sherali-Adams proof system. First, we show that Res(2), which manipulates 2-DNF formulas, is exponentially stronger than Sherali-Adams. This provides a sharp separation in power between Sherali-Adams and the bounded-depth Frege hierarchy, as Sherali-Adams polynomially simulates Resolution. Second, we prove that Sherali-Adams is not size-automatable unless P = NP. This resolves an open problem on the automatability of weak proof systems asked by many researchers. Both of these results are obtained by using a new technique for proving lower bounds on Sherali-Adams degree via polynomial interpolation.

Friday, October 30 2026, 20:16

Galactus the Devourer

from Ben Recht

On efficient algorithms, the exponential time hypothesis, and losing the plot.

Hi there, argmin readers! Today’s post is about the history of failing to rigorously define efficiency in computer science. Like my previous post on Fourier transforms, it’s more technical than usual, but I tried to make it as accessible as possible.

I’ve always been fascinated by this pernicious pattern in the practice of applied mathematics, which I wrote about here a few weeks ago:

Applied mathematicians often need to build a giant scaffolding of mathematical modeling to solve a problem. This scaffolding creates new mathematical puzzles that aren’t directly connected to the original problem of interest, but that entice problem solvers. You’ll then see dozens of follow-up papers solving the puzzles but forgetting the problem we cared about in the first place.

There is probably no better example of such forgetting than the field of complexity theory.

In 1965, Jack Edmonds came up with a clever way to find a maximum matching in a graph. He wanted to argue that his algorithm could be computed “efficiently,” as the procedure was a bit complicated and required a computer to run. But he found he didn’t have the language to describe what it would mean for an algorithm to be efficient. In Section 2 of his paper, named “Digression,” he wrote,

According to the dictionary, “efficient” means “adequate in operation or performance.” This is roughly the meaning I want—in the sense that it is conceivable for maximum matching to have no efficient algorithm. Perhaps a better word is “good.”

I am claiming, as a mathematical result, the existence of a good algorithm.

Edmonds noted that he could enumerate every possible solution to the matching problem and find the largest one. This would be a correct algorithm. However, the number of total matchings is exponentially large in the size of the graph. Blindly writing all possible matchings down and picking the one that achieved the maximum was a “not good” algorithm. You might call it a “bad” algorithm. His new procedure was vastly more efficient.

So Edmonds spent two pages constructing a definition of good. He demanded that everyone agree on a reasonable definition for a problem size. For his matching problem, he said the size was the number of vertices plus the number of edges in the graph. A good algorithm would then have a running time upper-bounded by some well-defined function of that size, no matter how large the instance. He demanded that the running-time bound hold for every conceivable size, because he feared someone might rig the problem by planting a few small instances that were unnaturally hard.

Edmonds wasn’t completely satisfied with this definition. Indeed, he wrote:

It would be unfortunate for any rigid criterion to inhibit the practical development of algorithms which are either not known or known not to conform nicely to the criterion. Many of the best algorithmic ideas known today would suffer by such theoretical pedantry. In fact, an outstanding open question is, essentially: “how good” is a particular algorithm for linear programming, the simplex method? And, on the other hand, many important algorithmic ideas in electrical switching theory are obviously not “good” in our sense.1

With this caveat noted, he concluded his digression by announcing the complexity of his new blossom algorithm: “An upper bound on the order of difficulty of the matching algorithm is n⁴, where n is the number of vertices in the graph.”

Edmonds did not make any further arguments for what he meant by good. He asserted that exponential time is bad. For most algorithm designers, linear time, proportional to the input length, was naturally excellent. Moreover, our grade-school multiplication algorithm took quadratic time (multiply two three-digit numbers out by hand like you were taught in school and count the number of digits you write down when you calculate it), and that seemed pretty good. Edmonds had a matching algorithm that required a quartic number of operations. To later formalists like Richard Karp, it seemed like a “polynomial” number of operations was a reasonable definition for goodness. Polynomial time is good. Exponential time is bad. This created the field of complexity theory, with the goal of proving that all sorts of algorithms run in polynomial time and characterizing the problems for which no polynomial-time algorithm existed.

But polynomial time probably haunted Edmonds. It is not a “good” algorithm if your procedure scales like n¹⁰⁰⁰, even if this is technically a polynomial. More absurdly,

is a polynomial bigger than the exponential function 2ⁿfor any n we could ever encounter. Polynomial was always an imperfect definition of good, but we ran with it. Thousands of papers later, we just have a zoo of confusing complexity results unmoored from the realities of computation.

Indeed, asymptotic thinking about how polynomials differ from exponentials enables the bad habit of designing “galactic algorithms,” conceptual procedures with running times so large they could never be implemented in our universe. I mentioned these in my earlier post about OpenAI’s Faster Fourier Transform.

This brings me back to our beneficent friends at OpenAI. This week they declared their latest breakthrough in theoretical computer science. They provided an algorithm for the satisfiability problem that ran in “subexponential time.” Satisfiability is one of the most well-known “NP complete problems, and I don’t want to go into the details of P vs NP today, but suffice it to say that you would win a Clay Millennium Prize if you could find a polynomial-time algorithm for satisfiability.2

Now, subexponential time does not mean polynomial time. It merely means that the running time can be bounded as

Where n is the problem size, and f is a “sublinear” function, meaning the limit of f(n)/n goes to zero as n goes to infinity.

Indeed, in the case of OpenAI’s new algorithm for the satisfiability problem, the bound was

For some unspecified constant numbers d and C. Ugly as this expression is, this was touted on Twitter as the biggest moment in theoretical computer science in 50 years. It was covered all over the tech press this week. OpenAI clearly wanted us to think it was a big fucking deal. And many theoretical computer scientists took to social media to agree.

Since it was such a big deal, I wanted to see if I could understand the algorithm a bit. Of course, as I’ve told all of you, we’re not supposed to read these papers. We’re supposed to ask ChatGPT what they say. And ChatGPT told me there was a catch.

The derived algorithm runs exhaustive search—the algorithm Edmonds declared bad in 1965—if the number of variables is less than 10^10,000,000,000,000,000,000,000,000.

This is a completely unfathomable number.3 There are something like 10^80 atoms in the universe. That’s 10^9,999,999,999,999,999,999,999,920 times smaller. Up until this barely finite threshold, the magic new algorithm calls an exponential-time subroutine. For any problem that we could ever consider, the procedure is exponential time. The “result” is that only in some bizarre, unknowable neighborhood of infinity, satisfiability problems are solved in subexponential time.

The claimed subexponential algorithm for satisfiability is definitive evidence that our definition of efficient was even worse than we thought it was 50 years ago. The answer tells us nothing about efficiency or goodness. Sometimes you need to answer your question to realize you wanted something else.

I’m not a betting man, but I have a prediction. It’s going to turn out that P=NP in some galactic sense, and all we’ll learn is that we spent fifty years answering the wrong question.

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1

It’s funny that Edmonds references the simplex algorithm in his musings here. The simplex algorithm is the most important algorithm ever invented, and we now know that, technically speaking, its running time is exponential. Most of the time it’s fast, but you can construct weird cases where it takes an exponential number of steps. And yet, it is deployed all over the globe, determining optimal programs for exponentially diverse applications in optimization. Edmonds foresaw this sharp disconnect between theory and practice in 1965!

2

What’s particularly ironic is that we have incredibly efficient algorithms to solve industrial-scale satisfiability problems. Conventional computers can solve problems with millions of variables and millions of clauses. This guy at Intel solved a problem with a billion variables and 4 billion clauses in a day.

3

I confirmed this crazy number by plugging the paper into Claude. ChatGPT and Claude hate each other, so if they agree, it must be true.

By Ben Recht

Friday, October 30 2026, 14:34

What Can Be Computed with Indefinite Causal Structure?

Authors: Omri Shmueli

Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit. We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in $\textbf{NP}$. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.

Authors: Omri Shmueli

Can quantum computation become more powerful when operations do not need to follow a definite causal order? We study this question in the process matrix formalism of Oreshkov et al. (Nature Communications, 2012), which allows quantum operations to be connected without a fixed global order. In the computational model, a quantum circuit succinctly describes a process matrix, and the algorithm is given access to its generated channel, with computational cost measured by the size of the generating circuit. We give polynomial-time algorithms in two settings. First, general process matrix computation suffices to solve every problem in $\textbf{NP}$. The valid processes used by our algorithm satisfy the purification postulate of Araújo et al. (Quantum, 2017), and their generated channels are unitary, although their generating circuits are not. Second, requiring the process matrices themselves to be pure, we give a polynomial-time algorithm for unique-sink orientation, a general problem capturing several longstanding computational problems in optimization. Our unitary generators are constructed explicitly and efficiently. Assuming that unique-sink orientation has no quantum polynomial-time algorithm, our result shows that indefinite causal order can provide computational power beyond $\textbf{BQP}$ within the process matrix formalism, even while preserving unitarity.

Friday, October 30 2026, 00:00

On the Quantum Chromatic Gap II

Authors: Lorenzo Ciardo, Antoine Mottet

We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's $d$-to-$1$ games: for every $d\geq 4$, every $ε>0$, and every $k\in\mathbb N$, there is a $d$-to-$1$ label-cover game of classical value less than $ε$ that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least $k$. As a consequence, there exist graphs whose quantum chromatic number $χ_q$ is $3$ and whose classical chromatic number $χ$ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which $χ$ grows exponentially in $χ_q$. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to $4$-to-$1$ games, we establish quantum completeness of this reduction, and we show that it preserves context radius.

Authors: Lorenzo Ciardo, Antoine Mottet

We confirm a conjecture of the first author [SODA'26] on a strong form of pseudo-telepathy for Khot's $d$-to-$1$ games: for every $d\geq 4$, every $ε>0$, and every $k\in\mathbb N$, there is a $d$-to-$1$ label-cover game of classical value less than $ε$ that admits a perfect quantum strategy using finite-dimensional shared entanglement with measurement contexts of radius at least $k$. As a consequence, there exist graphs whose quantum chromatic number $χ_q$ is $3$ and whose classical chromatic number $χ$ is arbitrarily large. This is the first improvement on the quantum chromatic gap since the work of Brassard, Cleve, and Tapp [Phys. Rev. Lett.'99], which produced graphs for which $χ$ grows exponentially in $χ_q$. To prove the conjecture, we proceed as follows. We start with a ``stretched'' version of the Mermin--Peres magic square, obtained by spacing out its equations and linking them with additional long equation wires. This operation increases the radius of the initial measurement contexts of a perfect quantum strategy for the game, while preserving classical unsatisfiability. We then apply parallel repetition to make the classical value small, and we turn the system into a pseudo-telepathic multi-layered Kneser hypergraph via a PCP construction of Dinur, Guruswami, Khot, and Regev [STOC'03]. Finally, we feed the resulting hypergraph into the reduction of Fei, Minzer, and Wang from hypergraph colouring to $4$-to-$1$ games, we establish quantum completeness of this reduction, and we show that it preserves context radius.

Friday, October 30 2026, 00:00

Directed Cut-Query Complexity of Finding a King

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

A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.

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

A king in a tournament is a vertex from which every other vertex is reachable by a directed path of length at most two. We study the deterministic query complexity of finding a king when a query specifies a set of vertices and returns the number of edges directed from that set to its complement. We prove that this complexity is $Θ(n)$ for $n$-vertex tournaments. To show this, we first show that every correct deterministic algorithm in this model must output a maximum-outdegree vertex, even though a king need not have maximum outdegree. We then give an adversary whose answers, after too few queries, remain consistent with a tournament in which the algorithm's output does not have maximum outdegree.

Friday, October 30 2026, 00:00

Cubic lower bounds for the permanent in odd characteristic, for Pfaffian representations, and for polynomial entries

Authors: Karthik Sheshadri

The determinantal complexity of a polynomial is the order of the smallest matrix whose entries are polynomials of degree at most one and whose determinant equals the polynomial. A lower bound for the permanent in this measure is the algebraic form of the question whether the permanent is harder to compute than the determinant. Over the complex numbers, the best known lower bound grows cubically in the size of the permanent. It holds even when the permanent only has to be a limit of such determinants. I prove that the cubic lower bound holds over every algebraically closed field of odd characteristic $p$, with a constant that depends on $p$. Hence the cubic bound does not depend on characteristic zero. I also prove the bound for Pfaffians, which are square roots of determinants of skew symmetric matrices. Lower bounds for Pfaffians imply the same bounds for determinants. Further, I prove the bound, divided by $a$, for matrices whose entries are polynomials of degree at most $a$. All three results come from one lower bound for smooth hypersurfaces. I prove it in every characteristic by counting intersection points together with their multiplicities.

Authors: Karthik Sheshadri

The determinantal complexity of a polynomial is the order of the smallest matrix whose entries are polynomials of degree at most one and whose determinant equals the polynomial. A lower bound for the permanent in this measure is the algebraic form of the question whether the permanent is harder to compute than the determinant. Over the complex numbers, the best known lower bound grows cubically in the size of the permanent. It holds even when the permanent only has to be a limit of such determinants. I prove that the cubic lower bound holds over every algebraically closed field of odd characteristic $p$, with a constant that depends on $p$. Hence the cubic bound does not depend on characteristic zero. I also prove the bound for Pfaffians, which are square roots of determinants of skew symmetric matrices. Lower bounds for Pfaffians imply the same bounds for determinants. Further, I prove the bound, divided by $a$, for matrices whose entries are polynomials of degree at most $a$. All three results come from one lower bound for smooth hypersurfaces. I prove it in every characteristic by counting intersection points together with their multiplicities.

Friday, October 30 2026, 00:00

Random ($\log{n}$)-CNF Formulas Are Hard for Depth-$2$ Frege with Modular Gates

Authors: Jonas Conneryd

We prove that, with high probability, refuting a random $k$-CNF formula on $n$ variables with ${M = n2^{k}}$ clauses for $ω_n(1) \leq k \leq O(\log{n})$ requires polynomial calculus degree at least ${n/2-O(n/\sqrt{k})}$ over every field. Through Pang's recent tradeoff between $\mathsf{AC}^0[p]$-Frege size and polynomial calculus degree [Pang 2026, arXiv:2610.00837v2], this implies a size lower bound of $\exp(Ω_p(\sqrt{k} \log{n}))$ for depth-$2$ $\mathsf{AC}^0[p]$-Frege. In this range of $k$, the bound is superpolynomial in $M$ while $M \leq \mathrm{poly}(n)$. We also prove a nearly matching degree upper bound of $n/2-n\ln(k)/{2k} +O(n/k)$, which shows that other formula families are needed to obtain lower bounds for full $\mathsf{AC}^0[p]$-Frege through Pang's tradeoff.

Authors: Jonas Conneryd

We prove that, with high probability, refuting a random $k$-CNF formula on $n$ variables with ${M = n2^{k}}$ clauses for $ω_n(1) \leq k \leq O(\log{n})$ requires polynomial calculus degree at least ${n/2-O(n/\sqrt{k})}$ over every field. Through Pang's recent tradeoff between $\mathsf{AC}^0[p]$-Frege size and polynomial calculus degree [Pang 2026, arXiv:2610.00837v2], this implies a size lower bound of $\exp(Ω_p(\sqrt{k} \log{n}))$ for depth-$2$ $\mathsf{AC}^0[p]$-Frege. In this range of $k$, the bound is superpolynomial in $M$ while $M \leq \mathrm{poly}(n)$. We also prove a nearly matching degree upper bound of $n/2-n\ln(k)/{2k} +O(n/k)$, which shows that other formula families are needed to obtain lower bounds for full $\mathsf{AC}^0[p]$-Frege through Pang's tradeoff.

Friday, October 30 2026, 00:00

Diagonalization and Undecidability in Semi-Thue Systems

Authors: Vesa Halava, Reino Niskanen

We revisit classical diagonalization argument in the setting of semi-Thue systems, viewing computation as nondeterministic rewriting rather than sequential machine execution. Semi-Thue systems naturally generate derivation trees, making existential acceptance, reachability, and infinite derivations fundamental computational notions. We formulate acceptance and decision directly in terms of derivation trees and study the associated assertion, word, and infiniteness problems. We prove that the complements of the assertion and word problems are not recursively enumerable by direct Post-style diagonal arguments. For the infiniteness problem, we give a constructive diagonalization based on a fixed universal semi-Thue system. Unlike the classical approach, the diagonal construction is carried out entirely within the rewriting formalism: the universal system remains an ordinary nondeterministic semi-Thue system while simulating encoded semi-Thue computations, including self-reference. These results show that undecidability in semi-Thue systems arises naturally from universality and self-reference in nondeterministic rewriting, rather than from machine states or control mechanisms. This perspective clarifies the relationship between rewriting systems, recursive enumerability, and classical diagonal arguments originating in the work of Post and Turing.

Authors: Vesa Halava, Reino Niskanen

We revisit classical diagonalization argument in the setting of semi-Thue systems, viewing computation as nondeterministic rewriting rather than sequential machine execution. Semi-Thue systems naturally generate derivation trees, making existential acceptance, reachability, and infinite derivations fundamental computational notions. We formulate acceptance and decision directly in terms of derivation trees and study the associated assertion, word, and infiniteness problems. We prove that the complements of the assertion and word problems are not recursively enumerable by direct Post-style diagonal arguments. For the infiniteness problem, we give a constructive diagonalization based on a fixed universal semi-Thue system. Unlike the classical approach, the diagonal construction is carried out entirely within the rewriting formalism: the universal system remains an ordinary nondeterministic semi-Thue system while simulating encoded semi-Thue computations, including self-reference. These results show that undecidability in semi-Thue systems arises naturally from universality and self-reference in nondeterministic rewriting, rather than from machine states or control mechanisms. This perspective clarifies the relationship between rewriting systems, recursive enumerability, and classical diagonal arguments originating in the work of Post and Turing.

Friday, October 30 2026, 00:00

Reachability in Five-Dimensional Vector Addition Systems Is PSPACE-Complete

Authors: Weijun Chen, Bo Fu, Yuxi Fu

Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.

Authors: Weijun Chen, Bo Fu, Yuxi Fu

Vector addition systems (VAS) are a fundamental model of concurrent computation in which transitions update nonnegative integer counters by adding integer vectors, provided all counters remain nonnegative. Closely related to Petri nets, VAS are the single-state counterpart of vector addition systems with states (VASS), which use finitely many states to restrict the available transitions. A classical simulation of $d$-VASS by $(d+3)$-VAS connects their reachability problems, but the complexity-theoretic picture of fixed-dimensional VAS remains largely unclear. While reachability in $2$-VASS is PSPACE-complete under binary encoding, the best known upper bound for $5$-VAS was only elementary. We prove that every reachable $5$-VAS instance has a reachability witness of length polynomial in its unary size. Consequently, $5$-VAS reachability is NL-complete under unary encoding and PSPACE-complete under binary encoding. We also establish a polynomial factorization of the reachability relation of every $d$-VAS with $d \ge 3$. For $5$-VAS, this strengthens Hopcroft and Pansiot's effective semilinearity of reachability sets to the full reachability relation. For every fixed $d\ge3$, it also yields an $\mathsf F_d$ upper bound for reachability in $(d+3)$-VAS.

Friday, October 30 2026, 00:00

Quantum Query Complexity of Tournament Solutions

Authors: Pragya Arora, Udit Kumar

A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among $N$ alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require $Θ(N^2)$ queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with $\tildeΘ(N)$ queries, and a Condorcet winner can be found with $\tilde Θ(\sqrt N)$ queries. Finding a strong king requires $Θ(N^{3/2})$ queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known $Θ(N^{3/2})$ bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.

Authors: Pragya Arora, Udit Kumar

A tournament solution records a set of winners from a tournament, that is, from the outcomes of all pairwise comparisons among $N$ alternatives. We study the quantum query complexity of computing tournament solutions when the outcome of each comparison has to be learned by a query. Classically, the picture is uniform: the Copeland set, the uncovered set and the top cycle all require $Θ(N^2)$ queries, and so does finding a single vertex of maximum out-degree. We show that the quantum query complexity of the top cycle can be computed with $\tildeΘ(N)$ queries, and a Condorcet winner can be found with $\tilde Θ(\sqrt N)$ queries. Finding a strong king requires $Θ(N^{3/2})$ queries, the same as finding a vertex of maximum out-degree. We also give a self-contained alternative proof of the known $Θ(N^{3/2})$ bound for finding a vertex of maximum degree, established by Gilani, Mande, Patro, and Raja. Our lower bounds reduce counting problems to tournament problems using Ambainis's adversary method. For the top cycle, we find a Hamiltonian path and locate the prefix forming the top cycle by divide and conquer.

Friday, October 30 2026, 00:00

Mathematics and Efficiency of Song's CVP Hardness Reduction

Authors: Shuhong Gao

Song (2026) proves that, for every fixed $0<ε<1$, the closest codeword problem (CCP) for binary linear codes under Hamming distance is NP-hard to approximate within $N^{1-ε}$, where $N$ is the code length, under deterministic polynomial-time many-one reductions. The code-to-lattice construction gives hardness within $N^{1/2-ε/2}$ for Euclidean CVP, with the same $N$ aslattice rank. This reaches every fixed Euclidean hardness exponent below one half. Aharonov and Regev (2005), with the constant refined by Gao (2026), place GapCVP at factor $cN^{1/2}$ in $\mathrm{NP}\cap\mathrm{coNP}$ for every fixed $c>2/(π\sqrt3)\approx0.3676$. NP-hardness at that scale would imply $\mathrm{NP}=\mathrm{coNP}$. This paper has two goals: to explain Song's theorem and its mathematics for graduate students, and to make the reduction more efficient. The reduction turns satisfiability into a system of quadratic equations and then into CCP for binary linear codes, and recovers assignments from recurrence relations of algebraic moments. We adapt the method to linear codes over any fixed finite field of characteristic $p$. A common recurrence for each block column gives a moment window $L=c_p(κ-1)+1$, where $c_p=\max\{2,p-1\}$ and $κ$ is the rank cutoff. Together with ordinary digit products and flexible sampling, this reduces interpolation degrees and output lengths. For CCP gap $N^{1-ε}$, our explicit schedule gives code length and lattice rank at most $n^{4/ε^2+O(1/ε)}$; Song's prescribed binary schedule gives $n^{8/ε^2+O(1/ε)}$. Here $n$ counts variables in a common quadratic system, equal to the number of original variables plus clauses for 3SAT. These are dimension bounds, not optimality claims or running-time exponents. The limiting hardness exponents remain Song's.

Authors: Shuhong Gao

Song (2026) proves that, for every fixed $0<ε<1$, the closest codeword problem (CCP) for binary linear codes under Hamming distance is NP-hard to approximate within $N^{1-ε}$, where $N$ is the code length, under deterministic polynomial-time many-one reductions. The code-to-lattice construction gives hardness within $N^{1/2-ε/2}$ for Euclidean CVP, with the same $N$ aslattice rank. This reaches every fixed Euclidean hardness exponent below one half. Aharonov and Regev (2005), with the constant refined by Gao (2026), place GapCVP at factor $cN^{1/2}$ in $\mathrm{NP}\cap\mathrm{coNP}$ for every fixed $c>2/(π\sqrt3)\approx0.3676$. NP-hardness at that scale would imply $\mathrm{NP}=\mathrm{coNP}$. This paper has two goals: to explain Song's theorem and its mathematics for graduate students, and to make the reduction more efficient. The reduction turns satisfiability into a system of quadratic equations and then into CCP for binary linear codes, and recovers assignments from recurrence relations of algebraic moments. We adapt the method to linear codes over any fixed finite field of characteristic $p$. A common recurrence for each block column gives a moment window $L=c_p(κ-1)+1$, where $c_p=\max\{2,p-1\}$ and $κ$ is the rank cutoff. Together with ordinary digit products and flexible sampling, this reduces interpolation degrees and output lengths. For CCP gap $N^{1-ε}$, our explicit schedule gives code length and lattice rank at most $n^{4/ε^2+O(1/ε)}$; Song's prescribed binary schedule gives $n^{8/ε^2+O(1/ε)}$. Here $n$ counts variables in a common quadratic system, equal to the number of original variables plus clauses for 3SAT. These are dimension bounds, not optimality claims or running-time exponents. The limiting hardness exponents remain Song's.

Friday, October 30 2026, 00:00

How Hard Is It to Compute Cover Time?

Authors: Ryan Aghazadeh

It is $\#\mathrm{P}\text{-hard}$. Even for connected simple undirected unweighted graphs, deciding whether an expected random-walk cover time is at most a given rational threshold is $\#\mathrm{P}\text{-hard}$ under polynomial-time Turing reductions, both from a specified starting vertex and when maximized over starting vertices. Both families have a universal vertex, called the hub, and diameter two. Deleting the hub leaves a planar bipartite graph, and every non-hub vertex has degree at most five. A shared construction encodes an Euler-tour count in rare covering walks. For specified starts, a number of fractional bits linear in the queried graph's vertex count suffices for recovery. For the maximum-start problem, a fixed cube attachment makes the maximizing starts known, and one interpolation of expected cover times recovers the count. Both reductions use polynomial-size graphs and polynomial-bit threshold queries. For general graphs, both comparison problems can be solved in polynomial time with access to an $\mathrm{NP}^{\mathrm{PP}}$ oracle.

Authors: Ryan Aghazadeh

It is $\#\mathrm{P}\text{-hard}$. Even for connected simple undirected unweighted graphs, deciding whether an expected random-walk cover time is at most a given rational threshold is $\#\mathrm{P}\text{-hard}$ under polynomial-time Turing reductions, both from a specified starting vertex and when maximized over starting vertices. Both families have a universal vertex, called the hub, and diameter two. Deleting the hub leaves a planar bipartite graph, and every non-hub vertex has degree at most five. A shared construction encodes an Euler-tour count in rare covering walks. For specified starts, a number of fractional bits linear in the queried graph's vertex count suffices for recovery. For the maximum-start problem, a fixed cube attachment makes the maximizing starts known, and one interpolation of expected cover times recovers the count. Both reductions use polynomial-size graphs and polynomial-bit threshold queries. For general graphs, both comparison problems can be solved in polynomial time with access to an $\mathrm{NP}^{\mathrm{PP}}$ oracle.

Friday, October 30 2026, 00:00

Upper Bounds for CSP Non-redundancy via Locally Decodable Codes

Authors: Joshua Brakensiek, David Conlon, Venkatesan Guruswami, Xiaoyu He, Jiaxi Nie, Maya Sankar, Hung-Hsun Hans Yu

In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.

Authors: Joshua Brakensiek, David Conlon, Venkatesan Guruswami, Xiaoyu He, Jiaxi Nie, Maya Sankar, Hung-Hsun Hans Yu

In the study of constraint satisfaction problems (CSPs), the non-redundancy of a relation measures the maximum number of clauses in a CSP instance which can be logically independent. In the last few years, CSP non-redundancy has found a number of applications including in query complexity, sparsification, kernelization and streaming. In this paper, we present a novel technique for upper-bounding the non-redundancy of some relations using locally decodable codes (LDCs) and locally correctable codes (LCCs). More precisely, known limitations on the structure of 2-query LDCs and LCCs can be translated into limitations on the size of certain non-redundant CSP instances. This technique builds on a technique used by Brakensiek and Guruswami [STOC 2025] to bound non-redundancy via techniques from matching vector codes. Using LDCs and LCCs, we give improved upper bounds on non-redundancy of the BCK predicate pioneered by Bessiere, Carbonnel, and Katsirelos [AAAI 2020] as well as its well-studied Boolean variant. We also give the first example of a Boolean predicate whose non-redundancy is provably not of the form $n^{k\pm o(1)}$ for some positive integer $k$.

Friday, October 30 2026, 00:00

Compressed Orientation Reachability Maps for Sub-Second Robotic Needle Entry Point Planning

Authors: Prabhu Nithin Gollapudi, Andreas Maier, Franziska Mathis-Ullrich, Water Zhao, Jia Wang, Siming Bayer

In the context of multi-stage trajectory planning for robotic percutaneous liver interventions, identifying a kinematically feasible entry point remains a critical computational bottleneck. Existing methods rely on exhaustive inverse kinematics or randomized searches, incurring delays of 13 to 27 seconds per target and limiting real-time intraoperative adaptation. This paper introduces an Orientation-Based Reachability Map (ORM) that precomputes the kinematic feasibility of a 6 Degrees of Freedom (DoF) serial manipulator across 188,873 workspace positions and encodes orientation feasibility into a compact, indexed 83 MB map. A Just-In-Time (JIT) compiled runtime pipeline replaces live optimization with near-instantaneous array lookups. Experimental evaluation demonstrates a reduction in planning latency from 13.5-26.9 s to 0.289-0.349 s per target while maintaining high geometric fidelity, achieving a mean convex hull volume overlap ratio of 0.933 and a mean alpha-shape volume ratio of 0.976 relative to the baseline reachable point cloud. The proposed ORM framework enables sub-second entry point planning and provides an efficient computational framework for robotic needle insertion in percutaneous liver ablation.

Authors: Prabhu Nithin Gollapudi, Andreas Maier, Franziska Mathis-Ullrich, Water Zhao, Jia Wang, Siming Bayer

In the context of multi-stage trajectory planning for robotic percutaneous liver interventions, identifying a kinematically feasible entry point remains a critical computational bottleneck. Existing methods rely on exhaustive inverse kinematics or randomized searches, incurring delays of 13 to 27 seconds per target and limiting real-time intraoperative adaptation. This paper introduces an Orientation-Based Reachability Map (ORM) that precomputes the kinematic feasibility of a 6 Degrees of Freedom (DoF) serial manipulator across 188,873 workspace positions and encodes orientation feasibility into a compact, indexed 83 MB map. A Just-In-Time (JIT) compiled runtime pipeline replaces live optimization with near-instantaneous array lookups. Experimental evaluation demonstrates a reduction in planning latency from 13.5-26.9 s to 0.289-0.349 s per target while maintaining high geometric fidelity, achieving a mean convex hull volume overlap ratio of 0.933 and a mean alpha-shape volume ratio of 0.976 relative to the baseline reachable point cloud. The proposed ORM framework enables sub-second entry point planning and provides an efficient computational framework for robotic needle insertion in percutaneous liver ablation.

Friday, October 30 2026, 00:00

Witnessed Boolean Contracts: Sound Local Distance Guarantees for Approximate Implicit CSG

Authors: Jeremy Pei

Hard min/max operations preserve the sign of exact implicit solids but generally destroy signed-distance semantics. Arbitrarily small, sign-preserving value perturbations can rotate the normalized gradient by nearly a right angle and displace a gradient-projected boundary witness by a finite amount. We formalize this as the absence of a value-only stability modulus for witness projection. We introduce a witnessed Boolean contract checking value and gradient errors, positive reach, active-branch separation, and a Lipschitz visibility margin at the approximate witness. A sharp normalization lemma bounds the normal-direction error. An enlarged neighborhood permits transfer of the inactive operand's bound between the approximate witness and the true foot point. Under the stated analytic and domain assumptions, passing the checks identifies a true point on the union boundary and bounds distance deficiency by the active operand's pointwise value error. Failed or unavailable checks return Unknown. The argument extends to finite hard unions by testing every inactive operand. An executable proof-chain audit finds no violations on 75,738 certified samples. A broader analytic stress test covers spheres, capsules, ring tori, three Boolean operators, and a four-leaf union. Of 96,000 narrow-band samples, 87,469 pass with zero oracle violations; per-case coverage ranges from 72.67% to 98.30%. Removing visibility admits 75 truly invalid witnesses, showing that branch separation alone is insufficient. Non-union cases remain empirical tests, not new theorems.

Authors: Jeremy Pei

Hard min/max operations preserve the sign of exact implicit solids but generally destroy signed-distance semantics. Arbitrarily small, sign-preserving value perturbations can rotate the normalized gradient by nearly a right angle and displace a gradient-projected boundary witness by a finite amount. We formalize this as the absence of a value-only stability modulus for witness projection. We introduce a witnessed Boolean contract checking value and gradient errors, positive reach, active-branch separation, and a Lipschitz visibility margin at the approximate witness. A sharp normalization lemma bounds the normal-direction error. An enlarged neighborhood permits transfer of the inactive operand's bound between the approximate witness and the true foot point. Under the stated analytic and domain assumptions, passing the checks identifies a true point on the union boundary and bounds distance deficiency by the active operand's pointwise value error. Failed or unavailable checks return Unknown. The argument extends to finite hard unions by testing every inactive operand. An executable proof-chain audit finds no violations on 75,738 certified samples. A broader analytic stress test covers spheres, capsules, ring tori, three Boolean operators, and a four-leaf union. Of 96,000 narrow-band samples, 87,469 pass with zero oracle violations; per-case coverage ranges from 72.67% to 98.30%. Removing visibility admits 75 truly invalid witnesses, showing that branch separation alone is insufficient. Non-union cases remain empirical tests, not new theorems.

Friday, October 30 2026, 00:00

Certified Small Well-Separated Pair Decompositions in Practice

Authors: Guilherme D. da Fonseca

Given n points and $ε>0$, a well-separated pair decomposition (WSPD) is a collection of dumbbells {X,Y}, where X and Y have diameter at most $ε$ times their distance, such that every pair of points has one point in X and the other in Y for some dumbbell. A partition WSPD covers every pair exactly once, and a cover WSPD at least once. Callahan and Kosaraju give a partition WSPD with $O(n/ε^d)$ dumbbells in $R^d$, but many point sets have far smaller WSPDs of both kinds. For $d\ge2$, the known $O(1)$-approximations of the minimum size WSPD that preserve the separation $ε$ have unspecified factors, above $10^7$ for $d=2$ by our estimates. For cover WSPDs, we give an $O(n^2\log n)$-time sweep approximation algorithm with a factor 40 in the plane for $ε\le1/16$. It extends to $R^d$ with a factor $2^{O(d)}$. The sweep is also a lower-bound tool: its number of steps is a lower bound on the WSPD size. For partition WSPDs, we analyze quadtree and fair split tree constructions: in the plane and for $ε\le1/5$, they give constant approximation factors, the smallest of which is 252 for a randomly shifted and scaled quadtree. In experiments with $ε=1/20$, the number of sweep steps is at least 0.91 OPT for n = 500, where the optimum OPT is known, and a local search raises it to a lower bound of 0.99 OPT. A greedy variant of the sweep, which is still a certified 40-approximation, is at most 1.28 times the lower bound for n = 10000. Followed by a new local-search heuristic that shrinks the WSPD, it is within 0.02% of the optimum on every instance with n = 500, and at most 1.10 times the lower bound for n = 10000. We then show how to convert the cover WSPDs into partition WSPDs. For n = 10000, the conversion gives on average 1.13 times the lower bound, against 1.05 for the cover WSPD itself and 1.55 for the Callahan-Kosaraju partition WSPD.

Authors: Guilherme D. da Fonseca

Given n points and $ε>0$, a well-separated pair decomposition (WSPD) is a collection of dumbbells {X,Y}, where X and Y have diameter at most $ε$ times their distance, such that every pair of points has one point in X and the other in Y for some dumbbell. A partition WSPD covers every pair exactly once, and a cover WSPD at least once. Callahan and Kosaraju give a partition WSPD with $O(n/ε^d)$ dumbbells in $R^d$, but many point sets have far smaller WSPDs of both kinds. For $d\ge2$, the known $O(1)$-approximations of the minimum size WSPD that preserve the separation $ε$ have unspecified factors, above $10^7$ for $d=2$ by our estimates. For cover WSPDs, we give an $O(n^2\log n)$-time sweep approximation algorithm with a factor 40 in the plane for $ε\le1/16$. It extends to $R^d$ with a factor $2^{O(d)}$. The sweep is also a lower-bound tool: its number of steps is a lower bound on the WSPD size. For partition WSPDs, we analyze quadtree and fair split tree constructions: in the plane and for $ε\le1/5$, they give constant approximation factors, the smallest of which is 252 for a randomly shifted and scaled quadtree. In experiments with $ε=1/20$, the number of sweep steps is at least 0.91 OPT for n = 500, where the optimum OPT is known, and a local search raises it to a lower bound of 0.99 OPT. A greedy variant of the sweep, which is still a certified 40-approximation, is at most 1.28 times the lower bound for n = 10000. Followed by a new local-search heuristic that shrinks the WSPD, it is within 0.02% of the optimum on every instance with n = 500, and at most 1.10 times the lower bound for n = 10000. We then show how to convert the cover WSPDs into partition WSPDs. For n = 10000, the conversion gives on average 1.13 times the lower bound, against 1.05 for the cover WSPD itself and 1.55 for the Callahan-Kosaraju partition WSPD.

Friday, October 30 2026, 00:00

PhysicsNeMo-Mesh: GPU-Native Meshes and Simulation Cases for Physics AI

Authors: Peter D. Sharpe, Corey Adams, Negin Sobhani, Sanjay Choudhry

PhysicsNeMo-Mesh is the mesh module of the open-source PhysicsNeMo library. It provides mesh processing similar to PyVista, but on the GPU, in batches, and inside PyTorch's automatic differentiation. Like xarray for unstructured meshes, it attaches semantics to data: fields on points, cells, and whole meshes have names and shapes, and a simulation case keeps the complete PDE problem specification, including boundary conditions and their prescribed values. It addresses three problems that plague machine learning for physical simulation. First, GPU utilization is bottlenecked by I/O, because samples are very large relative to the compute per forward pass. Second, because traditional preprocessing is expensive, it is necessarily done once, during curation, rather than differently for each model and workflow at the point of use. Finally, advanced equivariant and mesh-native architectures require information that curation ordinarily discards. Fast GPU mesh processing makes it practical to preprocess at the point of use: on one GB200 GPU, mesh operations run $6.2\text{--}1{,}068\times$ faster than VTK with 72 threads, and loading a whole case is $8.0\text{--}88\times$ faster than reading its VTK files ($8.0\times$ when both read the same fields). Overlapped with training, this preprocessing costs $2.1\%$ of throughput for a $127\,\text{ms}$ Transolver step. So one dataset serves many models and workflows: in PhysicsNeMo's public recipe, one stored DrivAerML dataset feeds Transolver, GeoTransolver, and GLOBE. PhysicsNeMo-Mesh combines the speed of an intermediate array store with the flexibility of an archival format such as VTK. When shapes and wall conditions both vary, per-boundary conditions cut a heat-conduction error from $87\%$ to $6\text{--}9\%$. And by keeping field ranks and named boundaries, it retains the semantic structure that next-generation architectures will need.

Authors: Peter D. Sharpe, Corey Adams, Negin Sobhani, Sanjay Choudhry

PhysicsNeMo-Mesh is the mesh module of the open-source PhysicsNeMo library. It provides mesh processing similar to PyVista, but on the GPU, in batches, and inside PyTorch's automatic differentiation. Like xarray for unstructured meshes, it attaches semantics to data: fields on points, cells, and whole meshes have names and shapes, and a simulation case keeps the complete PDE problem specification, including boundary conditions and their prescribed values. It addresses three problems that plague machine learning for physical simulation. First, GPU utilization is bottlenecked by I/O, because samples are very large relative to the compute per forward pass. Second, because traditional preprocessing is expensive, it is necessarily done once, during curation, rather than differently for each model and workflow at the point of use. Finally, advanced equivariant and mesh-native architectures require information that curation ordinarily discards. Fast GPU mesh processing makes it practical to preprocess at the point of use: on one GB200 GPU, mesh operations run $6.2\text{--}1{,}068\times$ faster than VTK with 72 threads, and loading a whole case is $8.0\text{--}88\times$ faster than reading its VTK files ($8.0\times$ when both read the same fields). Overlapped with training, this preprocessing costs $2.1\%$ of throughput for a $127\,\text{ms}$ Transolver step. So one dataset serves many models and workflows: in PhysicsNeMo's public recipe, one stored DrivAerML dataset feeds Transolver, GeoTransolver, and GLOBE. PhysicsNeMo-Mesh combines the speed of an intermediate array store with the flexibility of an archival format such as VTK. When shapes and wall conditions both vary, per-boundary conditions cut a heat-conduction error from $87\%$ to $6\text{--}9\%$. And by keeping field ranks and named boundaries, it retains the semantic structure that next-generation architectures will need.

Friday, October 30 2026, 00:00

All-even sets in $\mathbb{R}^2$ and $\mathbb{R}^3$

Authors: Daniel Kalmanovich, Gabriel Nivasch

Call a finite nonempty point set $S\subset\mathbb{R}^d$ all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$, the number of hyperplanes perpendicular to $\ell$ that intersect $S$ is even. In 2026, Francisco Santos presented a question of Juan Benigno Seoane-Sepúlveda on the existence of all-even sets in dimensions $3$ and higher, together with examples of such sets in dimension $2$. Call a set $S\in\mathbb{R}^d$ line-all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$ the number of lines parallel to $\ell$ that intersect $S$ is even. As Santos pointed out, every all-even set in $\mathbb{R}^3$ is line-all-even. We conjecture that there exist no line-all-even sets in $\mathbb{R}^3$ that affinely span $\mathbb{R}^3$ We show that our conjecture is equivalent to the statement that every planar all-even set satisfies a certain property, which we call height-rigidity. This property has a natural interpretation in linear algebra terms. Furthermore, we present a hierarchy of properties of planar point sets that imply height-rigidity. We construct several all-even sets in the plane, including some infinite families, and show that all the sets we constructed are height-rigid. In addition, we prove that every all-even set in the plane that can be covered by six or fewer parallel lines is height-rigid.

Authors: Daniel Kalmanovich, Gabriel Nivasch

Call a finite nonempty point set $S\subset\mathbb{R}^d$ all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$, the number of hyperplanes perpendicular to $\ell$ that intersect $S$ is even. In 2026, Francisco Santos presented a question of Juan Benigno Seoane-Sepúlveda on the existence of all-even sets in dimensions $3$ and higher, together with examples of such sets in dimension $2$. Call a set $S\in\mathbb{R}^d$ line-all-even if for every direction $\ell \in \mathbb{P}(\mathbb{R}^d)$ the number of lines parallel to $\ell$ that intersect $S$ is even. As Santos pointed out, every all-even set in $\mathbb{R}^3$ is line-all-even. We conjecture that there exist no line-all-even sets in $\mathbb{R}^3$ that affinely span $\mathbb{R}^3$ We show that our conjecture is equivalent to the statement that every planar all-even set satisfies a certain property, which we call height-rigidity. This property has a natural interpretation in linear algebra terms. Furthermore, we present a hierarchy of properties of planar point sets that imply height-rigidity. We construct several all-even sets in the plane, including some infinite families, and show that all the sets we constructed are height-rigid. In addition, we prove that every all-even set in the plane that can be covered by six or fewer parallel lines is height-rigid.

Friday, October 30 2026, 00:00

Larger Crossing Families

Authors: Joachim Orthaber, Patrick Schnider

We give the first explicit constant for crossing families of linear size by showing that every set of $n$ points in general position in the plane contains a crossing family of size approximately $\frac{n}{79}$. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large different pattern, namely non-convex 4-clusters: We show that every point set contains either a large non-convex 4-cluster or a large crossing family. The second condition is the existence of many crossings: We give a bound for the size of a crossing family in terms of the number of crossings of the complete geometric graph drawn on the point set.

Authors: Joachim Orthaber, Patrick Schnider

We give the first explicit constant for crossing families of linear size by showing that every set of $n$ points in general position in the plane contains a crossing family of size approximately $\frac{n}{79}$. We further identify some conditions where there are even larger crossing families. The first such condition forbids the existence of a large different pattern, namely non-convex 4-clusters: We show that every point set contains either a large non-convex 4-cluster or a large crossing family. The second condition is the existence of many crossings: We give a bound for the size of a crossing family in terms of the number of crossings of the complete geometric graph drawn on the point set.

Friday, October 30 2026, 00:00

Half-Optimal Induced Matchings in Random Graphs: Online and Low-Degree Algorithms

Authors: Abhishek Dhawan, Eren C. Kızıldağ, Neeladri Maitra, Cheng Mao, Timothy L. H. Wee

For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.

Authors: Abhishek Dhawan, Eren C. Kızıldağ, Neeladri Maitra, Cheng Mao, Timothy L. H. Wee

For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.

Friday, October 30 2026, 00:00

Retrieval Sequencing in a Compact Storage System with Side Access

Authors: Yağmur Gül

In a compact storage system with side access, bins are arranged in adjacent vertical stacks. In each retrieval cycle, bins blocking access are lifted to create a temporary tunnel through which a robot extracts one requested bin from the side. The lifted bins are then lowered again, and the energy expenditure of the cycle equals the number of lifted bins. Given a pick list known in advance, we seek a retrieval order that minimizes total energy expenditure. For every number of stacks, we prove that the total energy separates into an instance constant and a pairwise penalty, and that every energy-optimal order is laminar within each stack. For two stacks, we prove that the retrieval sequencing problem is solvable in polynomial time. For three stacks, we give an exact polynomial-time reduction to a merge problem and characterize which merge-problem instances arise from the storage model. We define a recursive class of retrieval orders and construct a polynomial-time dynamic program that minimizes energy expenditure over this class. Finally, we quantify the additional energy savings from batching, which allows several requested bins to be retrieved in one cycle.

Authors: Yağmur Gül

In a compact storage system with side access, bins are arranged in adjacent vertical stacks. In each retrieval cycle, bins blocking access are lifted to create a temporary tunnel through which a robot extracts one requested bin from the side. The lifted bins are then lowered again, and the energy expenditure of the cycle equals the number of lifted bins. Given a pick list known in advance, we seek a retrieval order that minimizes total energy expenditure. For every number of stacks, we prove that the total energy separates into an instance constant and a pairwise penalty, and that every energy-optimal order is laminar within each stack. For two stacks, we prove that the retrieval sequencing problem is solvable in polynomial time. For three stacks, we give an exact polynomial-time reduction to a merge problem and characterize which merge-problem instances arise from the storage model. We define a recursive class of retrieval orders and construct a polynomial-time dynamic program that minimizes energy expenditure over this class. Finally, we quantify the additional energy savings from batching, which allows several requested bins to be retrieved in one cycle.

Friday, October 30 2026, 00:00

Four-Coloring Planar Graphs in Linear Time

Authors: Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup

The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\log n)$ amortized time. Thus, we implement all $O(\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.

Authors: Yuta Inoue, Ken-ichi Kawarabayashi, Atsuyuki Miyashita, Bojan Mohar, Carsten Thomassen, Mikkel Thorup

The famous Four Color Theorem (4CT) states that any planar graph can be colored with four colors, but how do we find a 4-coloring? In our previous paper [FOCS'26], we presented a simple combinatorial algorithm to 4-color planar graphs in $O(n\log n)$ time. We improve it here to $O(n)$ time, which is optimal. The algorithm from our previous paper is based on proving that every planar triangulation has a linear number of reducible configurations or obstructing cycles. The bottleneck for the algorithm is that we need $O(\log n)$ Kempe changes to extend a recursive 4-coloring to the reducible configurations. Each Kempe change can involve a constant fraction of the graph and takes linear time. Here we show that all $O(\log n)$ Kempe changes can be identified and implemented in $O(n)$ time. The idea is to spend $O(n)$ time to preprocess the graph so that we can identify and simulate each good Kempe change in $O(n/\log n)$ amortized time. Thus, we implement all $O(\log n)$ Kempe changes in $O(n)$ total time. Based on this, we implement the whole 4-coloring in linear time. We note that creating such a representation for fast identification and simulation of the right Kempe changes is a challenge of data structures, unlike the previous paper that builds on a new graph theoretic understanding of reducible configurations.

Friday, October 30 2026, 00:00

Dimension Reduction and Coresets for Min-Sum $k$-Clustering

Authors: Asaf Dallal, Robert Krauthgamer

The min-sum $k$-clustering problem asks to partition an input of $n$ points into $k$ clusters so as to minimize the total pairwise distance within clusters. It is a natural alternative to center-based objectives when the data is not well captured by a small set of representative points. We study for this problem two standard sketching methods, dimension reduction and coresets, which represent the input more compactly while preserving the min-sum cost of every $k$-partition up to $1\pm\varepsilon$ factor. Our first contribution addresses dimension reduction in Euclidean space. The Johnson-Lindenstrauss (JL) Lemma readily implies that a random linear map to target dimension $t=O(\varepsilon^{-2}\log n)$ preserves, with high probability, the min-sum cost of all $k$-partitions. We show that target dimension $t=O(\varepsilon^{-2}(ddim\cdot\log(1/\varepsilon)+\log k+\log\log n))$ suffices for Euclidean inputs of doubling dimension at most $ddim$, and moreover this bound is nearly optimal. We also show that target dimension $t=O(\varepsilon^{-2}(\log(1/\varepsilon)+\log k+\logΔ))$ suffices for Euclidean inputs of aspect ratio at most $Δ$. Our second contribution is to design the first $\varepsilon$-coreset for metric min-sum $k$-clustering. It is actually a movement-based coreset, has size $(1/\varepsilon)^{O(ddim)} k\log n$ for inputs of doubling dimension at most $ddim$, and can be constructed in randomized near-linear-time. We further use these coresets to design streaming algorithms for the problem.

Authors: Asaf Dallal, Robert Krauthgamer

The min-sum $k$-clustering problem asks to partition an input of $n$ points into $k$ clusters so as to minimize the total pairwise distance within clusters. It is a natural alternative to center-based objectives when the data is not well captured by a small set of representative points. We study for this problem two standard sketching methods, dimension reduction and coresets, which represent the input more compactly while preserving the min-sum cost of every $k$-partition up to $1\pm\varepsilon$ factor. Our first contribution addresses dimension reduction in Euclidean space. The Johnson-Lindenstrauss (JL) Lemma readily implies that a random linear map to target dimension $t=O(\varepsilon^{-2}\log n)$ preserves, with high probability, the min-sum cost of all $k$-partitions. We show that target dimension $t=O(\varepsilon^{-2}(ddim\cdot\log(1/\varepsilon)+\log k+\log\log n))$ suffices for Euclidean inputs of doubling dimension at most $ddim$, and moreover this bound is nearly optimal. We also show that target dimension $t=O(\varepsilon^{-2}(\log(1/\varepsilon)+\log k+\logΔ))$ suffices for Euclidean inputs of aspect ratio at most $Δ$. Our second contribution is to design the first $\varepsilon$-coreset for metric min-sum $k$-clustering. It is actually a movement-based coreset, has size $(1/\varepsilon)^{O(ddim)} k\log n$ for inputs of doubling dimension at most $ddim$, and can be constructed in randomized near-linear-time. We further use these coresets to design streaming algorithms for the problem.

Friday, October 30 2026, 00:00

On Zeros and Algorithms for Disordered Systems II: The Antiferromagnetic Ising Model up to Kesten-Stigum

Authors: Ferenc Bencs, Brice Huang, Daniel Z. Lee, Kuikui Liu, Guus Regts

We give the first polynomial-time algorithm to estimate the partition function of the antiferromagnetic Ising model on the uniformly random $d$-regular graph to arbitrarily high accuracy for all inverse temperatures up to the Kesten-Stigum threshold. This is the entire regime in which the model is "replica symmetric," and is conjectured to be optimal for polynomial-time counting and sampling. To achieve our algorithmic result, we establish the absence of Fisher zeros for the partition function in a constant-width strip around the interval from $0$ up to the Kesten-Stigum threshold, with high probability over the randomness of the input graph. AI Disclosure: The overarching vision and proof architecture were developed solely by the authors without AI input. GPT 5.5 Pro was used, with extensive guidance from the authors, for only a few minor technical aspects of the proof. The authors independently verified correctness and wrote everything by hand. We take full responsibility for its contents. A more detailed statement can be found in Section 1.2.

Authors: Ferenc Bencs, Brice Huang, Daniel Z. Lee, Kuikui Liu, Guus Regts

We give the first polynomial-time algorithm to estimate the partition function of the antiferromagnetic Ising model on the uniformly random $d$-regular graph to arbitrarily high accuracy for all inverse temperatures up to the Kesten-Stigum threshold. This is the entire regime in which the model is "replica symmetric," and is conjectured to be optimal for polynomial-time counting and sampling. To achieve our algorithmic result, we establish the absence of Fisher zeros for the partition function in a constant-width strip around the interval from $0$ up to the Kesten-Stigum threshold, with high probability over the randomness of the input graph. AI Disclosure: The overarching vision and proof architecture were developed solely by the authors without AI input. GPT 5.5 Pro was used, with extensive guidance from the authors, for only a few minor technical aspects of the proof. The authors independently verified correctness and wrote everything by hand. We take full responsibility for its contents. A more detailed statement can be found in Section 1.2.

Friday, October 30 2026, 00:00

Approximating Minimum Spanning Trees in Metrics under Semi-Random Corruptions

Authors: Aditya Anand, Anupam Gupta, Rajesh Jayaram, Euiwoong Lee, Ali Vakilian

We study the problem of finding minimum spanning trees in semi-random corrupted metrics. In this problem proposed by Bateni, Dharangutte, Jayaram, and Wang (COLT'24), we start with a metric space $\mathcal{M} = (X, d)$. The adversary independently corrupts each pairwise distance with probability $p$. The algorithm only observes these possibly corrupted distances and must output a spanning tree whose expected cost (in the true metric) is multiplicatively close to the minimum spanning tree cost. Prior work showed that losing a factor of $Ω(\log n)$ is necessary, even if the algorithm may adaptively select $O(n/\log n)$ points and probe the true distances between them. However, no upper bounds were known. We study three regimes based on whether the adversary can only increase, decrease, or arbitrarily change the corrupted distances. We show the following: (1) In the increase-only setting, we give a polynomial-time $O(1)$-approximation algorithm for any constant $0 \le p < 1$. (2) In the decrease-only setting, we give an $O(\log n)$-approximation algorithm for any constant $0 \le p < 1$. Additionally, we show an $O(\log^4 n)$-approximation algorithm that runs in quasi-polynomial time $n^{O(\log n)}$, for any constant $p \leq 0.1$. Moreover, we show that a polynomial-time algorithm computing an $n^{1/3-\varepsilon}$-approximation is unlikely to exist. (3) In the general setting, where the adversary can both increase and decrease distances, we show an $O(\log n)$-approximation algorithm for every $0 \leq p < \frac{1}{4}$. Further, we show that no $o(n)$-approximation is possible for $\frac{1}{2} \leq p < 1$. Our algorithms answer the main open question in Bateni et al.'s work; moreover, the $O(\log n)$-approximations in the latter two settings are information-theoretically optimal.

Authors: Aditya Anand, Anupam Gupta, Rajesh Jayaram, Euiwoong Lee, Ali Vakilian

We study the problem of finding minimum spanning trees in semi-random corrupted metrics. In this problem proposed by Bateni, Dharangutte, Jayaram, and Wang (COLT'24), we start with a metric space $\mathcal{M} = (X, d)$. The adversary independently corrupts each pairwise distance with probability $p$. The algorithm only observes these possibly corrupted distances and must output a spanning tree whose expected cost (in the true metric) is multiplicatively close to the minimum spanning tree cost. Prior work showed that losing a factor of $Ω(\log n)$ is necessary, even if the algorithm may adaptively select $O(n/\log n)$ points and probe the true distances between them. However, no upper bounds were known. We study three regimes based on whether the adversary can only increase, decrease, or arbitrarily change the corrupted distances. We show the following: (1) In the increase-only setting, we give a polynomial-time $O(1)$-approximation algorithm for any constant $0 \le p < 1$. (2) In the decrease-only setting, we give an $O(\log n)$-approximation algorithm for any constant $0 \le p < 1$. Additionally, we show an $O(\log^4 n)$-approximation algorithm that runs in quasi-polynomial time $n^{O(\log n)}$, for any constant $p \leq 0.1$. Moreover, we show that a polynomial-time algorithm computing an $n^{1/3-\varepsilon}$-approximation is unlikely to exist. (3) In the general setting, where the adversary can both increase and decrease distances, we show an $O(\log n)$-approximation algorithm for every $0 \leq p < \frac{1}{4}$. Further, we show that no $o(n)$-approximation is possible for $\frac{1}{2} \leq p < 1$. Our algorithms answer the main open question in Bateni et al.'s work; moreover, the $O(\log n)$-approximations in the latter two settings are information-theoretically optimal.

Friday, October 30 2026, 00:00

Maximum Cardinality Matching in $n^{2+o(1)}$ Time

Authors: George Z. Li, Jason Li, Junkai Zhang

We consider the maximum cardinality matching problem in general graphs, and give an $n^{2+o(1)}$-time algorithm for the problem. Our work builds on recent work on combinatorial algorithms for maximum flow and extends the techniques to skew-symmetric flow problems which capture general matchings as a special case.

Authors: George Z. Li, Jason Li, Junkai Zhang

We consider the maximum cardinality matching problem in general graphs, and give an $n^{2+o(1)}$-time algorithm for the problem. Our work builds on recent work on combinatorial algorithms for maximum flow and extends the techniques to skew-symmetric flow problems which capture general matchings as a special case.

Friday, October 30 2026, 00:00

When Imputation Changes the Nearest Neighbor: Exact Certification of Similarity Retrieval for Incomplete Sensor Time Series

Authors: Jianru Shen

Sensor time series routinely reach similarity search with missing values, and the retrieved neighbours then depend on how the gaps were filled. We ask which retrieval conclusions hold for every admissible completion of an incomplete query. For a family of window-minimum distances that contains the $L_1$ distance as its separable end we give a pairwise certificate that one candidate is strictly closer than another, a guaranteed candidate set that contains every possible nearest neighbour, and a certificate that a nearest-neighbour label cannot change. The guarantees hold uniformly over the specified box and apply to the true query whenever it lies inside. At the separable end a closed form is known, but beyond it the worst-case gap need not be attained at a corner of the box. We prove a finite reduction to critical values, give an exact interval dynamic programme for the worst-case ranking gap, and a quadratic-time fast layer of valid bounds and feasible witnesses, supported by an exact criterion for when the feature-box relaxation is tight. On three traffic-sensor collections with real gap patterns, linear interpolation changes the nearest neighbour in 15 to 33 percent of the queries. At the full band the feature-box relaxation misses up to 14 percent of the exactly stable boundary pairs, whereas fixed-completion and Monte Carlo checks issue false certificates. The fast layer resolves about 90 percent of all tested boundary pairs without dynamic programming, and on paired network-scale queries the exact method is 82 to 110 times faster than the programme alone. Label certificates transfer to three public benchmarks, and the band width trades forecasting accuracy and certification coverage for lower imputation sensitivity, with dataset-dependent crossovers.

Authors: Jianru Shen

Sensor time series routinely reach similarity search with missing values, and the retrieved neighbours then depend on how the gaps were filled. We ask which retrieval conclusions hold for every admissible completion of an incomplete query. For a family of window-minimum distances that contains the $L_1$ distance as its separable end we give a pairwise certificate that one candidate is strictly closer than another, a guaranteed candidate set that contains every possible nearest neighbour, and a certificate that a nearest-neighbour label cannot change. The guarantees hold uniformly over the specified box and apply to the true query whenever it lies inside. At the separable end a closed form is known, but beyond it the worst-case gap need not be attained at a corner of the box. We prove a finite reduction to critical values, give an exact interval dynamic programme for the worst-case ranking gap, and a quadratic-time fast layer of valid bounds and feasible witnesses, supported by an exact criterion for when the feature-box relaxation is tight. On three traffic-sensor collections with real gap patterns, linear interpolation changes the nearest neighbour in 15 to 33 percent of the queries. At the full band the feature-box relaxation misses up to 14 percent of the exactly stable boundary pairs, whereas fixed-completion and Monte Carlo checks issue false certificates. The fast layer resolves about 90 percent of all tested boundary pairs without dynamic programming, and on paired network-scale queries the exact method is 82 to 110 times faster than the programme alone. Label certificates transfer to three public benchmarks, and the band width trades forecasting accuracy and certification coverage for lower imputation sensitivity, with dataset-dependent crossovers.

Friday, October 30 2026, 00:00

Minimum Label $k$-Cut on Planar Graphs: Complexity and Color Structure

Authors: Yuanhao Wang, Wei Wang

Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.

Authors: Yuanhao Wang, Wei Wang

Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.

Friday, October 30 2026, 00:00

The Tight Competitive Ratio of Greedy for Online Steiner Forest

Authors: Iman Gholami, Ilya Hajiaghayi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade

The online Steiner Forest problem is a fundamental network design problem in which terminal pairs arrive one at a time and an algorithm must irrevocably purchase edges connecting each pair. The natural greedy algorithm connects each terminal pair by a shortest path after making all previously purchased edges free. Awerbuch, Azar, and Bartal (SODA 1996) proved that this algorithm is $O(log^2 k)$-competitive and conjectured an $O(log k)$ bound, where $k$ is the number of terminal pairs. Bamas, Drygala, and Maggiori (SODA 2022) obtained an $O(\log \NumPairs\log\log \NumPairs)$ bound for greedy on a restricted class of inputs, leaving the general case open. In this paper, we resolve the competitive ratio of the greedy algorithm for online Steiner Forest, proving that it is tightly $Θ(log(k) log log(k))$. Specifically, we establish an $O(log(k) log log(k))$ upper bound for arbitrary instances and construct a matching $Ω(log(k) log log(k))$ lower bound. Our results refute the $O(log k)$ conjecture of Awerbuch, Azar, and Bartal and establish a tight $Θ(log log k)$ separation between the greedy algorithm and the best possible online algorithm for Steiner Forest.

Authors: Iman Gholami, Ilya Hajiaghayi, MohammadTaghi Hajiaghayi, Peyman Jabbarzade

The online Steiner Forest problem is a fundamental network design problem in which terminal pairs arrive one at a time and an algorithm must irrevocably purchase edges connecting each pair. The natural greedy algorithm connects each terminal pair by a shortest path after making all previously purchased edges free. Awerbuch, Azar, and Bartal (SODA 1996) proved that this algorithm is $O(log^2 k)$-competitive and conjectured an $O(log k)$ bound, where $k$ is the number of terminal pairs. Bamas, Drygala, and Maggiori (SODA 2022) obtained an $O(\log \NumPairs\log\log \NumPairs)$ bound for greedy on a restricted class of inputs, leaving the general case open. In this paper, we resolve the competitive ratio of the greedy algorithm for online Steiner Forest, proving that it is tightly $Θ(log(k) log log(k))$. Specifically, we establish an $O(log(k) log log(k))$ upper bound for arbitrary instances and construct a matching $Ω(log(k) log log(k))$ lower bound. Our results refute the $O(log k)$ conjecture of Awerbuch, Azar, and Bartal and establish a tight $Θ(log log k)$ separation between the greedy algorithm and the best possible online algorithm for Steiner Forest.

Friday, October 30 2026, 00:00

Nearly-Optimal Time-Sample Trade-off in Consensus Dynamics

Authors: Francesco d'Amore, Joel Rybicki, Nobutaka Shimizu, Takeharu Shiraga, Robin Vacus

How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\textsf{Majority} that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.

Authors: Francesco d'Amore, Joel Rybicki, Nobutaka Shimizu, Takeharu Shiraga, Robin Vacus

How does the number of sampled opinions affect the time required to reach consensus in distributed consensus dynamics? We study the synchronous $h$-\textsf{Majority}, in which every node independently samples $h$ nodes uniformly at random with replacement in each round and adopts a most frequent sampled opinion, breaking ties uniformly at random. We show that $h$-\textsf{Majority} achieves the nearly-optimal trade-off between the sample size and the consensus time, among a broad class of consensus dynamics. In particular, we prove the following results. For every sample size $h \ge 3$ and for every initial configuration, we prove that the process reaches consensus with high probability within $\widetilde{O}\left(1+\min\left\{\frac{1}{\min\{h^2p_{\max},1\}},\frac{\sqrt{n}}{h}\right\}\right)$ rounds, where $n$ is the number of nodes and $p_{\max}$ is the largest initial fraction of nodes supporting any single opinion. This is the first bound on the consensus time for $h$-\textsf{Majority} that holds for every $h\ge3$ and every initial configuration. When the initially most popular opinion has a sufficiently large lead over every other opinion, we show that all nodes adopt this opinion with high probability within $O(\log n/\min\{h^2p_{\max},1\})$ rounds. We establish a matching lower bound up to polylogarithmic factors for any fixed update rule that treats all opinions symmetrically and that selects either a sampled opinion or the node's current opinion.

Friday, October 30 2026, 00:00

The Value of Experiments in Assortment Optimization

Authors: Mohammadreza Ahmadnejadsaein, Amine Bennouna, Omar El Housni

Retailers can learn customer preferences by experimenting with different assortments. However, the cost of each experiment and the operational burden of repeatedly changing product offerings can make extensive experimentation impractical. We study how much a retailer can improve its final assortment decision using customer observations from one (or a very small number of) fixed experimental assortments. We consider the multinomial logit model, where product revenues are known but customer preferences are unknown. In this setting, we jointly design the experimental assortment and the rule for selecting a final assortment from observed customer choices to minimize worst-case regret: the expected revenue loss of the final assortment relative to an optimal assortment under the true preferences, maximized over possible customer preferences. We first characterize exactly the minimax regret and an optimal assortment when no experimentation is possible. We then show, surprisingly, that offering a single experimental assortment to just one customer can already substantially reduce regret, depending on how the products' revenues compare. We derive the exact minimax regret and an optimal experimental policy in this setting. When a single fixed experimental assortment is offered to many customers, we give a geometric characterization of the limiting minimax regret and construct an asymptotically optimal policy. Depending on product revenues, this single experiment can yield no improvement over no experimentation or nearly complete regret reduction. Finally, we extend our analysis to multiple experimental assortments chosen in advance, characterizing the limiting minimax regret, constructing asymptotically optimal policies, and identifying which collections of assortments allow worst-case regret to vanish as the number of customer observations grows.

Authors: Mohammadreza Ahmadnejadsaein, Amine Bennouna, Omar El Housni

Retailers can learn customer preferences by experimenting with different assortments. However, the cost of each experiment and the operational burden of repeatedly changing product offerings can make extensive experimentation impractical. We study how much a retailer can improve its final assortment decision using customer observations from one (or a very small number of) fixed experimental assortments. We consider the multinomial logit model, where product revenues are known but customer preferences are unknown. In this setting, we jointly design the experimental assortment and the rule for selecting a final assortment from observed customer choices to minimize worst-case regret: the expected revenue loss of the final assortment relative to an optimal assortment under the true preferences, maximized over possible customer preferences. We first characterize exactly the minimax regret and an optimal assortment when no experimentation is possible. We then show, surprisingly, that offering a single experimental assortment to just one customer can already substantially reduce regret, depending on how the products' revenues compare. We derive the exact minimax regret and an optimal experimental policy in this setting. When a single fixed experimental assortment is offered to many customers, we give a geometric characterization of the limiting minimax regret and construct an asymptotically optimal policy. Depending on product revenues, this single experiment can yield no improvement over no experimentation or nearly complete regret reduction. Finally, we extend our analysis to multiple experimental assortments chosen in advance, characterizing the limiting minimax regret, constructing asymptotically optimal policies, and identifying which collections of assortments allow worst-case regret to vanish as the number of customer observations grows.

Friday, October 30 2026, 00:00

Exact Competitive Ratios of Index Rules for Scheduling with Obligatory Testing

Authors: Felix Buld

We revisit scheduling with obligatory testing, where each job consists of two operations: a test with an initially known time and an operation whose processing time is revealed upon completion of the job's test. We develop an iterative competitive-ratio-preserving instance-reduction method that enables us to compute exact competitive ratios directly on highly structured instances for various index rules. Our main result is a clean analysis showing that the competitive ratio of the algorithm $1$-SORT is exactly the Golden Ratio $\varphi\approx 1.618$, improving the recent best-known upper bound of $1.861$ for arbitrary testing times. We further leverage this method to obtain near-tight bounds for a simple randomized algorithm on instances with uniform testing times.

Authors: Felix Buld

We revisit scheduling with obligatory testing, where each job consists of two operations: a test with an initially known time and an operation whose processing time is revealed upon completion of the job's test. We develop an iterative competitive-ratio-preserving instance-reduction method that enables us to compute exact competitive ratios directly on highly structured instances for various index rules. Our main result is a clean analysis showing that the competitive ratio of the algorithm $1$-SORT is exactly the Golden Ratio $\varphi\approx 1.618$, improving the recent best-known upper bound of $1.861$ for arbitrary testing times. We further leverage this method to obtain near-tight bounds for a simple randomized algorithm on instances with uniform testing times.

Friday, October 30 2026, 00:00

Zero-freeness for Random Regular Graphs up to the Replica Symmetry Breaking Threshold

Authors: Zejia Chen, Chunyang Wang, Xinyuan Zhang

We study antiferromagnetic spin systems on random regular graphs through the lens of the phase transitions characterized by the complex zeros of the partition function. Our motivating examples include the hardcore model and the antiferromagnetic Potts model (including proper $q$-colorings at $β=0$). For both models on random $d$-regular graphs, we establish, with high probability over the graph, zero-free regions throughout the interior of the replica-symmetric regime. Asymptotically, this regime extends up to tree occupation density $$α< α_{\mathrm{RSB}}(d) = (2\log d - 2\log\log d + 2 - 2\log 2 + o_d(1))/d$$ for the hardcore model and up to degree $$d < d_{\mathrm{RSB}}(q) = (2q-1)\log q - 2\log 2 + o_q(1)$$ for proper $q$-colorings. Leveraging the standard algorithmic implications of zero-freeness, we obtain deterministic polynomial-time algorithms that multiplicatively approximate the corresponding partition functions at fixed rational parameters throughout this regime, with high probability over the graph. Our algorithmic guarantees significantly improve upon prior work and extend beyond the parameter ranges reached by previously known search algorithms. To establish our results, we first develop reinforced small subgraph conditioning, which resolves short-cycle fluctuations in the complex plane by lifting classical small subgraph conditioning on random regular graphs to random holomorphic functions. Second, we overcome the standard second-moment barrier caused by the exponential blowup of the bare second moment inside the replica-symmetric phase, by showing that replica symmetry implies a nonpositive Bethe functional and upgrading planted overlap concentration to an exponentially small bound, which enables a typical-graph truncation that controls complex second moments up to the replica symmetry breaking threshold.

Authors: Zejia Chen, Chunyang Wang, Xinyuan Zhang

We study antiferromagnetic spin systems on random regular graphs through the lens of the phase transitions characterized by the complex zeros of the partition function. Our motivating examples include the hardcore model and the antiferromagnetic Potts model (including proper $q$-colorings at $β=0$). For both models on random $d$-regular graphs, we establish, with high probability over the graph, zero-free regions throughout the interior of the replica-symmetric regime. Asymptotically, this regime extends up to tree occupation density $$α< α_{\mathrm{RSB}}(d) = (2\log d - 2\log\log d + 2 - 2\log 2 + o_d(1))/d$$ for the hardcore model and up to degree $$d < d_{\mathrm{RSB}}(q) = (2q-1)\log q - 2\log 2 + o_q(1)$$ for proper $q$-colorings. Leveraging the standard algorithmic implications of zero-freeness, we obtain deterministic polynomial-time algorithms that multiplicatively approximate the corresponding partition functions at fixed rational parameters throughout this regime, with high probability over the graph. Our algorithmic guarantees significantly improve upon prior work and extend beyond the parameter ranges reached by previously known search algorithms. To establish our results, we first develop reinforced small subgraph conditioning, which resolves short-cycle fluctuations in the complex plane by lifting classical small subgraph conditioning on random regular graphs to random holomorphic functions. Second, we overcome the standard second-moment barrier caused by the exponential blowup of the bare second moment inside the replica-symmetric phase, by showing that replica symmetry implies a nonpositive Bethe functional and upgrading planted overlap concentration to an exponentially small bound, which enables a typical-graph truncation that controls complex second moments up to the replica symmetry breaking threshold.

Friday, October 30 2026, 00:00

Layering and reverse extension for weighted $k$-path vertex cover

Authors: Jiajing Wu, An Zhang, Shiming Li, Guohui Lin

We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.

Authors: Jiajing Wu, An Zhang, Shiming Li, Guohui Lin

We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.

Friday, October 30 2026, 00:00

Nearly Optimal Sparse Navigable Graphs in General Metrics

Authors: Hongyi Chen, Tianyi Zhang

We study approximation algorithms for constructing sparse navigable graphs in general metrics. Given an $n$-point metric $(P,d)$ and a parameter $α\ge1$, a directed graph on $P$ is $α$-navigable if, for every pair of distinct points $s,t\in P$, the source $s$ has an outgoing neighbor $p$ with $d(p,t)

Authors: Hongyi Chen, Tianyi Zhang

We study approximation algorithms for constructing sparse navigable graphs in general metrics. Given an $n$-point metric $(P,d)$ and a parameter $α\ge1$, a directed graph on $P$ is $α$-navigable if, for every pair of distinct points $s,t\in P$, the source $s$ has an outgoing neighbor $p$ with $d(p,t)

Friday, October 30 2026, 00:00

Counting approximate breakpoint medians of random genomes

Authors: Poly H. da Silva, Arash Jamshidpey, David Sankoff

Unsigned linear unichromosomal genomes with genes labeled 1, . . . , n are represented by permutations of these labels, whose adjacencies are the unordered pairs of genes occurring consecutively. The breakpoint distance counts the adjacencies of one genome that are absent from the other, and a breakpoint median minimizes the total breakpoint distance to a given set of genomes. Deciding whether a permutation is a median often requires a minimization over the whole symmetric group, which makes the median set resistant to exact counting. We replace this global condition by a local one: a c-approximate median is a genome having at most c adjacencies absent from all the input genomes. For a fixed genome we introduce its approximate median inverse, the set of ordered k-tuples of genomes ad- mitting it as a c-approximate median, and we construct and count this set explicitly. For k genomes chosen uniformly and independently at random, this gives the probability that a given genome is a c-approximate median and the expected number of such approximate medians. The relaxation is not only technical: approximate medians and near-optimal genomes co- incide up to a shift of the tolerance that remains bounded in probability as the number of genes grows. Finally, we derive an explicit upper bound on the probability that a given genome is an exact median, and on the expected number of exact breakpoint medians.

Authors: Poly H. da Silva, Arash Jamshidpey, David Sankoff

Unsigned linear unichromosomal genomes with genes labeled 1, . . . , n are represented by permutations of these labels, whose adjacencies are the unordered pairs of genes occurring consecutively. The breakpoint distance counts the adjacencies of one genome that are absent from the other, and a breakpoint median minimizes the total breakpoint distance to a given set of genomes. Deciding whether a permutation is a median often requires a minimization over the whole symmetric group, which makes the median set resistant to exact counting. We replace this global condition by a local one: a c-approximate median is a genome having at most c adjacencies absent from all the input genomes. For a fixed genome we introduce its approximate median inverse, the set of ordered k-tuples of genomes ad- mitting it as a c-approximate median, and we construct and count this set explicitly. For k genomes chosen uniformly and independently at random, this gives the probability that a given genome is a c-approximate median and the expected number of such approximate medians. The relaxation is not only technical: approximate medians and near-optimal genomes co- incide up to a shift of the tolerance that remains bounded in probability as the number of genes grows. Finally, we derive an explicit upper bound on the probability that a given genome is an exact median, and on the expected number of exact breakpoint medians.

Friday, October 30 2026, 00:00

Accumulating Pruning Constraints: New Exact Polynomial-Space Algorithms for Max 2-SAT

Authors: Konstantin Kutzkov

We present deterministic polynomial-space algorithms that solve Max 2-SAT on $n$ variables exactly in time $\Ostar(2^{(1-γ)n})$, for a fixed $γ>0$, on several classes of instances whose clause density may grow with $n$. Our main tool extends the classic DPLL branching algorithm. When the search gives a variable its less promising value, comparing the two values of the variable yields a pruning constraint that is satisfied by some optimal assignment. Accumulating these constraints along each branch prunes all but an exponentially small fraction of the assignments. The classes are formulas with many biased variables, semirandom 2-CNF formulas, and Max-Cut on regular expander graphs and on graphs generated by a stochastic block model. In a semirandom formula, the pairs of variables that may share a clause are arbitrary, and the clauses on these pairs are random. For semirandom formulas and block-model graphs, the algorithms are still exact on every input, and the time bound holds with high probability over the input. To our knowledge, these are the first polynomial-space algorithms with such a running time for nontrivial classes of dense Max 2-SAT instances. Beyond these classes, the pruning constraints can be added to any branching algorithm for Max 2-SAT, and we expect them to be of practical relevance for general formulas, in particular for sparse ones, where each constraint involves only a few variables.

Authors: Konstantin Kutzkov

We present deterministic polynomial-space algorithms that solve Max 2-SAT on $n$ variables exactly in time $\Ostar(2^{(1-γ)n})$, for a fixed $γ>0$, on several classes of instances whose clause density may grow with $n$. Our main tool extends the classic DPLL branching algorithm. When the search gives a variable its less promising value, comparing the two values of the variable yields a pruning constraint that is satisfied by some optimal assignment. Accumulating these constraints along each branch prunes all but an exponentially small fraction of the assignments. The classes are formulas with many biased variables, semirandom 2-CNF formulas, and Max-Cut on regular expander graphs and on graphs generated by a stochastic block model. In a semirandom formula, the pairs of variables that may share a clause are arbitrary, and the clauses on these pairs are random. For semirandom formulas and block-model graphs, the algorithms are still exact on every input, and the time bound holds with high probability over the input. To our knowledge, these are the first polynomial-space algorithms with such a running time for nontrivial classes of dense Max 2-SAT instances. Beyond these classes, the pruning constraints can be added to any branching algorithm for Max 2-SAT, and we expect them to be of practical relevance for general formulas, in particular for sparse ones, where each constraint involves only a few variables.

Friday, October 30 2026, 00:00

An Exact Parallel Dense Gauss-Seidel Algorithm with Trade-Offs between Parallel Execution Time and Preprocessing Time

Authors: Mugurel-Ionuţ Andreica, Pantelimon George Popescu, Eliana-Dina Tîrşa

The Gauss-Seidel iterative method is commonly used for solving systems of linear equations because of its good convergence properties. However, the Gauss-Seidel algorithm cannot be easily parallelized because of the data dependencies existing at each iteration. The best parallel execution time for dense systems of equations is $O(n)$ given that enough processors are available and no preprocessing is performed. In this paper we propose a new parallel Gauss-Seidel algorithm which is capable of achieving better parallel runtimes at the expense of more preprocessing time. The algorithm can be tuned in order to provide convenient trade-offs between the preprocessing time and the best parallel runtime achievable by the algorithm.

Authors: Mugurel-Ionuţ Andreica, Pantelimon George Popescu, Eliana-Dina Tîrşa

The Gauss-Seidel iterative method is commonly used for solving systems of linear equations because of its good convergence properties. However, the Gauss-Seidel algorithm cannot be easily parallelized because of the data dependencies existing at each iteration. The best parallel execution time for dense systems of equations is $O(n)$ given that enough processors are available and no preprocessing is performed. In this paper we propose a new parallel Gauss-Seidel algorithm which is capable of achieving better parallel runtimes at the expense of more preprocessing time. The algorithm can be tuned in order to provide convenient trade-offs between the preprocessing time and the best parallel runtime achievable by the algorithm.

Friday, October 30 2026, 00:00

Communication-Oblivious Permutation and Sorting

Authors: Qi Dong, Yilei Wang, Ke Yi

With the help of secure hardware such as trusted execution environments (TEEs), private data can be uploaded and computed in untrusted cloud clusters, so that an adversary controlling the cloud can only see the data in encrypted form. However, the communication pattern between the processors may still reveal sensitive information about the data, so it is important to develop parallel algorithms in the BSP model whose communication pattern is independent of the input data, a property known as \textit{communication obliviousness}. In this paper, we study such algorithms for three fundamental problems: permutation, sorting, and 0-1 sorting. For the first two problems, we prove a tight (up to lower-order terms) communication complexity of $(2-2/p)n\pm o(n)$. Here, $p$ is the number of processors and $n$ is the input size, assuming $2\le p$ and $p=o(n^{1/3})$. Our lower bound holds regardless of the number of rounds and local memory size, while for balanced permutation and sorting the upper bound is achieved in a constant number of rounds with load at most $n/p+o(n/p)$. For 0-1 sorting, we also derive a tight communication complexity of $C_{\mathrm{cut}}(p) \cdot n \pm o(n)$, where $C_{\mathrm{cut}}(p)$ is the optimal value of a linear program parameterized by $p$. Although there appears to be no closed-form for $C_{\mathrm{cut}}(p)$, we show that it is strictly smaller than $2-2/p$ for all $p\ge 3$, thus separating 0-1 sorting from general sorting in terms of their communication complexity.

Authors: Qi Dong, Yilei Wang, Ke Yi

With the help of secure hardware such as trusted execution environments (TEEs), private data can be uploaded and computed in untrusted cloud clusters, so that an adversary controlling the cloud can only see the data in encrypted form. However, the communication pattern between the processors may still reveal sensitive information about the data, so it is important to develop parallel algorithms in the BSP model whose communication pattern is independent of the input data, a property known as \textit{communication obliviousness}. In this paper, we study such algorithms for three fundamental problems: permutation, sorting, and 0-1 sorting. For the first two problems, we prove a tight (up to lower-order terms) communication complexity of $(2-2/p)n\pm o(n)$. Here, $p$ is the number of processors and $n$ is the input size, assuming $2\le p$ and $p=o(n^{1/3})$. Our lower bound holds regardless of the number of rounds and local memory size, while for balanced permutation and sorting the upper bound is achieved in a constant number of rounds with load at most $n/p+o(n/p)$. For 0-1 sorting, we also derive a tight communication complexity of $C_{\mathrm{cut}}(p) \cdot n \pm o(n)$, where $C_{\mathrm{cut}}(p)$ is the optimal value of a linear program parameterized by $p$. Although there appears to be no closed-form for $C_{\mathrm{cut}}(p)$, we show that it is strictly smaller than $2-2/p$ for all $p\ge 3$, thus separating 0-1 sorting from general sorting in terms of their communication complexity.

Friday, October 30 2026, 00:00

From Trees to Binary Trees: Approximation Algorithms and a PTAS