Theory of Computing Report

Authors: Qisheng Wang

In this paper, we present a unified framework for proving lower bounds for estimating functionals of quantum states. We therefore resolve several open problems by establishing lower bounds that match known upper bounds: we show that it requires $\widetildeΩ(N^2)$ samples to estimate the Uhlmann fidelity, trace distance, and von Neumann entropy. Moreover, they immediately imply matching query lower bounds of $\widetildeΩ(N)$ by quantum sample-to-query lifting. These lower bounds imply the near-optimality of a dozen quantum algorithms since 2016.

Authors: Qisheng Wang

In this paper, we present a unified framework for proving lower bounds for estimating functionals of quantum states. We therefore resolve several open problems by establishing lower bounds that match known upper bounds: we show that it requires $\widetildeΩ(N^2)$ samples to estimate the Uhlmann fidelity, trace distance, and von Neumann entropy. Moreover, they immediately imply matching query lower bounds of $\widetildeΩ(N)$ by quantum sample-to-query lifting. These lower bounds imply the near-optimality of a dozen quantum algorithms since 2016.

Tuesday, August 04 2026, 00:00

An Argmax Principle for Sum-of-Squares Relaxations on the Sphere

Authors: Fernando Jeronimo Granha, Pei Wu, Haochen Xu

We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as $Φ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-$O(\sqrt{n/ε})$ SoS analysis for approximating $h_{\mathrm{sep}}(P)$ in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from $\mathrm{QMA}(2)$ protocols. Second, for the matrix $2\to4$ norm, degree-$O(\sqrt n/ε)$ SoS gives a multiplicative $(1+ε)$ approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of $p\to q$ norms with even $q$. Finally, for degree-$d$ polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-$k$ SoS gives approximation ratio $O_d((n/k)^{d/2-1})$. The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.

Authors: Fernando Jeronimo Granha, Pei Wu, Haochen Xu

We develop an argmax principle for analyzing sum-of-squares relaxations of optimization problems over the unit sphere. Given a feasible pseudo-expectation, we form a polynomial of high-order pseudo-moments, such as $Φ_k(u)=\widetilde{\mathbb E}\langle x,u\rangle^{2k}$. Our guiding principle is that its maximizers are rounding candidates: their local and global optimality conditions reveal the reweighed pseudo-expectation inequalities governing SoS convergence. This viewpoint unifies several problems previously analyzed by rather different techniques. We obtain three results. First, for Best Separable State, we give a degree-$O(\sqrt{n/ε})$ SoS analysis for approximating $h_{\mathrm{sep}}(P)$ in the perfect-completeness regime, improving and simplifying Barak, Kothari and Steurer (STOC'17). The dependence is essentially tight for inverse-linear gap under the Exponential-Time Hypothesis, matching hardness from $\mathrm{QMA}(2)$ protocols. Second, for the matrix $2\to4$ norm, degree-$O(\sqrt n/ε)$ SoS gives a multiplicative $(1+ε)$ approximation. Barak et al. (STOC'12) previously gave a comparable-time constant-gap decision algorithm; our result gives a multiplicative guarantee and extends to a family of $p\to q$ norms with even $q$. Finally, for degree-$d$ polynomial optimization, we recover the convergence theorem of Bhattiprolu et al. (FOCS'17) with a shorter, more direct proof: degree-$k$ SoS gives approximation ratio $O_d((n/k)^{d/2-1})$. The paper introduces no new relaxation. Instead, the high-moment argmax gives a common way to read an SoS solution, unifying previously separate convergence analyses and yielding sharper bounds or simpler proofs.

Tuesday, August 04 2026, 00:00

Optimal Quantum de Finetti Theorems via Argmax Rounding

Authors: Fernando Granha Jeronimo, Pei Wu, Haochen Xu

We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state $ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))$, there is a probability measure $ν$ on the unit sphere such that \[ \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. \] By purification, the bosonic theorem also gives the optimal $O(d/N)$ upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026). More generally, $t$-site marginals satisfy $O(t\sqrt d/N)$ bosonic and $O(td/N)$ permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed $\varepsilon\in(0,1)$, we construct a channel with input dimension $D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))$ whose outputs are $\varepsilon$-close to separable states of local dimension $d$ and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic $\exp(\widetilde O(\sqrt d/\varepsilon))$-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing. Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate $Θ(N^{-1/2})$ when the dimension may grow.

Authors: Fernando Granha Jeronimo, Pei Wu, Haochen Xu

We prove optimal finite quantum de Finetti upper bounds. Given a bosonic state $ρ_N\in D(\mathrm{Sym}^N(\mathbb C^d))$, there is a probability measure $ν$ on the unit sphere such that \[ \left\| ρ_N^{(2)}-\int |u\rangle\langle u|^{\otimes 2}\,dν(u) \right\|_1 \le \frac{\sqrt{d-1}}{N-1}. \] By purification, the bosonic theorem also gives the optimal $O(d/N)$ upper bound for arbitrary exchangeable states. These results settle the dimension dependence left open by Christandl, König, Mitchison, and Renner (CMP 2007). The proof casts de Finetti approximation as sum-of-squares rounding and applies the argmax method of Jeronimo, Wu, and Xu (manuscript 2026). More generally, $t$-site marginals satisfy $O(t\sqrt d/N)$ bosonic and $O(td/N)$ permutation-invariant bounds. Our proof formulates de Finetti approximation as the integrality gap of a symmetric-extension semidefinite program and rounds an optimum by the argmax principle. The sharp bounds have several consequences. For every fixed $\varepsilon\in(0,1)$, we construct a channel with input dimension $D=\exp(O_\varepsilon(\sqrt d\log d))=\exp(o(d))$ whose outputs are $\varepsilon$-close to separable states of local dimension $d$ and whose image contains every such separable state, thereby refuting Watrous's disentangler conjecture. We also obtain deterministic $\exp(\widetilde O(\sqrt d/\varepsilon))$-time algorithms for explicit Best Separable State without perfect completeness and for trace-distance separability testing. Finally, spectral truncation gives the first dimension-free bosonic de Finetti theorem in Hilbert--Schmidt distance, with the optimal rate $Θ(N^{-1/2})$ when the dimension may grow.

Tuesday, August 04 2026, 00:00

Optimal Unambiguous DNFs and Alon-Saks-Seymour

Authors: Chirag Pabbaraju

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, Göös, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $Ω(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

Authors: Chirag Pabbaraju

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, Göös, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $Ω(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.

Tuesday, August 04 2026, 00:00

The Complete Extended Euclidean Scheme Is Not in Piecewise Arithmetic $\mathrm{AC}^0$

Authors: Amiel Ferman

We prove that the complete extended Euclidean scheme for pairs of monic univariate polynomials over a field of characteristic zero cannot be computed by polynomial-size, constant-depth piecewise arithmetic circuits in the select-gate model of Andrews and Wigderson. In fact, the lower bound already holds for the simpler task of outputting the complete padded list of nonzero Euclidean remainders. We show that a suitable Hankel determinant can be recovered from fixed coordinates of the complete Euclidean remainder sequence on a nonempty Zariski-open set. The connection is provided by a middle principal subresultant coefficient. A generic removal of select gates, followed by constant-depth division elimination, would therefore turn any piecewise constant-depth algorithm for the complete remainder sequence into an ordinary constant-depth circuit for Hankel determinants, contradicting the lower bound above. We also show that the same obstruction applies to several related outputs. It yields lower bounds for the complete polynomial continued-fraction expansion and for the complete profile of fixed-bound principal subresultant coefficients, since each of these outputs directly exposes the Hankel determinant used in the Euclidean reduction. In addition, we obtain a lower bound for normalized subdiagonal Pad'e approximation: even the normalized denominator alone suffices, through polynomially many parallel Pad'e computations and a telescoping product of determinantal ratios, to recover the same consecutive Hankel determinant. Consequently, none of these problems can be computed by polynomial-size, constant-depth piecewise arithmetic circuits.

Authors: Amiel Ferman

We prove that the complete extended Euclidean scheme for pairs of monic univariate polynomials over a field of characteristic zero cannot be computed by polynomial-size, constant-depth piecewise arithmetic circuits in the select-gate model of Andrews and Wigderson. In fact, the lower bound already holds for the simpler task of outputting the complete padded list of nonzero Euclidean remainders. We show that a suitable Hankel determinant can be recovered from fixed coordinates of the complete Euclidean remainder sequence on a nonempty Zariski-open set. The connection is provided by a middle principal subresultant coefficient. A generic removal of select gates, followed by constant-depth division elimination, would therefore turn any piecewise constant-depth algorithm for the complete remainder sequence into an ordinary constant-depth circuit for Hankel determinants, contradicting the lower bound above. We also show that the same obstruction applies to several related outputs. It yields lower bounds for the complete polynomial continued-fraction expansion and for the complete profile of fixed-bound principal subresultant coefficients, since each of these outputs directly exposes the Hankel determinant used in the Euclidean reduction. In addition, we obtain a lower bound for normalized subdiagonal Pad'e approximation: even the normalized denominator alone suffices, through polynomially many parallel Pad'e computations and a telescoping product of determinantal ratios, to recover the same consecutive Hankel determinant. Consequently, none of these problems can be computed by polynomial-size, constant-depth piecewise arithmetic circuits.

Tuesday, August 04 2026, 00:00

Alphabet-Preserving Lifting for the Log-Rank Conjecture

Authors: Zhao Song

For a Boolean communication matrix $M$, let $D(M)$ denote its deterministic communication complexity and let $r(M):={\mathrm{rank}}_{\mathbb{R}}(M)$. The log-rank conjecture asks whether $D(M)$ is polynomial in $\log r(M)$. The best known general upper bound, due to Sudakov and Tomon'25, is $D(M)=O(\sqrt{r(M)})$. On the lower-bound side, G{ö}{ö}s, Pitassi, and Watson'18 constructed explicit matrices satisfying $D(M)=Ω((\log r(M))^2/(\log\log r(M))^2)$. We improve the lower bound to $D(M)=Ω((\log r(M))^2/\log\log r(M))$. Our construction revisits their pointer function over its original non-Boolean alphabet and lifts it with an alphabet-valued Index gadget, via the multicolor simulation theorem stated by Roughgarden and Weinstein'16. Compared with the quantitatively explicit GPW bound, the alphabet-preserving lift removes one factor of $\log\log r$. We also give a self-contained proof of the multicolor simulation theorem in the parameter regime required by the construction.

Authors: Zhao Song

For a Boolean communication matrix $M$, let $D(M)$ denote its deterministic communication complexity and let $r(M):={\mathrm{rank}}_{\mathbb{R}}(M)$. The log-rank conjecture asks whether $D(M)$ is polynomial in $\log r(M)$. The best known general upper bound, due to Sudakov and Tomon'25, is $D(M)=O(\sqrt{r(M)})$. On the lower-bound side, G{ö}{ö}s, Pitassi, and Watson'18 constructed explicit matrices satisfying $D(M)=Ω((\log r(M))^2/(\log\log r(M))^2)$. We improve the lower bound to $D(M)=Ω((\log r(M))^2/\log\log r(M))$. Our construction revisits their pointer function over its original non-Boolean alphabet and lifts it with an alphabet-valued Index gadget, via the multicolor simulation theorem stated by Roughgarden and Weinstein'16. Compared with the quantitatively explicit GPW bound, the alphabet-preserving lift removes one factor of $\log\log r$. We also give a self-contained proof of the multicolor simulation theorem in the parameter regime required by the construction.

Tuesday, August 04 2026, 00:00

Symmetric Numerical Three-Dimensional Matching: Intractability and Inapproximability

Authors: Zhi-Long Chen, Nicholas G. Hall

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

Authors: Zhi-Long Chen, Nicholas G. Hall

Symmetric Numerical Three-Dimensional Matching (SN3DM) asks whether three disjoint labeled classes with identical weight multisets can be partitioned into class-transversal triples of one common target sum. Its theme is role recovery under marginal symmetry: identical numerical catalogues force the asymmetric source roles to be reconstructed from incidence structure alone. This tutorial develops three complementary hardness results for that symmetry restriction. Part I gives a unary-polynomial reduction from N3DM. Source roles become ports in one common occurrence set, a uniquely forced filler system reserves one main incidence per port, bipartite edge coloring restores the output-class labels, and a no-carry mixed-radix encoding packs four coordinates into positive integers. Hence SN3DM is strongly NP-complete. Part II studies Max-SN3DM, for which strong NP-hardness alone does not exclude a PTAS. Two numerical compilers lift Petrank's perfect-completeness gap for bounded 3DM to unary Max-N3DM, and a defect-stability lemma shows that a symmetric matching of size 13n - d yields a source matching of size at least n - 21d, where n is the multiset cardinality, and d is a symmetric defect. Hence, for some epsilon > 0, it is NP-hard to separate perfect instances from those of optimum at most (1- epsilon) times perfect, so no PTAS exists unless P = NP. Every maximal legal triple matching is a 3-approximation, placing the problem in APX. Part III supplies the approximation-preserving reduction Part II does not claim. An exact pair compiler and a one-live-port separation map degree-three Maximum 3DM to unary Max-SN3DM with OPT(Max-SN3DM) = Gamma + OPT(Max-3DM) for a fixed offset Gamma and one-for-one optimum-error transfer. The L-reduction has constants alpha = 764 and beta = 1, so Max-SN3DM is APX-complete. The two are incomparable; worked yes / no instances audit each construction.

Tuesday, August 04 2026, 00:00

Geometry-Informed Polynomial Time Quantum Approximation Schemes for Constrained Optimisation

Authors: Chinonso Onah, Kristel Michielsen

When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.

Authors: Chinonso Onah, Kristel Michielsen

When does a noisy quantum sampler yield an end-to-end polynomial-time optimization algorithm with performance guarantees? Building on finite-depth and finite-shot guarantees for Constraint-Enhanced QAOA, we show that inverse-polynomial ideal probability on the optimal set, together with independent sampling, polynomial-time feasibility repair, and scoring, produces an exact-hit fully polynomial randomized approximation scheme, which we call an FPRASq. This guarantee survives device noise within an instance-dependent window. For effective circuit depth linear in the product of layer count and problem size, preserving an inverse-depth fraction of the ideal optimal mass increases the required shot complexity by one power of the problem size. Beyond this window, deterministic repair guarantees feasibility and provides an instance-dependent approximation guarantee whenever the induced objective inflation is controlled. The resulting NP-HQ algorithm fits the Chen-Cotler-Huang-Li oracle model. On any NP-hard kernel-admissible promise family, reproducing its inverse-polynomial optimal overlap with a polynomial-time classical sampler would imply that NP is contained in BPP, even with identical repair and perfect access to the constraint structure. Thus, the separation lies in generating the sampling distribution. We further introduce Heavy-Hitter QAOA, which preserves these conditional guarantees while reducing the retained candidate set and classical post-processing cost by one power of the problem size. Hardware experiments on IBM Eagle r3 processors cover instances with up to one hundred logical variables and match or improve every tested QOptlib reference tour.

Tuesday, August 04 2026, 00:00

Block Sensitivity can exceed Spectral Sensitivity Squared

Authors: Alexander Meiburg

The spectral sensitivity $λ(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(λ(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\ge 14011$ and $λ(f)\le 89.0162$, so that $bs(f)\geλ(f)^{2.127}$, and hence by composition a family with $λ(f_n)\to\infty$ and $bs(f_n)=Ω(λ(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.

Authors: Alexander Meiburg

The spectral sensitivity $λ(f)$ of a Boolean function is the largest eigenvalue of the adjacency matrix of its sensitivity graph. It lower-bounds every standard measure of query complexity, and Aaronson, Ben-David, Kothari, Rao and Tal, who introduced it, asked whether block sensitivity is at most quadratic in it: is $bs(f)=O(λ(f)^{2})$? We show that it is not. We construct a total Boolean function on $2017584$ variables with $bs(f)\ge 14011$ and $λ(f)\le 89.0162$, so that $bs(f)\geλ(f)^{2.127}$, and hence by composition a family with $λ(f_n)\to\infty$ and $bs(f_n)=Ω(λ(f_n)^{2.127})$. The function is the indicator of a union of $k$ subcubes indexed by the vertices of a doubly regular tournament, and the freedom left in the construction is fixed by the Lovász local lemma. The main result has been formally verified in Lean. We also give numerical evidence that a member of the same family on $1255$ variables reaches an exponent near $2.20$, and exhibit a member on $30$ variables whose exponent already exceeds $2$ and whose spectral sensitivity can be computed exactly.

Tuesday, August 04 2026, 00:00

Sharp Hardness for MAX-3-CUT and Quantum MAX-CUT

Authors: Steven Heilman

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $α_3+ε$ for every $ε>0$, where $α_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $α_{\rm BOV}+ε$ for every $ε>0$, where $α_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

Authors: Steven Heilman

Assuming the Unique Games Conjecture, we show it is NP-hard to approximate MAX-3-CUT within a multiplicative factor of $α_3+ε$ for every $ε>0$, where $α_3\approx.83600811464$ is the approximation ratio of Frieze-Jerrum's polynomial-time algorithm from 1995. That is, we prove sharp hardness of approximation for MAX-3-CUT. This result resolves a conjecture of Khot-Kindler-Mossel-O'Donnell from 2004 by proving the three candidate Plurality is Stablest Conjecture for correlations in $[-1/2,2/5]$ and generalizes the Majority is Stablest Theorem of Mossel-O'Donnell-Oleszkiewicz [Annals of Math, 2010]. With a similar strategy we prove: assuming the Unique Games Conjecture, it is NP-hard to approximate the product-state value of Quantum MAX-CUT within a multiplicative factor of $α_{\rm BOV}+ε$ for every $ε>0$, where $α_{\rm BOV}\approx 0.9563372685$ is the approximation ratio of the Briët-de Oliveira Filho-Vallentin algorithm. This sharp hardness result completes the conjectured hardness of Hwang-Neeman-Parekh-Thompson-Wright from 2021 by proving their $S^{k-1}$-valued Borell inequality for correlations in $[-.5843,.5843]$ for all $k\geq3$.

Tuesday, August 04 2026, 00:00

On the Complexity of Computing Outputs of a Metric Turing Machine

Authors: Yaroslav Ivanashev

The classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ contain functions that compute the median solution for certain types of problems. In this paper, for these classes we introduce analogous classes of functions that compute the k-th solution, where k is an order function that depends on the input. We prove that the classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ are polynomial-time 1-Turing inter-reducible with the corresponding classes, where the order function is from FP or FP$^{\text{#P}}$. For MedP we also prove that it coincides with the corresponding classes, where the order function is from FP or #P. For several inclusions between function classes we give equivalent inclusions between language classes. In particular, we establish inclusion relations between MaxP and median classes MidP, MedP, and $\small{\overline{\text{MedP}}}$. We also prove that NPSV$_{\text{t}} \subseteq$ MaxP $\subseteq$ FP$^{\text{NP}}$ and both inclusions are proper if and only if NP $\neq$ coNP.

Authors: Yaroslav Ivanashev

The classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ contain functions that compute the median solution for certain types of problems. In this paper, for these classes we introduce analogous classes of functions that compute the k-th solution, where k is an order function that depends on the input. We prove that the classes MidP, MedP, and $\small{\overline{\text{MedP}}}$ are polynomial-time 1-Turing inter-reducible with the corresponding classes, where the order function is from FP or FP$^{\text{#P}}$. For MedP we also prove that it coincides with the corresponding classes, where the order function is from FP or #P. For several inclusions between function classes we give equivalent inclusions between language classes. In particular, we establish inclusion relations between MaxP and median classes MidP, MedP, and $\small{\overline{\text{MedP}}}$. We also prove that NPSV$_{\text{t}} \subseteq$ MaxP $\subseteq$ FP$^{\text{NP}}$ and both inclusions are proper if and only if NP $\neq$ coNP.

Tuesday, August 04 2026, 00:00

No low-degree tests for quantum states

Authors: Omar Alrabiah, Srinivasan Arunachalam, Sabee Grewal, John Wright

We study the problem of testing low-degree phase states, namely m-qudit quantum states of the form $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} ω^{f(x)} |x>$, where $f$ is a degree-$d$ polynomial. In contrast to the classical setting, where low-degree polynomials admit highly efficient classical testers, it is not known whether analogous quantum tests exist. We show that no such quantum low-degree test exists: any tester requires $Ω(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ copies to determine whether a given state is a degree-$d$ phase state or is far from every such state. Our results follow from a general framework that relates quantum testing of codeword phase states to classical decoding properties of the dual code, which allows us to leverage known bounds on the tolerance of high-rate Reed--Muller codes to random errors.

Authors: Omar Alrabiah, Srinivasan Arunachalam, Sabee Grewal, John Wright

We study the problem of testing low-degree phase states, namely m-qudit quantum states of the form $q^{-m/2} \sum_{x \in \mathbb{F}_q^m} ω^{f(x)} |x>$, where $f$ is a degree-$d$ polynomial. In contrast to the classical setting, where low-degree polynomials admit highly efficient classical testers, it is not known whether analogous quantum tests exist. We show that no such quantum low-degree test exists: any tester requires $Ω(\binom{\lfloor m/2\rfloor}{\lfloor (d-1)/2 \rfloor})$ copies to determine whether a given state is a degree-$d$ phase state or is far from every such state. Our results follow from a general framework that relates quantum testing of codeword phase states to classical decoding properties of the dual code, which allows us to leverage known bounds on the tolerance of high-rate Reed--Muller codes to random errors.

Tuesday, August 04 2026, 00:00

Algorithms for Connectivity Maintenance and Barrier Coverage on a Closed Cycle

Authors: Shimin Li

The problem of maintaining connectivity of a wireless network on a closed cycle is studied in this paper. In the initial input, we have $n$ points located on a closed cycle. The points can move along the cycle, and if the distance between two points is at most a given value $r$, we say these two points are connected. The goal of the problem is to move the points along the cycle such that any adjacent pair of points is directly connected--i.e., there exist two paths between them in opposite directions along the cycle--while minimizing the maximum movement over all points. This problem is motivated by applications in mobile wireless networks, including sensors, vehicles, and satellites operating on closed orbits. It is also applicable to barrier or border coverage problems, where sensors are deployed along a closed boundary and coverage is achieved through repositioning along the cycle. We present a linear time optimal algorithm for this problem. Then we refine the algorithm to obtain a lexicographically optimal solution without increasing the time complexity.

Authors: Shimin Li

The problem of maintaining connectivity of a wireless network on a closed cycle is studied in this paper. In the initial input, we have $n$ points located on a closed cycle. The points can move along the cycle, and if the distance between two points is at most a given value $r$, we say these two points are connected. The goal of the problem is to move the points along the cycle such that any adjacent pair of points is directly connected--i.e., there exist two paths between them in opposite directions along the cycle--while minimizing the maximum movement over all points. This problem is motivated by applications in mobile wireless networks, including sensors, vehicles, and satellites operating on closed orbits. It is also applicable to barrier or border coverage problems, where sensors are deployed along a closed boundary and coverage is achieved through repositioning along the cycle. We present a linear time optimal algorithm for this problem. Then we refine the algorithm to obtain a lexicographically optimal solution without increasing the time complexity.

Tuesday, August 04 2026, 00:00

Computational Thresholds for Balanced and Fixed-Slice Independent Sets in Bipartite Graphs

Authors: Ijay Narang, Will Perkins, Yuzhou Wang, Timothy L. H. Wee

Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $Δ$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=α_L n$ and $|I\cap R|=α_R n$. We show that if the overall density $α$ lies in the interval $(\frac{1}Δ, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $Δ$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $λ>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $λ$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $Δ\ge 3$, if $λ<λ_c(Δ)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $λ>λ_c(Δ)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.

Authors: Ijay Narang, Will Perkins, Yuzhou Wang, Timothy L. H. Wee

Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $Δ$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=α_L n$ and $|I\cap R|=α_R n$. We show that if the overall density $α$ lies in the interval $(\frac{1}Δ, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $Δ$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $λ>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $λ$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $Δ\ge 3$, if $λ<λ_c(Δ)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $λ>λ_c(Δ)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.

Tuesday, August 04 2026, 00:00

A quantitative container characterization of one-sided testability

Authors: Gaia Carenini, Cameron Seth, Yuichi Yoshida

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in modern combinatorics. Combining this equivalence with the Alon-Shapira notion of semi-hereditariness yields a quantitative characterization of arbitrary graph properties. The correspondence is effective in both directions and provides explicit translations between tester complexity and container parameters. Our proof is regularity-free and extends uniformly to every fixed finite relational signature of bounded arity, including digraphs, coloured graphs, and hypergraphs. As applications, we obtain quantitative closure results for partition properties and testers for properties defined by the existence of a linearly large induced substructure.

Authors: Gaia Carenini, Cameron Seth, Yuichi Yoshida

We give a quantitative combinatorial characterization of size-oblivious one-sided testability in the dense graph model, resolving a question of Alon, Fischer, Newman, and Shapira. For hereditary graph properties, we prove that one-sided testability is quantitatively equivalent to the existence of suitable hypergraph containers, a central and widely used tool in modern combinatorics. Combining this equivalence with the Alon-Shapira notion of semi-hereditariness yields a quantitative characterization of arbitrary graph properties. The correspondence is effective in both directions and provides explicit translations between tester complexity and container parameters. Our proof is regularity-free and extends uniformly to every fixed finite relational signature of bounded arity, including digraphs, coloured graphs, and hypergraphs. As applications, we obtain quantitative closure results for partition properties and testers for properties defined by the existence of a linearly large induced substructure.

Tuesday, August 04 2026, 00:00

A Subsampling Theorem for Constraint Satisfaction Problems with Large Arity

Authors: Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Gabriele Farina, Giulio Malavolta

Subsampling theorems for constraint satisfaction problems (CSPs) guarantee that the value of the CSP is approximately preserved after restricting it to small random subsets of variables. We provide the first subsampling theorem for CSPs, which requires a sample size that is polynomial in the arity $k$ and error $\varepsilon$, and polylogarithmic in the alphabet size $q$. This improves upon the subsampling theorem of Barak, Hardt, Holenstein, and Steurer (SODA '11), which achieves a polynomial dependency on $\varepsilon$ and polylogarithmic in $q$ only in the constant-arity regime. Our subsampling theorem has applications in interactive proofs and property testing. In interactive proofs, it provides a key missing ingredient for the proof of Aaronson, Impagliazzo, and Moshkovitz (CCC '14) that $\textsf{AM}(\textsf{poly})=\textsf{AM}$ (where $\textsf{AM}(k)$ is the class of languages decidable by Arthur-Merlin protocols with $k$ non-communicating Merlins with independent questions). In property testing, it yields the first one-sided tester for satisfiability with sample size polynomial in the arity $k$ and the error $\varepsilon^{-1}$, and polylogarithmic in the alphabet size $q$.

Authors: Martino Bernasconi, Matteo Castiglioni, Andrea Celli, Gabriele Farina, Giulio Malavolta

Subsampling theorems for constraint satisfaction problems (CSPs) guarantee that the value of the CSP is approximately preserved after restricting it to small random subsets of variables. We provide the first subsampling theorem for CSPs, which requires a sample size that is polynomial in the arity $k$ and error $\varepsilon$, and polylogarithmic in the alphabet size $q$. This improves upon the subsampling theorem of Barak, Hardt, Holenstein, and Steurer (SODA '11), which achieves a polynomial dependency on $\varepsilon$ and polylogarithmic in $q$ only in the constant-arity regime. Our subsampling theorem has applications in interactive proofs and property testing. In interactive proofs, it provides a key missing ingredient for the proof of Aaronson, Impagliazzo, and Moshkovitz (CCC '14) that $\textsf{AM}(\textsf{poly})=\textsf{AM}$ (where $\textsf{AM}(k)$ is the class of languages decidable by Arthur-Merlin protocols with $k$ non-communicating Merlins with independent questions). In property testing, it yields the first one-sided tester for satisfiability with sample size polynomial in the arity $k$ and the error $\varepsilon^{-1}$, and polylogarithmic in the alphabet size $q$.

Tuesday, August 04 2026, 00:00

Exact Algorithms for Minimum Steiner Point Trees

Authors: Eungyu Woo, Donghoon Shin

Given distinct terminals $P\subset R^2$ and $R>0$, the Steiner tree problem with minimum number of Steiner points and bounded edge length asks for a straight line tree spanning $P$, with every edge of length at most $R$, that minimizes the number of Steiner points. Length is measured in a fixed $L_p$ metric with $p\in Q_{\ge 1}\cup\{\infty\}$. The optimum $k$ is not bounded by $n$, even in two-terminal case. We give a deterministic exact algorithm that computes an optimal implicit representation in $n^{O(n)}$ time, independent of $k$, in the computation model of Section~\ref{subseccomputation}. The representation consists of a full Steiner topology, exact branch coordinates, and a segment count for each topology edge. Subdivision requires additional time $Θ(n+k)$. For each full Steiner topology, the feasible segment count vectors are the integer points of a convex projection in $O(n)$ dimensions. A continuous relaxation restricts the integer optimum to $2n-3$ consecutive values. Exact semialgebraic routines and a flatness recursion in integral lattice coordinates decide these values. Together with the parameterized bottleneck algorithm of Bandyapadhyay et al., this gives the value bound $\min\{n^{O(n)}, k^{O(k)}n^{O(1)}\}$ for every fixed metric considered here.

Authors: Eungyu Woo, Donghoon Shin

Given distinct terminals $P\subset R^2$ and $R>0$, the Steiner tree problem with minimum number of Steiner points and bounded edge length asks for a straight line tree spanning $P$, with every edge of length at most $R$, that minimizes the number of Steiner points. Length is measured in a fixed $L_p$ metric with $p\in Q_{\ge 1}\cup\{\infty\}$. The optimum $k$ is not bounded by $n$, even in two-terminal case. We give a deterministic exact algorithm that computes an optimal implicit representation in $n^{O(n)}$ time, independent of $k$, in the computation model of Section~\ref{subseccomputation}. The representation consists of a full Steiner topology, exact branch coordinates, and a segment count for each topology edge. Subdivision requires additional time $Θ(n+k)$. For each full Steiner topology, the feasible segment count vectors are the integer points of a convex projection in $O(n)$ dimensions. A continuous relaxation restricts the integer optimum to $2n-3$ consecutive values. Exact semialgebraic routines and a flatness recursion in integral lattice coordinates decide these values. Together with the parameterized bottleneck algorithm of Bandyapadhyay et al., this gives the value bound $\min\{n^{O(n)}, k^{O(k)}n^{O(1)}\}$ for every fixed metric considered here.

Tuesday, August 04 2026, 00:00

The Condition-Number Barrier in Sparse Least Squares

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

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right), \] where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

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

In [AS21], Axiotis and Sviridenko conjectured that the linear dependence on the restricted condition number in sparse convex optimization cannot be improved by a polynomial-time algorithm. We establish their conjectured lower bound for least-squares objectives, conditional on the randomized exact-volume Small-Set Expansion Hypothesis in the weighted regular-graph formulation of Raghavendra, Steurer, and Tulsiani [RST12]. Concretely, for every fixed $γ\in(0,1]$, there is no randomized polynomial-time algorithm that, with probability at least $2/3$, returns a vector $x$ such that, writing $s=\lVert x\rVert_0$, \[ \lVert Ax-b\rVert_2^2 \leq \min_{\lVert z\rVert_0\leq k}\lVert Az-b\rVert_2^2+\varepsilon \quad\text{and}\quad s=O\!\left(k\,κ_{s+k}^{\,1-γ}\right), \] where $κ_r$ is the restricted condition number at sparsity level $r$. The result holds even on rational instances with $A$ of full column rank. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

Tuesday, August 04 2026, 00:00

Pairwise-Independent Dithering for Single-Stage Hadamard Quantization

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

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3π\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

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

Quantizing high-dimensional vectors is fundamental to similarity search, distributed learning, and model compression. Feng, Indyk, Kapralov, Krachun, and Prokhorov established sharp guarantees for an unbiased dithered quantizer based on a randomized Hadamard transform [FIK+26]. Their $1/d$-scale inner-product estimator, however, uses a second randomized transform and residual quantization, increasing both communication and the leading constant in the proved bound. We show that this extra stage is unnecessary: pairwise-independent dithers across Hadamard coordinates suffice. The resulting unbiased single-stage estimator uses $b$ bits per coordinate and achieves \[ \mathbb{E}\!\left[ \left|\left\langle y,\widehat{x}-x\right\rangle\right|^2 \right] \leq \left(\frac{3π\sqrt{3}}{2}+o(1)\right) \frac{\lVert y\rVert_2^2}{d\,4^b}, \] as $b\to\infty$, with a dimension-free $o(1)$ term uniform over unit inputs and fixed queries. Compared with the two-stage construction of Feng et al., it eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$. The proof was first obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors have verified the proof and edited it for clarity of presentation.

Tuesday, August 04 2026, 00:00

Approximating two-terminal network reliability

Authors: Weiming Feng, Yucheng Fu, Heng Guo

We present a fully polynomial-time randomised approximation scheme (FPRAS) for the two-terminal reliability problem on general graphs, both directed and undirected. We also show that the complementary unreliability question is \BIS-hard. The key idea of the algorithm was discovered by GPT-5.6 Sol Ultra.

Authors: Weiming Feng, Yucheng Fu, Heng Guo

We present a fully polynomial-time randomised approximation scheme (FPRAS) for the two-terminal reliability problem on general graphs, both directed and undirected. We also show that the complementary unreliability question is \BIS-hard. The key idea of the algorithm was discovered by GPT-5.6 Sol Ultra.

Tuesday, August 04 2026, 00:00

Solving the Shortest Vector Problem in time $2^{0.6039n}$ Time via Mid-point Hessian

Authors: Minki Hhan

We present randomized algorithms for the shortest vector problem (SVP). For the $n$-dimensional lattice $\mathcal L$, our algorithms solve SVP in time $2^{0.6039n+o(n)}$ classically and $2^{0.5411n+o(n)}$ quantumly and space $2^{0.5n+o(n)}$, improving the previous best algorithm running in $2^{n+o(n)}$ time and space of Aggarwal, Dadush, Regev, and Stephens-Davidowitz [STOC'15]. Our algorithms heavily use the property of the Hessian of the periodic Gaussian function at the half shortest vector: For a shortest vector $v \in \mathcal L$, the Hessian at $v/2$ has the eigenvector close to $v$, which can be used to recover $v$ using the (preprocessing) bounded distance decoding algorithm. Given the periodicity modulo $\mathcal L$, the candidate midpoints are indexed by the parity classes in $\mathcal L/2\mathcal L$. Our algorithm searches for the class of a shortest vector by estimating the corresponding Hessians using discrete Gaussian samples. We optimize the algorithm using random sublattice cosets and various sampling technique, achieving the final complexity. The optimization techniques may be of independent interest.

Authors: Minki Hhan

We present randomized algorithms for the shortest vector problem (SVP). For the $n$-dimensional lattice $\mathcal L$, our algorithms solve SVP in time $2^{0.6039n+o(n)}$ classically and $2^{0.5411n+o(n)}$ quantumly and space $2^{0.5n+o(n)}$, improving the previous best algorithm running in $2^{n+o(n)}$ time and space of Aggarwal, Dadush, Regev, and Stephens-Davidowitz [STOC'15]. Our algorithms heavily use the property of the Hessian of the periodic Gaussian function at the half shortest vector: For a shortest vector $v \in \mathcal L$, the Hessian at $v/2$ has the eigenvector close to $v$, which can be used to recover $v$ using the (preprocessing) bounded distance decoding algorithm. Given the periodicity modulo $\mathcal L$, the candidate midpoints are indexed by the parity classes in $\mathcal L/2\mathcal L$. Our algorithm searches for the class of a shortest vector by estimating the corresponding Hessians using discrete Gaussian samples. We optimize the algorithm using random sublattice cosets and various sampling technique, achieving the final complexity. The optimization techniques may be of independent interest.

Tuesday, August 04 2026, 00:00

Recovering Planted Colorings in Sublinear Time

Authors: Weronika Wrzos-Kaminska

We give a sublinear algorithm for the planted $k$-coloring problem. Given an expander $G$ with a planted coloring, the goal is to efficiently determine the color class of a given vertex. We work in the adversarial planted coloring model of David and Feige [STOC 2016], where an adversary chooses a $d$-regular spectral $λ$-expander $G$ on $n$ vertices and plants a balanced $k$-coloring by partitioning the vertices into $k$ equal parts and deleting all edges within each part. This model generalizes the earlier random graph models studied by Blum and Spencer [J. Algorithms 1995] and Alon and Kahale [STOC 1994]. We give the first sublinear-time algorithm for recovering planted colorings in this model. The algorithm has preprocessing time and space $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, and produces a data structure that answers color queries in time $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, such that the resulting labeling agrees with the planted coloring on all but an $O(\sqrt{λ/d})$ fraction of vertices, up to a permutation of the $k$ colors. The algorithm gives sublinear-time inner product access to the bottom eigenspace of the normalized adjacency matrix, which allows us to adapt the classical spectral approach of Alon and Kahale in sublinear time.

Authors: Weronika Wrzos-Kaminska

We give a sublinear algorithm for the planted $k$-coloring problem. Given an expander $G$ with a planted coloring, the goal is to efficiently determine the color class of a given vertex. We work in the adversarial planted coloring model of David and Feige [STOC 2016], where an adversary chooses a $d$-regular spectral $λ$-expander $G$ on $n$ vertices and plants a balanced $k$-coloring by partitioning the vertices into $k$ equal parts and deleting all edges within each part. This model generalizes the earlier random graph models studied by Blum and Spencer [J. Algorithms 1995] and Alon and Kahale [STOC 1994]. We give the first sublinear-time algorithm for recovering planted colorings in this model. The algorithm has preprocessing time and space $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, and produces a data structure that answers color queries in time $\widetilde O\left(n^{1/2+O(1/\log(d/λ))}\right)$, such that the resulting labeling agrees with the planted coloring on all but an $O(\sqrt{λ/d})$ fraction of vertices, up to a permutation of the $k$ colors. The algorithm gives sublinear-time inner product access to the bottom eigenspace of the normalized adjacency matrix, which allows us to adapt the classical spectral approach of Alon and Kahale in sublinear time.

Tuesday, August 04 2026, 00:00

Noisy k-means++ is Not too Noisy

Authors: Poojan Shah

The celebrated $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) achieves an $O(\log k)$ expected approximation for the classical $k$-means problem using $D^2$-sampling, a technique now ubiquitous in clustering algorithm design. Bhattacharya et al. (ESA 2020) introduced $\varepsilon$-noisy $k$-means++, where sampling probabilities may incur an adversarial multiplicative error of $(1\pm\varepsilon)$, but obtained only an $O(\log^2 k)$ guarantee. Grunau et al. (ESA 2023) recovered the asymptotic $O(\log k)$ guarantee, but their analysis loses a constant factor of roughly $147{,}638$ even as $\varepsilon\to0$, leaving open whether $k$-means++ is highly sensitive to even a small amount of noise. They asked whether a bound within $1+O(\varepsilon)$ of the classical guarantee is possible. We resolve this affirmatively, proving an expected approximation guarantee of $8(\ln k+2)\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^4 = (1+O(\varepsilon))\,8(\ln k+2)$. We complement the upper bound with two separations. First, a noisy version of the Arthur and Vassilvitskii lower-bound instance incurs a $1+Ω(\varepsilon)$ loss over exact $k$-means++, so linear dependence on the noise is necessary. Second, pointwise multiplicative control is qualitatively essential: replacing it with per-round total variation closeness admits no finite approximation guarantee, even for $k=2$.

Authors: Poojan Shah

The celebrated $k$-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) achieves an $O(\log k)$ expected approximation for the classical $k$-means problem using $D^2$-sampling, a technique now ubiquitous in clustering algorithm design. Bhattacharya et al. (ESA 2020) introduced $\varepsilon$-noisy $k$-means++, where sampling probabilities may incur an adversarial multiplicative error of $(1\pm\varepsilon)$, but obtained only an $O(\log^2 k)$ guarantee. Grunau et al. (ESA 2023) recovered the asymptotic $O(\log k)$ guarantee, but their analysis loses a constant factor of roughly $147{,}638$ even as $\varepsilon\to0$, leaving open whether $k$-means++ is highly sensitive to even a small amount of noise. They asked whether a bound within $1+O(\varepsilon)$ of the classical guarantee is possible. We resolve this affirmatively, proving an expected approximation guarantee of $8(\ln k+2)\left(\frac{1+\varepsilon}{1-\varepsilon}\right)^4 = (1+O(\varepsilon))\,8(\ln k+2)$. We complement the upper bound with two separations. First, a noisy version of the Arthur and Vassilvitskii lower-bound instance incurs a $1+Ω(\varepsilon)$ loss over exact $k$-means++, so linear dependence on the noise is necessary. Second, pointwise multiplicative control is qualitatively essential: replacing it with per-round total variation closeness admits no finite approximation guarantee, even for $k=2$.

Tuesday, August 04 2026, 00:00

Reconfiguration of Temporal Networks under Reachability Constraints

Authors: Othon Michail, George Skretas, Georg Tennigkeit, Shaily Verma

Temporal networks model dynamic systems in which edges represent interactions and labels specify when these interactions occur. Examples include transportation networks, time-sensitive communication networks, and industrial control systems. In many such applications, an existing temporal network must be transformed into a desired one through a sequence of atomic modifications while maintaining essential functionality throughout the transformation. We formalize the time label reconfiguration problem and provide a theoretical framework for reasoning about such transformation processes. As temporal reachability is a central functionality in many temporal networks, we study a reconfiguration problem on directed temporal graphs subject to temporal reachability constraints. We are given a static graph with a designated set of sources, along with two labeling functions indicating an availability time for every edge. The goal is to transform one labeling into the other by changing the label of a single edge at a time while maintaining temporal reachability of the sources throughout. Our results reveal a sharp complexity transition: the problem is polynomial-time solvable for a single source but becomes PSPACE-hard with two sources. We also show that if the static graph is acyclic or an almost-tournament graph, then all valid labelings can be reconfigured into each other. Our proofs are constructive and yield polynomial-time algorithms.

Authors: Othon Michail, George Skretas, Georg Tennigkeit, Shaily Verma

Temporal networks model dynamic systems in which edges represent interactions and labels specify when these interactions occur. Examples include transportation networks, time-sensitive communication networks, and industrial control systems. In many such applications, an existing temporal network must be transformed into a desired one through a sequence of atomic modifications while maintaining essential functionality throughout the transformation. We formalize the time label reconfiguration problem and provide a theoretical framework for reasoning about such transformation processes. As temporal reachability is a central functionality in many temporal networks, we study a reconfiguration problem on directed temporal graphs subject to temporal reachability constraints. We are given a static graph with a designated set of sources, along with two labeling functions indicating an availability time for every edge. The goal is to transform one labeling into the other by changing the label of a single edge at a time while maintaining temporal reachability of the sources throughout. Our results reveal a sharp complexity transition: the problem is polynomial-time solvable for a single source but becomes PSPACE-hard with two sources. We also show that if the static graph is acyclic or an almost-tournament graph, then all valid labelings can be reconfigured into each other. Our proofs are constructive and yield polynomial-time algorithms.

Tuesday, August 04 2026, 00:00

Randomized Algorithms for Learning Partitions with Near Optimal Query Complexity in Constant Rounds

Authors: Deeparnab Chakrabarty, Aditi Dudeja, David Saulpic

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $Θ(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $Ω(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $Θ(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

Authors: Deeparnab Chakrabarty, Aditi Dudeja, David Saulpic

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $Θ(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $Ω(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $Θ(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

Tuesday, August 04 2026, 00:00

Minimal-to-Maximal Conversion Search Is Not Output-Polynomial

Authors: Bennet Hörmann, Martin Schirneck

The Transversal Hypergraph problem is to enumerate (list) all inclusion-wise minimal hitting sets of a given hypergraph $\mathcal{H}$. It is the most important open question in enumeration whether this problem admits an output-polynomial algorithm whose running time scales polynomially with the size of $\mathcal{H}$ and the number of solutions. Currently, Minimal-to-Maximal Conversion Search (MMCS) by Murakami and Uno [DAM 2014] is the most efficient algorithm for real-world instances, but there are no worst-case performance guarantees known for it. We prove that MMCS is in fact not output-polynomial. The lower bound construction motivates a more detailed analysis of how certain heuristic choices in the algorithm design affect the running time. We conduct a thorough analysis of those heuristics and, based on this, propose new extension. We then show in extensive running time experiments that this new heuristic further improves practical performance.

Authors: Bennet Hörmann, Martin Schirneck

The Transversal Hypergraph problem is to enumerate (list) all inclusion-wise minimal hitting sets of a given hypergraph $\mathcal{H}$. It is the most important open question in enumeration whether this problem admits an output-polynomial algorithm whose running time scales polynomially with the size of $\mathcal{H}$ and the number of solutions. Currently, Minimal-to-Maximal Conversion Search (MMCS) by Murakami and Uno [DAM 2014] is the most efficient algorithm for real-world instances, but there are no worst-case performance guarantees known for it. We prove that MMCS is in fact not output-polynomial. The lower bound construction motivates a more detailed analysis of how certain heuristic choices in the algorithm design affect the running time. We conduct a thorough analysis of those heuristics and, based on this, propose new extension. We then show in extensive running time experiments that this new heuristic further improves practical performance.

Tuesday, August 04 2026, 00:00

Approximating the Trace Distance Between Product Quantum States

Authors: Kun He, Dimitrios Myrisiotis, Junhong Nie, Zongqi Wan

We study the trace distance \[D_{\mathrm{tr}}(ρ,σ) =\frac12\|ρ-σ\|_1, ρ=\bigotimes_{i=1}^nρ_i,\quad σ=\bigotimes_{i=1}^nσ_i, \] when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is $\#\mathsf P$-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.

Authors: Kun He, Dimitrios Myrisiotis, Junhong Nie, Zongqi Wan

We study the trace distance \[D_{\mathrm{tr}}(ρ,σ) =\frac12\|ρ-σ\|_1, ρ=\bigotimes_{i=1}^nρ_i,\quad σ=\bigotimes_{i=1}^nσ_i, \] when the two exponentially large states are specified by their local factors. We give a deterministic approximation within a universal constant factor for rational product inputs. Its running time is polynomial in the number of factors, the local dimension, and the input bit length. In the opposite direction, exact computation is $\#\mathsf P$-hard even for diagonal qubit states, by the corresponding hardness of total variation distance between product distributions. The proof uses local Uhlmann-optimal purifications to reduce the problem to estimating the product-fidelity defect and the trace norm of a structured first-order operator. Although this operator acts on an exponentially large space, we approximate its trace norm by a local convex surrogate that admits a polynomial-size classical conic formulation. A square-function estimate shows that the surrogate upper-bounds this trace norm. Conversely, duality and local dephasing reduce the reverse comparison to a head--tail inequality for independent centered random variables, showing that the surrogate is at most a dimension-free constant times the same norm.

Tuesday, August 04 2026, 00:00

Online Algorithms via Minimax and Posterior Matching

Authors: Thomas Kesselheim, Marco Molinaro, Kalen Patton, Sahil Singla

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

Authors: Thomas Kesselheim, Marco Molinaro, Kalen Patton, Sahil Singla

Competitive analysis is central to the study of online algorithms, but upper bounds are often highly problem-specific. We develop a more unifying methodology via the minimax viewpoint. Guided by Yao's principle, we reduce worst-case competitive analysis to Bayesian online design under an arbitrary correlated prior over arrival sequences. For such a prior, let $X^*$ be the hindsight-optimal fractional solution for the realized instance, and let $X^{(t)}=\mathbb E[X^*\mid \mathcal F_t]$ be its posterior process. Our guiding rule is posterior matching: at each time $t$, choose the feasible online action that tracks the current posterior $X^{(t)}$ as closely as the online constraints permit. We show that this single principle yields optimal or near-optimal guarantees for several classical online fractional problems, including set cover, load balancing, matching and more general resource-allocation problems, recovering or improving state-of-the-art bounds in these settings with norm/concave objectives. Via known rounding reductions, it also yields randomized integral guarantees for weighted paging, MTS on star metrics, and ski-rental. At a technical level, our analysis reduces competitive guarantees to key probabilistic inequalities for the vector martingales generated by the posterior of the offline optimum. The resulting framework gives a reusable route from Bayesian online design under arbitrary correlated priors to information-theoretic worst-case competitive guarantees.

Tuesday, August 04 2026, 00:00

MEM-finding with run-length compressed strings

Authors: Paola Bonizzoni, Davide Cozzi, Travis Gagie, Younan Gao, Ragnar Groot Koerkamp

We show how to store a text $T [1..n]$ consisting of $ρ_T$ runs in $O (ρ_T + χ)$ space, where $χ$ is the size of the smallest suffixient set for $T$, such that when we are given a pattern $P [1..m]$ consisting of $ρ_P$ runs we can find the maximal exact matches (MEMs) of $P$ with respect to $T$ in $O (ρ_P \log m)$ time plus constant time for each edge we would fully or partially descend in the suffix tree for $T$ while finding those MEMs. We then adapt and optimize our result to finding set-maximal exact matches (SMEMs) of query haplotypes with respect to stored haplotype panels.

Authors: Paola Bonizzoni, Davide Cozzi, Travis Gagie, Younan Gao, Ragnar Groot Koerkamp

We show how to store a text $T [1..n]$ consisting of $ρ_T$ runs in $O (ρ_T + χ)$ space, where $χ$ is the size of the smallest suffixient set for $T$, such that when we are given a pattern $P [1..m]$ consisting of $ρ_P$ runs we can find the maximal exact matches (MEMs) of $P$ with respect to $T$ in $O (ρ_P \log m)$ time plus constant time for each edge we would fully or partially descend in the suffix tree for $T$ while finding those MEMs. We then adapt and optimize our result to finding set-maximal exact matches (SMEMs) of query haplotypes with respect to stored haplotype panels.

Tuesday, August 04 2026, 00:00

Context Compaction Theory

Authors: Hayder Tirmazi, Sam Markelon, Allison Bishop, Michael Mitzenmacher

Large Language Models (LLMs) have a bounded context window. The context window is the maximum input size an LLM can consume for a single inference. AI agents rely on a process called context compaction to fit their state within the context window when calling an LLM. Despite its ubiquity, context compaction has received essentially no formal analysis. In this paper, we initiate a formal study of context compaction. We first introduce a framework consisting of two games that capture the two algorithmic strategies for context compaction used by contemporary AI agents in practice. The Context Selection Game models context compaction algorithms that select a subset of an agent's accumulated state to retain. The Context Generation Game models context compaction algorithms that summarize an agent's state by an arbitrary message of bounded length. We then prove an equivalence between the Context Generation Game and one-way communication complexity. The minimum context compaction budget for answering a set of queries within a target error is equal to the one-way communication complexity of the induced communication problem at the same error. Known bounds from communication complexity therefore transfer directly to context compaction. We also show that the Context Selection Game corresponds to a restricted class of one-way communication protocols. Any gap between selection and generation is therefore a gap between two classes of communication protocols. We prove that there exists a set of queries for which generation needs strictly less budget than selection. The equivalence between the Context Generation Game and one-way communication also lets us measure how well a deployed context compaction algorithm performs on a query relative to the optimal strategy. As an example, we present a case study that evaluates Anthropic's context compaction endpoint on set membership queries.

Authors: Hayder Tirmazi, Sam Markelon, Allison Bishop, Michael Mitzenmacher

Large Language Models (LLMs) have a bounded context window. The context window is the maximum input size an LLM can consume for a single inference. AI agents rely on a process called context compaction to fit their state within the context window when calling an LLM. Despite its ubiquity, context compaction has received essentially no formal analysis. In this paper, we initiate a formal study of context compaction. We first introduce a framework consisting of two games that capture the two algorithmic strategies for context compaction used by contemporary AI agents in practice. The Context Selection Game models context compaction algorithms that select a subset of an agent's accumulated state to retain. The Context Generation Game models context compaction algorithms that summarize an agent's state by an arbitrary message of bounded length. We then prove an equivalence between the Context Generation Game and one-way communication complexity. The minimum context compaction budget for answering a set of queries within a target error is equal to the one-way communication complexity of the induced communication problem at the same error. Known bounds from communication complexity therefore transfer directly to context compaction. We also show that the Context Selection Game corresponds to a restricted class of one-way communication protocols. Any gap between selection and generation is therefore a gap between two classes of communication protocols. We prove that there exists a set of queries for which generation needs strictly less budget than selection. The equivalence between the Context Generation Game and one-way communication also lets us measure how well a deployed context compaction algorithm performs on a query relative to the optimal strategy. As an example, we present a case study that evaluates Anthropic's context compaction endpoint on set membership queries.

Tuesday, August 04 2026, 00:00

Dense Language Generation Made Simple: Deterministic, Randomized, and Multi-Order Algorithms

Authors: Ziyi Cai, Shuangping Li, Yiheng Shen, Kangning Wang, Peng Zhang

Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.

Authors: Ziyi Cai, Shuangping Li, Yiheng Shen, Kangning Wang, Peng Zhang

Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.

Tuesday, August 04 2026, 00:00

Active Regression for Single-Index Models with Unknown Link Functions

Authors: Chansophea Wathanak In, Yi Li, Wai Ming Tai, Xuan Wu

This paper studies active regression for single-index models under general $\ell_p$-loss with an unknown $1$-Lipschitz link function $f$, formulated as $\min_{f,x} \|f(Ax)-b\|_p^p$ with full access to $A$ but coordinate-query access to $b$. Prior work established upper bounds for known link functions for all $p\geq 1$ and for unknown link functions only in the $p=2$ case, together with lower bounds for $p\leq 2$. This work addresses the more challenging setting of unknown link functions and general $p \geq 1$. A non-adaptive sampling algorithm is presented that achieves a $(1+ε)$-approximation using $O(d^{p/2\vee 1}/ε^{p\vee 2}\operatorname{poly}\log(n/ε))$ queries. Nearly tight lower bounds are also established for $p>2$. These results close much of the remaining gap in active $\ell_p$-regression for single-index models.

Authors: Chansophea Wathanak In, Yi Li, Wai Ming Tai, Xuan Wu

This paper studies active regression for single-index models under general $\ell_p$-loss with an unknown $1$-Lipschitz link function $f$, formulated as $\min_{f,x} \|f(Ax)-b\|_p^p$ with full access to $A$ but coordinate-query access to $b$. Prior work established upper bounds for known link functions for all $p\geq 1$ and for unknown link functions only in the $p=2$ case, together with lower bounds for $p\leq 2$. This work addresses the more challenging setting of unknown link functions and general $p \geq 1$. A non-adaptive sampling algorithm is presented that achieves a $(1+ε)$-approximation using $O(d^{p/2\vee 1}/ε^{p\vee 2}\operatorname{poly}\log(n/ε))$ queries. Nearly tight lower bounds are also established for $p>2$. These results close much of the remaining gap in active $\ell_p$-regression for single-index models.

Tuesday, August 04 2026, 00:00

Testing Full Quartet Consistency: Adaptive Reconstruction, Random Verification, and Constant-Query Testability

Authors: Chuang-Chieh Lin

We study dense property testing for full systems of resolved quartet topologies on $n$ taxa: determining whether a system is induced by a phylogenetic tree or is $\varepsilon$-far from every tree-induced system. Our main result is an explicit polynomial-time adaptive one-sided-error tester. It reconstructs a candidate tree through anchored quartet queries and verifies the candidate using uniformly random quartet queries. With error probability $δ$, it uses $O\!\left(n\log n+\varepsilon^{-1}\log(1/δ)\right)$ queries. We also give a non-adaptive cached-anchor variant using $\binom{n-1}{3}+O\!\left(\varepsilon^{-1}\log(1/δ)\right)$ queries. Both improve the previous explicit $O(n^3/\varepsilon)$ query bound. Since the input contains $\binom{n}{4}=Θ(n^4)$ quartet entries, both testers use $o\!\left(\binom{n}{4}\right)$ queries for fixed~$\varepsilon$ and~$δ$. We additionally encode full quartet systems, equivariantly under relabeling, as directed, three-colored $4$-ary structures. Hereditary directed-hypergraph testing then yields an $n$-independent one-sided-error tester, although its dependence on $\varepsilon$ is quantitatively impractical. Finally, we prove lower bounds. In ordinary property testing, every adaptive randomized tester, even with two-sided error, requires asymptotically at least $\ln((1-δ)/δ)/\ln(1/(1-\varepsilon))$ queries as $n\to\infty$. Every one-sided-error tester requires $\ln(1/δ)/\ln(1/(1-\varepsilon))$ queries, matching the random-verification term up to rounding. For the stronger reconstruct-or-reject task, our upper bounds are optimal up to constant factors: the adaptive and non-adaptive complexities are $Θ(n\log n+\varepsilon^{-1}\log(1/δ))$ and $Θ(n^3+\varepsilon^{-1}\log(1/δ))$, respectively.

Authors: Chuang-Chieh Lin

We study dense property testing for full systems of resolved quartet topologies on $n$ taxa: determining whether a system is induced by a phylogenetic tree or is $\varepsilon$-far from every tree-induced system. Our main result is an explicit polynomial-time adaptive one-sided-error tester. It reconstructs a candidate tree through anchored quartet queries and verifies the candidate using uniformly random quartet queries. With error probability $δ$, it uses $O\!\left(n\log n+\varepsilon^{-1}\log(1/δ)\right)$ queries. We also give a non-adaptive cached-anchor variant using $\binom{n-1}{3}+O\!\left(\varepsilon^{-1}\log(1/δ)\right)$ queries. Both improve the previous explicit $O(n^3/\varepsilon)$ query bound. Since the input contains $\binom{n}{4}=Θ(n^4)$ quartet entries, both testers use $o\!\left(\binom{n}{4}\right)$ queries for fixed~$\varepsilon$ and~$δ$. We additionally encode full quartet systems, equivariantly under relabeling, as directed, three-colored $4$-ary structures. Hereditary directed-hypergraph testing then yields an $n$-independent one-sided-error tester, although its dependence on $\varepsilon$ is quantitatively impractical. Finally, we prove lower bounds. In ordinary property testing, every adaptive randomized tester, even with two-sided error, requires asymptotically at least $\ln((1-δ)/δ)/\ln(1/(1-\varepsilon))$ queries as $n\to\infty$. Every one-sided-error tester requires $\ln(1/δ)/\ln(1/(1-\varepsilon))$ queries, matching the random-verification term up to rounding. For the stronger reconstruct-or-reject task, our upper bounds are optimal up to constant factors: the adaptive and non-adaptive complexities are $Θ(n\log n+\varepsilon^{-1}\log(1/δ))$ and $Θ(n^3+\varepsilon^{-1}\log(1/δ))$, respectively.

Tuesday, August 04 2026, 00:00

A Consistency-Robustness Framework for Robust Optimization: Integrating Predictions into Robust Scheduling

Authors: Yasser Alghouass, Eric Balkanski, Nicole Megow, Vineet Goyal

Robust optimization protects against uncertainty by optimizing for the worst case over a prescribed uncertainty set. This protection can be overly conservative when forecasts, historical data, or learned predictions indicate a more likely scenario. We introduce a framework for robust optimization with predictions. The input consists of an uncertainty set together with a distinguished predicted scenario, and the goal is to compute a single solution that is both consistent, meaning near-optimal for the predicted scenario, and robust, meaning competitive with the classical min-max robust optimum. Unlike in standard learning-augmented algorithms, the prediction does not merely estimate the realized input; it creates a separate benchmark, the predicted optimum, which must be balanced against the min-max robust optimum. We study this framework for makespan scheduling with uncertain processing times and give a structural classification across standard uncertainty models and machine environments. For interval uncertainty, we obtain a smooth $(1+1/λ,1+λ)$ consistency-robustness tradeoff for restricted-assignment and related machines. Furthermore, we prove that unrelated machines admit no constant tradeoff. For budgeted uncertainty, we obtain a $(1+1/λ,2+λ)$ tradeoff for restricted assignment. Our analysis is based on a duality-based reduction to an interval-like upper envelope. We complement this with a lower bound showing that related machines admit no constant tradeoff even when only one job may deviate. For arbitrary uncertainty sets, we obtain constant tradeoffs for identical machines via a support-function block construction, and prove impossibility for restricted assignment. Our results show that the possibility of combining consistency and robustness in robust scheduling depends critically on the interaction between the uncertainty model and the machine environment.

Authors: Yasser Alghouass, Eric Balkanski, Nicole Megow, Vineet Goyal

Robust optimization protects against uncertainty by optimizing for the worst case over a prescribed uncertainty set. This protection can be overly conservative when forecasts, historical data, or learned predictions indicate a more likely scenario. We introduce a framework for robust optimization with predictions. The input consists of an uncertainty set together with a distinguished predicted scenario, and the goal is to compute a single solution that is both consistent, meaning near-optimal for the predicted scenario, and robust, meaning competitive with the classical min-max robust optimum. Unlike in standard learning-augmented algorithms, the prediction does not merely estimate the realized input; it creates a separate benchmark, the predicted optimum, which must be balanced against the min-max robust optimum. We study this framework for makespan scheduling with uncertain processing times and give a structural classification across standard uncertainty models and machine environments. For interval uncertainty, we obtain a smooth $(1+1/λ,1+λ)$ consistency-robustness tradeoff for restricted-assignment and related machines. Furthermore, we prove that unrelated machines admit no constant tradeoff. For budgeted uncertainty, we obtain a $(1+1/λ,2+λ)$ tradeoff for restricted assignment. Our analysis is based on a duality-based reduction to an interval-like upper envelope. We complement this with a lower bound showing that related machines admit no constant tradeoff even when only one job may deviate. For arbitrary uncertainty sets, we obtain constant tradeoffs for identical machines via a support-function block construction, and prove impossibility for restricted assignment. Our results show that the possibility of combining consistency and robustness in robust scheduling depends critically on the interaction between the uncertainty model and the machine environment.

Tuesday, August 04 2026, 00:00

Sublinear Time Eigenvector Approximation via Column Sampling

Authors: Rajarshi Bhattacharjee, Cameron Musco, Dominic Rutkowski

We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/ε^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $λ$ of $A$ with $|λ| \ge εn$, outputs an approximate eigenvector $v$ satisfying $\|Av - λv\|_2 \le εn$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/ε^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $ε\|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$'s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $ε\|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. '22] and [Swartworth and Woodruff '25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.

Authors: Rajarshi Bhattacharjee, Cameron Musco, Dominic Rutkowski

We study sublinear time sampling methods for approximating the outlying eigenvectors of large matrices. Our main result is an algorithm that uniformly samples just $\tilde{O}(\log n/ε^4)$ columns of a symmetric matrix $A \in \mathbb{R}^{n \times n}$ with entries bounded in magnitude by $1$, and, for any eigenvalue $λ$ of $A$ with $|λ| \ge εn$, outputs an approximate eigenvector $v$ satisfying $\|Av - λv\|_2 \le εn$. For approximating just the eigenvector of the largest magnitude eigenvalue, our algorithm samples only $\tilde{O}(\log n/ε^2)$ columns. Given the ability to sample rows and columns of $A$ proportional to their squared norms, we give a similar result with an improved error bound of $ε\|A\|_F$. For top eigenvector approximation, we show our bound is tight up to logarithmic terms. A key feature of our algorithms is that the output eigenvectors are spanned by a small number of $A$'s columns, and individual entries can be computed rapidly, in poly(log n, 1/epsilon) time per entry. This makes them applicable in the quantum-inspired algorithms framework of [Tang, STOC 2019], where we give the first sublinear time classical algorithms for eigenvector approximation with additive error $ε\|A\|_F$. Finally, we present an alternative approach, based on a truncated Nystrom method, that, while not allowing poly(log n, 1/epsilon) time entrywise computation of the approximate eigenvectors, achieves near optimal sample complexity for general symmetric matrices, and improved bounds for positive semidefinite matrices. Technically, our bounds build on recent work on approximating the outlying eigenvalues of symmetric matrices via random sampling in [Bhattacharjee et al. '22] and [Swartworth and Woodruff '25]. We demonstrate for the first time that these approaches extend to the problem of eigenvector estimation.

Tuesday, August 04 2026, 00:00

An Analysis of Brent's Insertion Method for Hash Tables

Authors: William Kuszmaul

In 1968, Richard P.~Brent introduced a new way of building a hash table that, at least empirically, achieves a remarkable property: Even if the hash table is filled to 100\% full, the expected time to query a \emph{random key out of those present} is $O(1)$. Despite the simplicity of Brent's method, the guarantees of the method have never been formally analyzed. This is due to the subtle issue of handling \emph{spoiled randomness}. The algorithm will sometimes try to use hash functions $h_j$ on keys $y$ that it has already probed in the past. When the algorithm does this, we cannot treat the hash function as random, because its random bits have already affected the state of the table. This issue makes Brent's method surprisingly subtle to reason about formally. In this note, we give a simple and formal analysis of Brent's hash table. The analysis can be taught in a graduate randomized algorithms course, and provides a nice example of how to deal with subtle issues in a probabilistic analysis (namely, the issue of spoiled randomness).

Authors: William Kuszmaul

In 1968, Richard P.~Brent introduced a new way of building a hash table that, at least empirically, achieves a remarkable property: Even if the hash table is filled to 100\% full, the expected time to query a \emph{random key out of those present} is $O(1)$. Despite the simplicity of Brent's method, the guarantees of the method have never been formally analyzed. This is due to the subtle issue of handling \emph{spoiled randomness}. The algorithm will sometimes try to use hash functions $h_j$ on keys $y$ that it has already probed in the past. When the algorithm does this, we cannot treat the hash function as random, because its random bits have already affected the state of the table. This issue makes Brent's method surprisingly subtle to reason about formally. In this note, we give a simple and formal analysis of Brent's hash table. The analysis can be taught in a graduate randomized algorithms course, and provides a nice example of how to deal with subtle issues in a probabilistic analysis (namely, the issue of spoiled randomness).

Tuesday, August 04 2026, 00:00

Faster Deterministic Integer Root Finding for Integer Polynomials

Authors: Itamar Nir

We give a deterministic algorithm for finding all integer roots of a square-free polynomial $f\in\mathbb Z[x]$ of degree $n$ with $\lVert f\rVert_\infty<2^b$. The running time is $$ \tilde{O}(n^{3/2}b), $$ improving the $\tilde{O}(n^2b)$ bound of Harvey and Hittmeir (Research in Number Theory, 2022). The algorithm follows the classical $p$-adic framework: find roots modulo a prime $p$, lift them modulo a high power of $p$, and verify the lifted candidates. The main new idea is to avoid searching for a prime for which $f\bmod p$ is square-free. Instead, we find a prime for which the total multiplicity of repeated roots modulo $p$ is small. This requires lifting repeated roots, which we handle using a weighted lifting tree. We also give a faster deterministic candidate-verification algorithm: given $n$ candidate integers smaller in absolute value than $2^b$, we decide which are roots of $f$ in $$ \tilde{O}(nb+\min(n^2,nb^2)) $$ bit operations. Together, these ingredients give the first deterministic subquadratic-in-$n$ improvement for integer root finding in the square-free case.

Authors: Itamar Nir

We give a deterministic algorithm for finding all integer roots of a square-free polynomial $f\in\mathbb Z[x]$ of degree $n$ with $\lVert f\rVert_\infty<2^b$. The running time is $$ \tilde{O}(n^{3/2}b), $$ improving the $\tilde{O}(n^2b)$ bound of Harvey and Hittmeir (Research in Number Theory, 2022). The algorithm follows the classical $p$-adic framework: find roots modulo a prime $p$, lift them modulo a high power of $p$, and verify the lifted candidates. The main new idea is to avoid searching for a prime for which $f\bmod p$ is square-free. Instead, we find a prime for which the total multiplicity of repeated roots modulo $p$ is small. This requires lifting repeated roots, which we handle using a weighted lifting tree. We also give a faster deterministic candidate-verification algorithm: given $n$ candidate integers smaller in absolute value than $2^b$, we decide which are roots of $f$ in $$ \tilde{O}(nb+\min(n^2,nb^2)) $$ bit operations. Together, these ingredients give the first deterministic subquadratic-in-$n$ improvement for integer root finding in the square-free case.

Tuesday, August 04 2026, 00:00

Support Contraction for Well-Supported Nash Equilibria

Authors: Zhengyang Liu

We introduce the support contraction, a structural procedure for computing well-supported Nash equilibria (WSNE) in bimatrix games. On a common action rectangle, the procedure alternately retains the support of a row maximin strategy for the row-payoff matrix and the support of a column maximin strategy for the column-payoff matrix. Each restriction preserves the value of the matrix game that selected it and can only increase the other value. At a stable rectangle, full support of the two maximizing strategies and complementary slackness make every surviving action tight. Crossing the two minimax strategies then gives an exact Nash equilibrium of the retained subgame, while the two zero-sum values bound deviations to deleted actions. Support contraction gives a deterministic polynomial-time algorithm that computes a $1/2$-WSNE of every rational bimatrix game with payoffs in $[0,1]$, with no additive slack, and a deterministic $O(ε^{-2}\log^2 n)$-bit two-party protocol for a $(1/2+ε)$-WSNE. The same support-contraction certificate, preceded by randomized one-sided localization, gives an $O(ε^{-2}n\log n)$ payoff-query algorithm for the same guarantee, improving the $ε^{-4}$ dependence to $ε^{-2}$.

Authors: Zhengyang Liu

We introduce the support contraction, a structural procedure for computing well-supported Nash equilibria (WSNE) in bimatrix games. On a common action rectangle, the procedure alternately retains the support of a row maximin strategy for the row-payoff matrix and the support of a column maximin strategy for the column-payoff matrix. Each restriction preserves the value of the matrix game that selected it and can only increase the other value. At a stable rectangle, full support of the two maximizing strategies and complementary slackness make every surviving action tight. Crossing the two minimax strategies then gives an exact Nash equilibrium of the retained subgame, while the two zero-sum values bound deviations to deleted actions. Support contraction gives a deterministic polynomial-time algorithm that computes a $1/2$-WSNE of every rational bimatrix game with payoffs in $[0,1]$, with no additive slack, and a deterministic $O(ε^{-2}\log^2 n)$-bit two-party protocol for a $(1/2+ε)$-WSNE. The same support-contraction certificate, preceded by randomized one-sided localization, gives an $O(ε^{-2}n\log n)$ payoff-query algorithm for the same guarantee, improving the $ε^{-4}$ dependence to $ε^{-2}$.

Tuesday, August 04 2026, 00:00

Robust Polynomial Freiman-Ruzsa from Corrupted Set Observations via a Sharp Persistent-Subset BSG Compiler

Authors: Cheng Peng

We study structural inference from an exact, adversarially corrupted set observation over $\mathbb F_2^n$. A hidden nonempty set $A$ satisfies $|A+A|\leq K|A|$, while the algorithm receives deterministic membership and exact uniform-sampling access only to $B$, where $|A\triangle B|\leqη|A|$. Since corruption can destroy the doubling of $B$, sharp existential BSG and clean-input Algorithmic PFR do not directly compose in this model. Our main result is a promise-free sharp persistent-subset BSG compiler. Given sample-and-query access to $S$ and $α$, it returns either $\mathsf{FAIL}$ or a descriptor defining one fixed subset $Y\subseteq S$ with $|Y|\geq c\sqrtα\,|S|$ and $|Y+Y|\leq Cα^{-4}|Y|$. Without an energy promise, every nonfailure output is valid except with the prescribed soundness probability; high energy guarantees success with high probability. The descriptor gives persistent membership under adaptive queries, and a finite-horizon bridge gives conditionally exact product samples. Thus sharp retained mass, promise-free validity, persistence, and exact finite sampling form one composable interface. Combined with certified size-oblivious Algorithmic PFR and deterministic lifting, the compiler yields a randomized FPT-form algorithm for $η=O(K^{-1/2})$, outputting $V$ with $|V|\leq|A|$ and $\mathcal N_V(A)\leq K^{O(1)}$. For every supplied $η<1$, an iterated residual algorithm outputs a common list serving every compatible hidden set with polynomial covering budget; samples are polynomial, while membership-query and running-time complexity are XP. Finally, every nonempty compatibility class admits one common subspace nonconstructively, whereas an exact two-subspace construction forces common covering cost $Θ((1-η)^{-1/2})$.

Authors: Cheng Peng

We study structural inference from an exact, adversarially corrupted set observation over $\mathbb F_2^n$. A hidden nonempty set $A$ satisfies $|A+A|\leq K|A|$, while the algorithm receives deterministic membership and exact uniform-sampling access only to $B$, where $|A\triangle B|\leqη|A|$. Since corruption can destroy the doubling of $B$, sharp existential BSG and clean-input Algorithmic PFR do not directly compose in this model. Our main result is a promise-free sharp persistent-subset BSG compiler. Given sample-and-query access to $S$ and $α$, it returns either $\mathsf{FAIL}$ or a descriptor defining one fixed subset $Y\subseteq S$ with $|Y|\geq c\sqrtα\,|S|$ and $|Y+Y|\leq Cα^{-4}|Y|$. Without an energy promise, every nonfailure output is valid except with the prescribed soundness probability; high energy guarantees success with high probability. The descriptor gives persistent membership under adaptive queries, and a finite-horizon bridge gives conditionally exact product samples. Thus sharp retained mass, promise-free validity, persistence, and exact finite sampling form one composable interface. Combined with certified size-oblivious Algorithmic PFR and deterministic lifting, the compiler yields a randomized FPT-form algorithm for $η=O(K^{-1/2})$, outputting $V$ with $|V|\leq|A|$ and $\mathcal N_V(A)\leq K^{O(1)}$. For every supplied $η<1$, an iterated residual algorithm outputs a common list serving every compatible hidden set with polynomial covering budget; samples are polynomial, while membership-query and running-time complexity are XP. Finally, every nonempty compatibility class admits one common subspace nonconstructively, whereas an exact two-subspace construction forces common covering cost $Θ((1-η)^{-1/2})$.

Tuesday, August 04 2026, 00:00

Tenure-track position in Computer Science at the Gran Sasso Science Institute (GSSI)

from Luca Aceto

The Gran Sasso Science Institute (GSSI) in L’Aquila, Italy, invites applications for a full-time tenure-track researcher position in Computer Science.

The GSSI Computer Science group is among the top-ranked in Italy and has been recognized as a national Department of Excellence. Its main areas of research include Algorithms, Artificial Intelligence, Formal Methods, and Software Engineering. Research also addresses applications in fields such as robotics, human-centric systems, cyber-physical systems, the Internet of Things, space, and smart cities.

We welcome candidates whose research complements or expands these areas. Excellent theoretical, applied, and interdisciplinary profiles are strongly encouraged to apply.

The successful candidate will be expected to establish an independent and internationally recognized research programme, supervise PhD students, collaborate with postdoctoral researchers, strengthen international research networks, and attract competitive research funding. The GSSI is a School of Advanced Studies. Teaching responsibilities will include doctoral and postdoctoral courses and seminars, delivered in English.

Key Details & Requirements:
Contract: Six-year, full-time tenure-track appointment
Location: L’Aquila, Italy
Eligibility: PhD or equivalent qualification and at least two years of documented postdoctoral research experience
Language of Instruction: English
Application Deadline: 17 August 2026, 23:59 Italian time

About L’Aquila: Located in the mountains of central Italy, L’Aquila offers an excellent quality of life, a welcoming international academic environment, and immediate access to nature and outdoor activities. The city is also well positioned for reaching Rome and the Adriatic coast.

Application Process:
Applications must be submitted through the PICA platform: lnkd.in/gncZKSDn

Applicants holding a non-Italian PhD may apply while the Italian recognition procedure is pending, provided that they submit evidence that a recognition request has been filed. See more here: lnkd.in/gyCbShDa

Official call (in Italian): lnkd.in/gbnZUa7b

By Luca Aceto

The Gran Sasso Science Institute (GSSI) in L’Aquila, Italy, invites applications for a full-time tenure-track researcher position in Computer Science.

The GSSI Computer Science group is among the top-ranked in Italy and has been recognized as a national Department of Excellence. Its main areas of research include Algorithms, Artificial Intelligence, Formal Methods, and Software Engineering. Research also addresses applications in fields such as robotics, human-centric systems, cyber-physical systems, the Internet of Things, space, and smart cities.

We welcome candidates whose research complements or expands these areas. Excellent theoretical, applied, and interdisciplinary profiles are strongly encouraged to apply.

The successful candidate will be expected to establish an independent and internationally recognized research programme, supervise PhD students, collaborate with postdoctoral researchers, strengthen international research networks, and attract competitive research funding. The GSSI is a School of Advanced Studies. Teaching responsibilities will include doctoral and postdoctoral courses and seminars, delivered in English.

Key Details & Requirements:
Contract: Six-year, full-time tenure-track appointment
Location: L’Aquila, Italy
Eligibility: PhD or equivalent qualification and at least two years of documented postdoctoral research experience
Language of Instruction: English
Application Deadline: 17 August 2026, 23:59 Italian time

About L’Aquila: Located in the mountains of central Italy, L’Aquila offers an excellent quality of life, a welcoming international academic environment, and immediate access to nature and outdoor activities. The city is also well positioned for reaching Rome and the Adriatic coast.

Application Process:
Applications must be submitted through the PICA platform: https://lnkd.in/gncZKSDn

Applicants holding a non-Italian PhD may apply while the Italian recognition procedure is pending, provided that they submit evidence that a recognition request has been filed. See more here: https://lnkd.in/gyCbShDa

Official call (in Italian): https://lnkd.in/gbnZUa7b

By Luca Aceto

Monday, August 03 2026, 22:12

Quantum Algorithms for Modular Factorials

Authors: Yann Tal

We give a bounded-error quantum algorithm that, given a prime $p$, a divisor $q\mid(p-1)$, and an integer $0

Authors: Yann Tal

We give a bounded-error quantum algorithm that, given a prime $p$, a divisor $q\mid(p-1)$, and an integer $0

Monday, August 03 2026, 00:00

Strong invariants and Tverberg numbers in convexity spaces

Authors: Minho Cho, Andreas F. Holmsen, Attila Jung, Hong Liu

Helly, Carathéodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carathéodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.

Authors: Minho Cho, Andreas F. Holmsen, Attila Jung, Hong Liu

Helly, Carathéodory, and Radon numbers encode three kinds of finite certificates in a convexity space: for the emptiness of an intersection, for membership in a convex hull, and for the existence of intersecting hulls. We study exact versions of these certificates, in which a subfamily must preserve the whole intersection or a subset must preserve the whole hull. Our first main result shows that, for finite configurations in an arbitrary convexity space, five a priori different boundedness conditions are equivalent: VC-dimension, strong Helly number, strong Carathéodory number, comatching number, and strong Radon number (with the expected additive-one shift). We also obtain equivalent layered Tverberg-type decompositions and colorful consequences. The common mechanism is exposed by the bipartite incidence graph between points and a generating family. For finite spaces, the unique minimal generator yields a natural dual convexity space; we characterize double dualization and prove that the strong parameters are duality invariant. The same model gives a polynomial-size, $O(t^4)$, realization of Bukh's counterexample to the Calder-Eckhoff partition conjecture. Finally, we obtain the first Tverberg bound for separable convexity spaces that is simultaneously linear in the number of parts and polynomial in the Radon number. If an $S_3$-separable convexity space has Helly number $h$ and its halfspaces have VC-dimension $d$, then $r_t=O(dh\log h)\,t$; in particular, Radon number $r$ gives $r_t=O(r^2\log r)\,t$. The bound attains the weak-Eckhoff scale $O(rt)$ whenever the Helly number is bounded. For axis-parallel box convexity in $\mathbb{R}^k$, gives the optimal order $r_t=O(rt)$ uniformly in every dimension. This appears to be the first dimension-uniform estimate of weak-Eckhoff order for box convexity, whereas the previous direct theory was confined to dimension three.

Monday, August 03 2026, 00:00

Enumeration and Classification of Triangle-Maximal Pseudoline Arrangements

Authors: Roman Parpalak, Denis Utkin

We describe algorithms for the exhaustive enumeration and classification of simple arrangements of $n$ pseudolines ($n$ odd) maximizing the number of triangular faces. The depth-first search enumerates reduced words for the longest permutation $w_0$ by branching only on the even-indexed generators, using pruning constraints imposed by the geometry of optimal arrangements. The approach handles both perfect arrangements with a regular triangular pattern and unavoidable deviations from it for $n \equiv 1 \pmod 6$. The output is classified into a hierarchy of equivalence classes: by commutation, by Euclidean transformations, and by projective transformations. For each projective class we recover its full symmetry group $G \subseteq S_{n+1}$ together with the orbit-stabilizer profile of its Euclidean subclasses. Completeness of the search and classification is proved: every wiring diagram is reached. We report full enumerations; e.g. for $n=27$, 85,562,064 wiring diagrams partitioned into 56,646 projective classes. For larger $n$ (up to $n=93$), where exhaustive enumeration is out of reach, we report partial (first-hit) results.

Authors: Roman Parpalak, Denis Utkin

We describe algorithms for the exhaustive enumeration and classification of simple arrangements of $n$ pseudolines ($n$ odd) maximizing the number of triangular faces. The depth-first search enumerates reduced words for the longest permutation $w_0$ by branching only on the even-indexed generators, using pruning constraints imposed by the geometry of optimal arrangements. The approach handles both perfect arrangements with a regular triangular pattern and unavoidable deviations from it for $n \equiv 1 \pmod 6$. The output is classified into a hierarchy of equivalence classes: by commutation, by Euclidean transformations, and by projective transformations. For each projective class we recover its full symmetry group $G \subseteq S_{n+1}$ together with the orbit-stabilizer profile of its Euclidean subclasses. Completeness of the search and classification is proved: every wiring diagram is reached. We report full enumerations; e.g. for $n=27$, 85,562,064 wiring diagrams partitioned into 56,646 projective classes. For larger $n$ (up to $n=93$), where exhaustive enumeration is out of reach, we report partial (first-hit) results.

Monday, August 03 2026, 00:00

Differentiable Approximations for Distance Queries

Authors: Ahmed Abdelkader, David M. Mount

The widespread use of gradient-based optimization has motivated the adaptation of various classical algorithms into differentiable solvers compatible with learning pipelines. In this paper, we investigate the enhancement of traditional geometric query problems such that the result consists of both the geometric function as well as its gradient. Specifically, we study the fundamental problem of distance queries against a set of points $P$ in $\mathbb{R}^d$, which also underlies various similarity measures for learning algorithms. The main result of this paper is a multiplicative $(1+\varepsilon)$-approximation of the Euclidean distance to $P$ which is differentiable at all points in $\mathbb{R}^d \setminus P$ with asymptotically optimal bounds on the norms of its gradient and Hessian, from a data structure with storage and query time matching state-of-the-art results for approximate nearest-neighbor searching. The approximation is realized as a regularized distance through a partition-of-unity framework, which efficiently blends multiple local approximations, over a suitably defined covering of space, into a smooth global approximation. In order to obtain the local distance approximations in a manner that facilitates blending, we develop a new approximate Voronoi diagram based on a simple point-location data structure, simplifying away both the lifting transformation and ray shooting.

Authors: Ahmed Abdelkader, David M. Mount

The widespread use of gradient-based optimization has motivated the adaptation of various classical algorithms into differentiable solvers compatible with learning pipelines. In this paper, we investigate the enhancement of traditional geometric query problems such that the result consists of both the geometric function as well as its gradient. Specifically, we study the fundamental problem of distance queries against a set of points $P$ in $\mathbb{R}^d$, which also underlies various similarity measures for learning algorithms. The main result of this paper is a multiplicative $(1+\varepsilon)$-approximation of the Euclidean distance to $P$ which is differentiable at all points in $\mathbb{R}^d \setminus P$ with asymptotically optimal bounds on the norms of its gradient and Hessian, from a data structure with storage and query time matching state-of-the-art results for approximate nearest-neighbor searching. The approximation is realized as a regularized distance through a partition-of-unity framework, which efficiently blends multiple local approximations, over a suitably defined covering of space, into a smooth global approximation. In order to obtain the local distance approximations in a manner that facilitates blending, we develop a new approximate Voronoi diagram based on a simple point-location data structure, simplifying away both the lifting transformation and ray shooting.

Monday, August 03 2026, 00:00

Convex Approximation and the Hilbert Geometry

Authors: Ahmed Abdelkader, David M. Mount

The efficient representation of convex bodies in multi-dimensional spaces is a fundamental problem in computational geometry. Several key developments were recently brought about using a number of constructions utilizing Macbeath regions. In this paper, we present a novel intrinsic approach for approximate membership testing, where we carry out the entire development based on structures derived from the Hilbert metric associated with a convex body $K$ in $\mathbb{R}^d$. First, we revisit the construction of economical Delone sets, deriving the size bound based on the notion of volume entropy. Second, we design a new query structure based on a simple covering by ellipsoids, where queries are answered by ray shooting. As an added bonus, the intrinsic viewpoint facilitates finger searching, where the query time can be bounded by the distance traveled in the Hilbert metric.

Authors: Ahmed Abdelkader, David M. Mount

The efficient representation of convex bodies in multi-dimensional spaces is a fundamental problem in computational geometry. Several key developments were recently brought about using a number of constructions utilizing Macbeath regions. In this paper, we present a novel intrinsic approach for approximate membership testing, where we carry out the entire development based on structures derived from the Hilbert metric associated with a convex body $K$ in $\mathbb{R}^d$. First, we revisit the construction of economical Delone sets, deriving the size bound based on the notion of volume entropy. Second, we design a new query structure based on a simple covering by ellipsoids, where queries are answered by ray shooting. As an added bonus, the intrinsic viewpoint facilitates finger searching, where the query time can be bounded by the distance traveled in the Hilbert metric.

Monday, August 03 2026, 00:00

The Kikuchi Hierarchy is Sharp for $k$XOR

Authors: Alexander Schmidhuber, Matthew B. Hastings

Planted noisy $k$XOR and the strong refutation of random $k$XOR are governed by a conjectured trade-off between signal strength and time: Level $\ell$ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*} m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses} \quad\Longleftrightarrow\quad \text{solvable in time }n^{O(\ell)}, \end{equation*} where $ρ$ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse $k$XOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time. We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity $k\ge3$. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree $O_k(\ell)$. We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery. The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.

Authors: Alexander Schmidhuber, Matthew B. Hastings

Planted noisy $k$XOR and the strong refutation of random $k$XOR are governed by a conjectured trade-off between signal strength and time: Level $\ell$ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*} m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses} \quad\Longleftrightarrow\quad \text{solvable in time }n^{O(\ell)}, \end{equation*} where $ρ$ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse $k$XOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time. We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity $k\ge3$. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree $O_k(\ell)$. We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery. The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.

Monday, August 03 2026, 00:00

Approximation Algorithms for Geometric Maximum Coverage

Authors: Sujoy Bhore, Timothy M. Chan, Pasin Manurangsi

We study the maximum coverage problem for geometric set systems: given a set of points, a set of geometric objects, and a number $k$, select $k$ objects maximizing the number of points inside their union. - We present a polynomial-time approximation algorithm with approximation factor strictly better than $1-1/e$ for any set system with linear 2-shallow cell complexity (or any set system that can be decomposed into a constant number of such set systems). The result also holds for the weighted maximum coverage problem, where objects have weights and we want to select objects with total weight within a given budget. The result applies to many types of geometric objects, including pseudodisks in 2D, fat axis-aligned rectangles in 2D, similar-size fat triangles in 2D, axis-aligned unit cubes in 3D. - For small $k$, we obtain a $(1-ε)$-approximation algorithm more generally for any set system with constant VC dimension, running in time exponential in $\tilde{O}(k/ε)$. This simplifies and improves Badanidiyuru, Kleinberg, and Lee's parameterized approximation scheme [SoCG'12] running in time exponential in $\tilde{O}(k^2/ε^5)$. - A continuous version of the geometric maximum coverage problem asks for $k$ objects maximizing the volume of their union. We give better approximation algorithms for this problem for certain families of objects; e.g., we obtain an EPTAS for fat convex objects in any constant dimension. - We complement our algorithms with several hardness results, e.g., APX-hardness for fat axis-aligned rectangles in 2D, $(1-1/e+ε)$-approximation hardness for axis-aligned boxes in a dimension dependent on $ε$, and a lower bound ruling out $n^{\mathop{\rm poly}(1/ε)}$-time PTASs for the continuous problem for axis-aligned boxes in 3D.

Authors: Sujoy Bhore, Timothy M. Chan, Pasin Manurangsi

We study the maximum coverage problem for geometric set systems: given a set of points, a set of geometric objects, and a number $k$, select $k$ objects maximizing the number of points inside their union. - We present a polynomial-time approximation algorithm with approximation factor strictly better than $1-1/e$ for any set system with linear 2-shallow cell complexity (or any set system that can be decomposed into a constant number of such set systems). The result also holds for the weighted maximum coverage problem, where objects have weights and we want to select objects with total weight within a given budget. The result applies to many types of geometric objects, including pseudodisks in 2D, fat axis-aligned rectangles in 2D, similar-size fat triangles in 2D, axis-aligned unit cubes in 3D. - For small $k$, we obtain a $(1-ε)$-approximation algorithm more generally for any set system with constant VC dimension, running in time exponential in $\tilde{O}(k/ε)$. This simplifies and improves Badanidiyuru, Kleinberg, and Lee's parameterized approximation scheme [SoCG'12] running in time exponential in $\tilde{O}(k^2/ε^5)$. - A continuous version of the geometric maximum coverage problem asks for $k$ objects maximizing the volume of their union. We give better approximation algorithms for this problem for certain families of objects; e.g., we obtain an EPTAS for fat convex objects in any constant dimension. - We complement our algorithms with several hardness results, e.g., APX-hardness for fat axis-aligned rectangles in 2D, $(1-1/e+ε)$-approximation hardness for axis-aligned boxes in a dimension dependent on $ε$, and a lower bound ruling out $n^{\mathop{\rm poly}(1/ε)}$-time PTASs for the continuous problem for axis-aligned boxes in 3D.

Monday, August 03 2026, 00:00

Online Shadow Tomography Matching the Classical Bounds

Authors: Sitan Chen, Ryan O'Donnell, Angelos Pelecanos, John Wright

In \emph{Online Shadow Tomography}, we are given copies of an unknown $d$-dimensional quantum state $ρ$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\Tr(A^{(t)}ρ)$ to within $\pm ε$. This is the direct quantum generalization of the classical problem of \emph{Adaptive Data Analysis}. %The ``offline'' case, in which $A^{(1)}, \ldots, A^{(m)}$ are given upfront, is also a well-studied problem. The main goal is to minimize the number of copies, $n$, required. Prior results for online Shadow Tomography were suboptimal in all three parameters $m, d, ε$, lagging behind the best known and classical rates~\cite{bassily2021algorithmic}, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. The bound on the left is the first to achieve $o(\log^2 m)$-dependence together with $\poly(\log(d)/\eps)$; moreover, it improves all three exponents even in the \emph{Offline} Shadow Tomography setting. The bound on the right is known to be optimal among bounds independent of~$d$, and improves the best prior result by a $\sqrt{m} \log m$ factor. The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron--Stein decomposition.

Authors: Sitan Chen, Ryan O'Donnell, Angelos Pelecanos, John Wright

In \emph{Online Shadow Tomography}, we are given copies of an unknown $d$-dimensional quantum state $ρ$, an adversary (adaptively) proposes a sequence of bounded observables $A^{(1)},\ldots,A^{(m)}$, and after each $A^{(t)}$ is given we must estimate $\Tr(A^{(t)}ρ)$ to within $\pm ε$. This is the direct quantum generalization of the classical problem of \emph{Adaptive Data Analysis}. %The ``offline'' case, in which $A^{(1)}, \ldots, A^{(m)}$ are given upfront, is also a well-studied problem. The main goal is to minimize the number of copies, $n$, required. Prior results for online Shadow Tomography were suboptimal in all three parameters $m, d, ε$, lagging behind the best known and classical rates~\cite{bassily2021algorithmic}, for which there is some evidence of optimality. In this work, we finally close this gap, giving a pair of algorithms matching the classical rates. The bound on the left is the first to achieve $o(\log^2 m)$-dependence together with $\poly(\log(d)/\eps)$; moreover, it improves all three exponents even in the \emph{Offline} Shadow Tomography setting. The bound on the right is known to be optimal among bounds independent of~$d$, and improves the best prior result by a $\sqrt{m} \log m$ factor. The key to our proof is a new framework for quantifying post-measurement damage, based on the quantum Efron--Stein decomposition.

Monday, August 03 2026, 00:00

Spectrum Estimation is Almost as Hard as Tomography