Society for Industrial and Applied Mathematics: SIAM Journal on Computing: Table of Contents

Quantum Time-Space Tradeoffs for Matrix Problems

Paul Beame — 2026-05-20T07:00:00Z

SIAM Journal on Computing, Volume 55, Issue 3, Page 469-519, June 2026.
Abstract. We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices [math], including the discrete Fourier transform matrix, we prove that quantum circuits with at most [math] input queries and [math] qubits of memory require [math] to compute matrix-vector product [math] for [math]. We similarly prove that matrix multiplication for [math] binary matrices requires [math]. Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for [math] Boolean matrix multiplication to [math] from [math]. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.

The Power of Proportional Fairness for Nonclairvoyant Polytope Scheduling

Sven Jäger — 2026-03-02T08:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 247-279, April 2026.
Abstract. The polytope scheduling problem (PSP) was introduced by Im, Kulkarni, and Munagala [J. ACM, 65 (2018), pp. 3:1–3:33] as a very general abstraction of resource allocation over time, where jobs can receive processing rates subject to arbitrary packing constraints. It captures many well-studied problems, including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the proportional fairness (PF) algorithm, which was analyzed for PSP by Im, Kulkarni, and Munagala. We drastically improve the analysis of PF for both the general PSP and several of its important special cases subject to the objective of minimizing the sum of weighted completion times. We reduce the upper bound on the competitive ratio from 128 to 27 for general PSP and to 4 for the prominent class of monotone PSP. For certain heterogeneous machine environments, we even close the substantial gap to the lower bound of 2 for nonclairvoyant scheduling. Our analysis also gives the first polynomial-time improvement over the nearly 30-year-old bounds on the competitive ratio of the doubling framework, which was introduced by Hall, Shmoys, and Wein (SODA 1996) for clairvoyant online preemptive scheduling on unrelated machines. Somewhat surprisingly, we achieve this improvement by a nonclairvoyant algorithm, thereby demonstrating that nonclairvoyance is not a (significant) hurdle. Our improvements are based on exploiting monotonicity properties of PSP, providing tight dual fitting arguments on structured instances, and showing new algebraic properties of the optimal objective value for scheduling on unrelated machines. Finally, we establish new connections between PF and matching markets and thereby provide new insights on equilibria and their computational complexity.

Space Complexity of Vertex Connectivity Oracles

Seth Pettie — 2026-03-02T08:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 280-306, April 2026.
Abstract. A [math]-vertex connectivity oracle for an undirected graph [math] is a data structure that, given [math], reports [math], where [math] is the pairwise vertex connectivity between [math]. There are three main measures of efficiency: construction time, query time, and space. Prior work of Izsak and Nutov [Inform. Process. Lett., 112 (2012), pp. 39–43] produced a data structure of [math] words, which can even be encoded as a [math]-bit labeling scheme, that can answer vertex connectivity queries in [math] time. The construction time is polynomial but unspecified. In this paper, we address the top three complexity measures. (1) Space: We prove that any [math]-vertex connectivity oracle requires [math] bits of space for any [math]. This proves that the Iszak–Nutov data structure is optimal up to polylogarithmic factors for every [math] and that the sparsifiers of Nagamochi and Ibaraki [Algorithmica, 7 (1992), pp. 583–596] are optimal compression schemes for [math]-vertex connectivity up to a logarithmic factor. In particular, whereas all edge connectivities can be efficiently compressed (as a weighted [math]-edge Gomory–Hu tree), vertex connectivity admits no asymptotic compression: [math] bits are necessary. We design a variation on Izsak and Nutov’s data structure that uses [math] words of space. (2) Query time: We answer queries in [math] time, improving on the [math] time bound of Izsak and Nutov [Inform. Process. Lett., 112 (2012), pp. 39–43]. The main idea is to build instances of [math] data structures, with additional structure based on affine planes. This structure allows for query time that is linear in the output size, which evades some conditional lower bounds that are polynomial in the query set sizes [, ]. (3) Construction time: Our data structure can be constructed in the time of [math] max-flow computations, namely, [math] time, using the recent near-linear time flow algorithm of []. The main technical contribution here is a fast algorithm to compute a [math]-approximate Gomory–Hu tree for element connectivity in the time of [math] max-flow computations. Element connectivity is a notion that generalizes edge and vertex connectivity.

The Minimal Faithful Permutation Degree of Groups Without Abelian Normal Subgroups

Bireswar Das — 2026-03-03T08:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 307-331, April 2026.
Abstract. The minimal faithful permutation degree [math] of a finite group [math] is the smallest integer [math] for which there is an injective homomorphism [math] from [math] to [math]. The main result of this paper is a randomized polynomial-time algorithm for computing the minimal faithful permutation degree groups without abelian normal subgroups. Additionally, we show that: 1. For any primitive permutation group [math], [math] can be computed in quasi-polynomial time. 2. For a group [math] given by its Cayley table, [math] can be computed in [math].

Order-Competitive Ratio

Liyan Chen — 2026-03-04T08:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 332-361, April 2026.
Abstract. We introduce a new measure for the performance of online algorithms in Bayesian settings, where the input is drawn from a known prior, but the realizations are revealed one-by-one in an online fashion. Our new measure is called an order-competitive ratio. It is defined as the worst case (over all distribution sequences) ratio between the performance of the best order-unaware and order-aware algorithms, and quantifies the loss that is incurred due to lack of knowledge of the arrival order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, this loss has been overlooked thus far. We study the order-competitive ratio in the paradigmatic prophet inequality problem, for the two common objective functions of (i) maximizing the expected value, and (ii) maximizing the probability of obtaining the largest value; and with respect to two families of algorithms, namely, (i) adaptive algorithms, and (ii) single-threshold algorithms. We provide tight bounds for all four combinations, with respect to deterministic algorithms, and preliminary results for randomized algorithms. Our analysis requires new ideas and departs from standard techniques. In particular, our adaptive algorithms inevitably go beyond single-threshold algorithms. In contrast to the classic competitive ratio measure, where the optimal performance is obtained by deterministic single-threshold algorithms, our results for order-competitive ratio capture the intuition that adaptive algorithms may be more powerful than single-threshold ones, and randomized algorithms outperform deterministic ones.

Edge-Disjoint Paths in Expanders: Online with Removals

Nemanja Draganić — 2026-03-09T07:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 362-375, April 2026.
Abstract. We consider the problem of finding edge-disjoint paths between given pairs of vertices in a sufficiently strong [math]-regular expander graph [math] with [math] vertices. In particular, we describe a deterministic, polynomial time algorithm which maintains an initially empty collection of edge-disjoint paths [math] in [math] and fulfills any series of two types of requests: (1) Given two vertices [math] and [math] such that each appears as an endpoint in [math] paths in [math] and, additionally, [math], the algorithm finds a path of length at most [math] connecting [math] and [math] which is edge-disjoint from all other paths in [math], and adds it to [math]. (2) Remove a given path [math] from [math]. Importantly, each request is processed before seeing the next one. The upper bound on the length of found paths and the constraints are the best possible up to a constant factor. This establishes the first online algorithm for finding edge-disjoint paths in expanders which also allows removals, significantly strengthening a long list of previous results on the topic. We obtain the same result in the case [math] is directed.

Laplace Transform–Based Quantum Eigenvalue Transformation via Linear Combination of Hamiltonian Simulation

Dong An — 2026-03-17T07:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 376-409, April 2026.
Abstract. Eigenvalue transformations, which include solving time-dependent differential equations as a special case, have a wide range of applications in scientific and engineering computation. While quantum algorithms for singular value transformations are well studied, eigenvalue transformations are distinct, especially for nonnormal matrices. We propose an efficient quantum algorithm for performing a class of eigenvalue transformations that can be expressed as a certain type of matrix Laplace transformation. This allows us to significantly extend the recently developed linear combination of Hamiltonian simulation method [D. An, J.-P. Liu, and L. Lin, Phys. Rev. Lett., 131 (2023), 150603; D. An, A. M. Childs, and L. Lin, Commun. Math. Phys. 407, 19 (2026)] to represent a wider class of eigenvalue transformations, such as powers of the matrix inverse, [math], and the exponential of the matrix inverse, [math]. The latter can be interpreted as the solution of a mass-matrix differential equation of the form [math]. We demonstrate that our eigenvalue transformation approach can solve this problem without explicitly inverting [math], thereby reducing the computational complexity.

An FPT Algorithm for the Embeddability of Graphs Into Two-Dimensional Simplicial Complexes

Éric Colin de Verdière — 2026-04-07T07:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 410-450, April 2026.
Abstract. We consider the embeddability problem of a graph [math] into a two-dimensional simplicial complex [math]: Given [math] and [math], decide whether [math] admits a topological embedding into [math]. The problem is NP-hard, even in the restricted case where [math] is homeomorphic to a surface. We prove that the problem is fixed-parameter tractable in the size of the two-dimensional complex, by providing an [math]-time algorithm. If [math] embeds into [math], we can compute a representation of an embedding in the same amount of time. Moreover, we show that several known problems reduce to this one, such as the crossing number and the planarity number problems, and, under some conditions, the embedding extension problem. Our approach is to reduce to the case where [math] has bounded branchwidth via an irrelevant vertex method, and to apply dynamic programming. We do not rely on any component of the existing linear-time algorithms for embedding graphs on a fixed surface, but only on algorithms from graph minor theory. However, by combining our results with a linear-time algorithm for embedding graphs on surfaces and with a very recent result for the irrelevant vertex method, we can decide whether [math] embeds into [math] in [math] time, for some function [math].

A Subquadratic Upper Bound on Hurwitz’s Problem and Related Noncommutative Polynomials

Pavel Hrubeš — 2026-04-30T07:00:00Z

SIAM Journal on Computing, Volume 55, Issue 2, Page 451-467, April 2026.
Abstract. For every [math], we construct a sum-of-squares identity [math], where [math] are bilinear forms with complex coefficients and [math]. Previously, such a construction was known with [math]. The same bound holds over any field of positive characteristic. As an application to complexity of noncommutative computation, we show that the polynomial [math] in [math] noncommuting variables can be computed by a noncommutative arithmetic circuit of size [math]. This holds over any field of characteristic different from two. The same bound applies to noncommutative versions of the elementary symmetric polynomial of degree four and the rectangular permanent of a [math] matrix.

Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time

V. Arvind — 2026-06-01T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [A. Garg, L. Gurvits, R. M. de Oliveira, and A. Wigderson, A deterministic polynomial time algorithm for non-commutative rational identity testing, in Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, I. Dinur, ed., IEEE Computer Society, 2016, pp. 109–117], [M. A. Forbes and A. Shpilka, Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs, in Proceedings of the 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, IEEE Computer Society, 2013, pp. 243–252], [M. Hamada and H. Hirai, SIAM J. Appl. Algebra Geom., 5 (2021), pp. 455–478], and a randomized polynomial-time algorithm [H. Derksen and V. Makam, Adv. Math., 310 (2017), pp. 44–63] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result from [H. Derksen and V. Makam, Adv. Math., 310 (2017), pp. 44–63]. Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work [A. Garg, L. Gurvits, R. M. de Oliveira, and A. Wigderson, A deterministic polynomial time algorithm for non-commutative rational identity testing, in Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, New Brunswick, NJ, I. Dinur, ed., 2016, IEEE Computer Society, 2016, pp. 109–117], these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [V. Arvind, A. Chatterjee, and P. Mukhopadhyay, Black-box identity testing of noncommutative rational formulas of inversion height two in deterministic quasipolynomial time, in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2022, LIPIcs 245, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022, 23]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting.

Separating MAX 2-AND, MAX DI-CUT, and MAX CUT

Joshua Brakensiek — 2026-05-28T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Assuming the unique games conjecture (UGC), the best approximation ratio that can be obtained in polynomial time for the max cut problem is [math], obtained by the celebrated SDP-based approximation algorithm of Goemans and Williamson. The current best approximation algorithm for max di-cut, i.e., the max cut problem in directed graphs, achieves a ratio of about 0.87401, leaving open the question of whether max di-cut can be approximated as well as max cut. We obtain a slightly improved algorithm for max di-cut and a new UGC-hardness for it, showing that [math], where [math] is the best approximation ratio that can be obtained in polynomial time for max di-cut under UGC. Our new upper bound shows that max di-cut cannot be approximated as well as max cut, which separates max di-cut from max cut and resolves a question raised by Feige and Goemans. A natural generalization of max di-cut is the max [math]-and problem in which each constraint is of the form [math], where [math] and [math] are literals, i.e., variables or their negations (in max di-cut each constraint is of the form [math] where [math] and [math] are variables). Austrin separated max [math]-and from max cut by showing that [math] and conjectured that max [math]-and and max di-cut have the same approximation ratio. Our new lower bound on max di-cut refutes this conjecture, completing the separation of the three problems max [math]-and, max di-cut, and max cut. We also obtain a new lower bound for max [math]-and, showing that [math]. Our upper bound on max di-cut is achieved via a simple, analytical proof. The new lower bounds on max di-cut and max [math]-and, i.e., the new approximation algorithms, use experimentally discovered distributions of rounding functions which are then verified via computer-assisted proofs. Code for the project is available at https://github.com/jbrakensiek/max-dicut.

The Orthogonal Vectors Conjecture and Nonuniform Circuit Lower Bounds

R. Ryan Williams — 2025-10-14T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. A line of work has shown how nontrivial uniform algorithms for analyzing circuits can be used to derive nonuniform circuit lower bounds. We show how the nonexistence of nontrivial circuit-analysis algorithms can also imply nonuniform circuit lower bounds. Our connections yield new win-win circuit lower bounds, and suggest a potential approach to refuting the Orthogonal Vectors Conjecture in the [math]-dimensional case, which would be sufficient for refuting the Strong Exponential Time Hypothesis (SETH). For example, we show that at least one of the following holds: [math] There is an [math] such that for infinitely many [math], read-once 2-DNFs on [math] variables cannot be simulated by nonuniform [math]-size depth-two exact threshold circuits. It is already a notorious open problem to prove that the class [math] does not have polynomial-size depth-two exact threshold circuits, so such a lower bound would be a significant advance in low-depth circuit complexity. In fact, a stronger lower bound holds in this case: the [math] disjointness matrix (well-studied in communication complexity) cannot be expressed by a linear combination of [math] structured matrices that we call “equality matrices.” [math] For every [math] and every [math], orthogonal vectors on [math] vectors in [math] dimensions can be solved in [math] uniform deterministic time. This case would provide a strong refutation of the Orthogonal Vectors Conjecture, and of SETH: for example, CNF-SAT on [math] variables and [math] clauses could be solved in [math] time. Moreover, this case would imply nonuniform circuit lower bounds for [math], against Valiant series-parallel circuits. Inspired by this connection, we give evidence from SAT/SMT solvers that the first item (in particular, the disjointness lower bound) may be false in its full generality. In particular, we present a systematic approach to solving orthogonal vectors via constant-sized decompositions of the disjointness matrix, which already yields interesting new algorithms. For example, using a linear combination of 6 equality matrices that express [math] disjointness, we derive an [math] time and [math] space algorithm for orthogonal vectors on [math] vectors in [math] dimensions. We show similar results for counting pairs of orthogonal vectors.

Hardness of Packing, Covering and Partitioning Simple Polygons with Unit Squares

Mikkel Abrahamsen — 2025-11-25T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. It has been known since the early 1980s that packing unit squares into a polygon with holes is NP-hard [Fowler, Paterson, Tanimoto, Inf. Process. Lett., 1981], but the version without holes was conjectured to be polynomial-time solvable more than two decades ago [Baur and Fekete, Algorithmica, 2001]. We show that packing axis-aligned unit squares into a simple polygon [math] is NP-hard even when [math] is an orthogonal and orthogonally convex polygon with half-integer coordinates. Our results can be extended to show hardness of covering simple polygons with unit squares and partitioning simple polygons into pieces that fit inside a unit square.

Deterministic Algorithm and Faster Algorithm for Submodular Maximization Subject to a Matroid Constraint

Niv Buchbinder — 2025-12-02T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We study the problem of maximizing a monotone submodular function subject to a matroid constraint, and present for it a deterministic nonoblivious local search algorithm that has an approximation guarantee of [math] (for any [math]) and query complexity of [math], where [math] is the size of the ground set and [math] is the rank of the matroid. Our algorithm vastly improves over the previous state-of-the-art 0.5008-approximation deterministic algorithm, and in fact shows that there is no separation between the approximation guarantees that can be obtained by deterministic and randomized algorithms for the problem considered. The query complexity of our algorithm can be improved to [math] using randomization, which is nearly linear for [math], and is always at least as good as the previous state-of-the-art algorithms.

Faster Isomorphism Testing of p-Groups of Frattini Class 2

Gábor Ivanyos — 2025-12-03T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The finite group isomorphism problem asks to decide whether two finite groups of order [math] are isomorphic. Improving the classical [math]-time algorithm for group isomorphism is a long-standing open problem. It is generally regarded that [math]-groups of class 2 and exponent [math] form a bottleneck case for group isomorphism in general. The recent breakthrough by Sun [Proceedings of the 55th Annual ACM Symposium on Theory of Computing, 2023, pp. 433–440] presents an [math]-time algorithm for this group class. In this paper, we improve Sun’s algorithm by presenting an [math]-time algorithm for this group class. We also extend our result to the more general [math]-groups of Frattini class 2 for any odd prime [math]. Our algorithm is obtained by sharpening the key technical ingredients in Sun’s algorithm and building connections with other research topics. One intriguing connection is with the maximal and noncommutative ranks of matrix spaces, which have recently received considerable attention in algebraic complexity and computational invariant theory. Results from the theory of tensor isomorphism complexity class [J. A. Grochow and Y. Qiao, SIAM J. Comput., 52 (2023), pp. 568–617] are utilized to simplify the algorithm and to achieve the extension to [math]-groups of Frattini class 2.

Gapped Clique Homology on Weighted Graphs Is [math]-Hard and Contained in QMA

Robbie King — 2026-01-02T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We study the complexity of a classic problem in computational topology, the homology problem: given a description of some space [math] and an integer [math], decide if [math] contains a [math]-dimensional hole. The setting and statement of the homology problem are completely classical, yet we find that the complexity is characterized by quantum complexity classes. Our result can be seen as an aspect of a connection between homology and supersymmetric quantum mechanics established by Witten (1982). We consider clique complexes, motivated by the practical application of topological data analysis (TDA). The clique complex of a graph is the simplicial complex formed by declaring every [math]-clique in the graph to be a [math]-simplex. Our main result is that deciding whether the clique complex of a weighted graph has a hole or not, given a suitable promise on the gap, is [math]-hard and contained in QMA. Our main innovation is a technique to lower bound the eigenvalues of the combinatorial Laplacian operator. For this, we invoke a tool from algebraic topology known as spectral sequences. In particular, we exploit a connection between spectral sequences and Hodge theory outlined by Forman (1994). Spectral sequences will play a role analogous to perturbation theory for combinatorial Laplacians. In addition, we develop the simplicial surgery technique used in prior work by Crichigno and Kohler (2022). Our result provides some suggestion that the quantum TDA algorithm due to Lloyd, Garnerone, and Zanardi (2016) cannot be dequantized, although proving this would require studying the complexity of a different problem related to the combinatorial Laplacian of clique complexes.

Constant-Depth Arithmetic Circuits for Linear Algebra Problems

Robert Andrews — 2026-01-26T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We design polynomial-size, constant-depth (namely, [math]) arithmetic formulae for the greatest common divisor (GCD) of two polynomials as well as the related problems of the discriminant, resultant, and Bézout coefficients; squarefree decomposition; and the inversion of structured matrices, such as Sylvester and Bézout matrices. Our GCD algorithm extends to any number of polynomials. Previously, the best-known arithmetic formulae for these problems required superpolynomial size regardless of depth. These results are based on new algorithmic techniques to compute various symmetric functions in the roots of polynomials and to manipulate the multiplicities of these roots without having access to them. These techniques allow [math] computation of a large class of linear and polynomial algebra problems, which include the above as special cases. We extend these techniques to problems whose inputs are multivariate polynomials, which are represented by constant-depth arithmetic circuits. Here, too, we solve problems such as computing the GCD and squarefree decomposition in [math].

Near-Tight Bounds for 3-Query Locally Correctable Binary Linear Codes via Rainbow Cycles

Omar Alrabiah — 2026-04-15T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We prove that a binary linear code of blocklength [math] that is locally correctable with 3 queries against a fraction [math] of adversarial errors must have dimension at most [math]. This is almost tight in view of quadratic Reed–Muller codes being a 3-query locally correctable code (LCC) with dimension [math]. Our result improves, for the binary field case, the [math] bound obtained in the recent breakthrough of [P. K. Kothari and P. Manohar, in Proceedings of the 56th Annual ACM Symposium on Theory of Computing, ACM, New York, 2024, pp. 776–787] and the more recent improvement to [math] of [T. Yankovitz, in Proceedings of the 2024 IEEE 65th Annual IEEE Symposium on Foundations of Computer Science, IEEE, 2024, pp. 1786–1801]. Previous bounds for 3-query linear LCCs proceed by constructing a 2-query locally decodable code (LDC) from the 3-query linear LCC/LDC and applying the strong bounds known for the former. Our approach is more direct and proceeds by bounding the covering radius of the dual code, borrowing inspiration from [E. Iceland and A. Samorodnitsky, Discrete Comput. Geom., 63 (2020), pp. 560–576]. That is, we show that if [math] is an arbitrary encoding map [math] for the 3-query LCC, then all vectors in [math] can be written as a [math]-sparse linear combination of the [math]’s, which immediately implies [math]. The proof of this fact proceeds by iteratively reducing the size of any arbitrary linear combination of at least [math] of the [math]’s. We achieve this using the recent breakthrough result of [N. Alon et al., Proc. Lond. Math. Soc. (3), 130 (2025), e70044] on the existence of rainbow cycles in properly edge-colored graphs, applied to graphs capturing the linear dependencies underlying the local correction property.

Reverse Mathematics of Complexity Lower Bounds

Lijie Chen — 2026-04-29T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are necessary to prove a given theorem. In this work, we systematically explore the reverse mathematics of complexity lower bounds. We explore reversals in the setting of bounded arithmetic, with Cook’s theory [math] as the base theory, and show that several natural lower bound statements about communication complexity, error correcting codes, and Turing machines are equivalent to widely investigated combinatorial principles such as the weak pigeonhole principle for polynomial-time functions and its variants. As a consequence, complexity lower bounds can be formally seen as fundamental mathematical axioms with far-reaching implications. The proof-theoretic equivalence between complexity lower bound statements and combinatorial principles yields several new implications for the (un)provability of lower bounds. Among other results, we derive the following consequences: (1) Under a plausible cryptographic assumption, the classical single-tape Turing machine [math]-time lower bound for Palindrome is unprovable in Jeřábek’s theory [math]. The conditional unprovability of this simple lower bound goes against the intuition shared by some researchers that most complexity lower bounds could be established in [math]. (2) While [math] proves one-way communication lower bounds for set disjointness, it does not prove one-way communication lower bounds for Equality, under a plausible cryptographic assumption. (3) An amplification phenomenon connected to the (un)provability of some lower bounds, under which a quantitatively weak [math] lower bound is provable if and only if a stronger (and often tight) [math] lower bound is provable. (4) Feasibly definable randomized algorithms can be feasibly defined deterministically ([math] is [math]-conservative over [math]) if and only if one-way communication complexity lower bound for set disjointness are provable in [math].

A Nearly Quadratic-Time FPTAS for Knapsack

Lin Chen — 2025-08-20T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We investigate the classic Knapsack problem and propose a fully polynomial-time approximation scheme (FPTAS) that runs in [math] time. This improves upon the [math]-time algorithm by Deng, Jin, and Mao [Proceedings of the 2023 Annual ACM-SIAM Symposium on Discrete Algorithms, 2023]. Our algorithm is the best possible (up to a subpolynomial factor) conditioned on the conjecture that [math]-convolution has no truly subquadratic-time algorithm since this conjecture implies that Knapsack has no [math]-time FPTAS for any constant [math].

Online Edge Coloring Is (Nearly) as Easy as Offline

Joakim Blikstad — 2025-08-20T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The classic theorem of Vizing [Diskret. Analiz., 3 (1964), pp. 25–30] asserts that any graph of maximum degree [math] can be edge colored (offline) using no more than [math] colors (with [math] being a trivial lower bound). In the online setting, Bar-Noy, Motwani, and Naor [Inform. Process. Lett., 44 (1992), pp. 251–253] conjectured that a [math]-edge-coloring can be computed online in [math]-vertex graphs of maximum degree [math]. Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., Aggarwal et al. [Proceedings of FOCS, 2003, pp. 502–512], Bahmani, Mehta, and Motwani (SODA’10), Cohen, Peng, and Wajc [Proceedings of FOCS, 2019, pp. 1–25], Bhattacharya, Grandoni, and Wajc [Proceedings of SODA, 2021, pp. 2830–2842], Kulkarni et al. [Proceedings of STOC, 2022, pp. 2958–2977]). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn [J. Combin. Theory Ser. A, 73 (1996), pp. 1–59] and of the recent “local” edge coloring result of Christiansen [Proceedings of STOC, 2023, pp. 1013–1026].

Ghost Value Augmentation for [math]-Edge-Connectivity

D Ellis Hershkowitz — 2025-09-08T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give a poly-time algorithm for the [math]-edge-connected spanning subgraph ([math]-ECSS) problem that returns a solution of cost no greater than the cheapest [math]-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of 2 for [math]-ECSS whenever the optimal value of [math]-ECSS is close to that of [math]-ECSS. This is a property that holds for the closely related problem [math]-edge-connected spanning multisubgraph ([math]-ECSM), which is identical to [math]-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a [math]-approximation algorithm for [math]-ECSM, which resolves a conjecture of Pritchard and improves upon a recent [math]-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for [math]-ECSM, showing that our approximation ratio is tight up to the constant factor in [math], unless [math].

Relaxed Local Correctability from Local Testing

Vinayak M. Kumar — 2025-09-29T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We construct the first asymptotically good relaxed locally correctable codes with polylogarithmic query complexity, bringing the upper bound polynomially close to the lower bound of Gur and Lachish [SIAM J. Comput., 50 (2021), pp. 788–813]. Our result follows from showing that a high-rate locally testable code can boost the block length of a smaller relaxed locally correctable code, while preserving the correcting radius and incurring only a modest additive cost in rate and query complexity. We use the locally testable code’s tester to check if the amount of corruption in the input is low; if so, we can “zoom-in” to a suitable substring of the input and recurse on the smaller code’s local corrector. Hence, iterating this operation with a suitable family of locally testable codes due to Dinur et al. [Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pp. 357–374] yields asymptotically good codes with relaxed local correctability, arbitrarily large block length, and polylogarithmic query complexity. Our codes asymptotically inherit the rate and distance of any locally testable code used in the final invocation of the operation. Therefore, our framework also yields nonexplicit relaxed locally correctable codes with polylogarithmic query complexity that have rate and distance approaching the Gilbert–Varshamov bound.

Tree Evaluation is in Space [math][math]

James Cook — 2025-10-10T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The Tree Evaluation Problem ([math]) [Cook et al., 2009] is a central candidate for separating polynomial time ([math]) from logarithmic space ([math]) via composition. While space lower bounds of [math] are known for multiple restricted models, it was recently shown by Cook and Mertz (2020) that [math] can be solved in space [math]. Thus its status as a candidate hard problem for [math] remains a mystery. Our main result is to improve the space complexity of [math] to [math], thus greatly strengthening the case that tree evaluation is in fact in [math]. We show two consequences of these results. First, we show that the KRW conjecture [Karchmer, Raz, and Wigderson, 1995] implies [math]; this itself would have many implications, such as branching programs not being efficiently simulable by formulas. Our second consequence is to increase our understanding of amortized branching programs, also known as catalytic branching programs; we show that every function [math] on [math] bits can be computed by such a program of length [math] and width [math].

[math][math] Passes Are Optimal for Semistreaming Maximal Independent Set

Sepehr Assadi — 2025-10-13T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. In the semistreaming model for processing massive graphs, an algorithm makes multiple passes over the edges of a given [math]-vertex graph and is tasked with computing the solution to a problem using [math] space. Semistreaming algorithms for a maximal independent set (MIS) that run in [math] passes have been known for almost a decade; however, the best lower bounds can only rule out single-pass algorithms. We close this large gap by proving that the current algorithms are optimal: Any semistreaming algorithm for finding an MIS with constant probability of success requires [math] passes. This settles the complexity of this fundamental problem in the semistreaming model and constitutes one of the first optimal multipass lower bounds in this model. We establish our result by proving an optimal round versus communication trade-off for the (multiparty) communication complexity of MIS. The key ingredient of this result is a new technique, called hierarchical embedding, for performing round elimination: we show how to pack many but small hard [math]-round instances of the problem into a single [math]-round instance, in a way that enforces any [math]-round protocol to effectively solve all these [math]-round instances also. These embeddings are obtained via a novel application of results from extremal graph theory—in particular dense graphs with many disjoint unique shortest paths—together with a newly designed graph product, and are analyzed via information-theoretic tools such as direct-sum and message compression arguments.

One-Way Functions and Zero Knowledge

Shuichi Hirahara — 2026-01-26T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The fundamental theorem of Goldreich, Micali, and Wigderson [] shows that the existence of a one-way function is sufficient for constructing computational zero knowledge ([math]) proofs for all languages in [math]. We prove its converse, thereby establishing characterizations of one-way functions based on the worst-case complexities of zero knowledge. Specifically, we prove that the following are equivalent: (i) A one-way function exists. (ii) [math] and [math] is hard in the worst case. (iii) [math] is hard in the worst case and the problem [math] of approximating circuit complexity is in [math]. The characterization above also holds for statistical and computational zero-knowledge argument systems. We further extend this characterization to a proof system with knowledge complexity [math]. We complement this result by showing that [math] admits an interactive proof system with knowledge complexity [math] under the existence of an exponentially hard auxiliary-input one-way function (which is a weaker primitive than an exponentially hard one-way function). We also characterize the existence of an adversary-robust-often nonuniformly computable one-way function by the nondeterministic hardness of [math] under the weak assumption that [math]. There are two applications of our results. First, we simplify the proof of the recent characterization of a one-way function by [math]-hardness of a meta-computational problem and the worst-case hardness of [math] given by Hirahara []. Second, we show that if [math] has a laconic zero-knowledge argument system, then there exists a public-key encryption scheme whose security can be based on the worst-case hardness of [math]. This improves previous results which assume the existence of an indistinguishability obfuscation.

Agnostic Proper Learning of Monotone Functions: Beyond the Black-Box Correction Barrier

Jane Lange — 2025-01-23T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give the first agnostic, efficient, proper learning algorithm for monotone Boolean functions. Given [math] uniformly random examples of an unknown function [math], our algorithm outputs a hypothesis [math] that is monotone and [math]-close to [math], where [math] is the distance from [math] to the closest monotone function. The running time of the algorithm (and consequently the size and evaluation time of the hypothesis) is also [math], nearly matching the lower bound of [E. Blais et al., Proceedings of the 19th International Workshop on Randomness and Computation, 2015, pp. 512–527]. We also give an algorithm for estimating up to additive error [math] the distance of an unknown function [math] to monotone using a run-time of [math]. Previously, for both of these problems, sample-efficient algorithms were known, but these algorithms were not run-time efficient. Our work thus closes this gap in our knowledge between the run-time and sample complexity. This work builds upon the improper learning algorithm of [N. H. Bshouty and C. Tamon, J. ACM, 43 (1996), pp. 747–770] and the proper semiagnostic learning algorithm of [J. Lange, R. Rubinfeld, and A. Vasilyan, Proceedings of the 63rd Annual Symposium on Foundations of Computer Science, IEEE, 2022, pp. 75–86], which obtains a nonmonotone Boolean-valued hypothesis, then “corrects” it to monotone using query-efficient local computation algorithms on graphs. This black-box correction approach can achieve no error better than [math] information-theoretically; we bypass this barrier by (a) augmenting the improper learner with a convex optimization step, and (b) learning and correcting a real-valued function before rounding its values to Boolean. Our real-valued correction algorithm solves the “poset sorting” problem of Lange, Rubinfeld, and Vasilyan for functions over general posets with non-Boolean labels.

Parallel Repetition for the GHZ Game: Exponential Decay

Mark Braverman — 2025-02-04T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We show that the value of the [math]-fold repeated GHZ game is at most [math], improving upon the polynomial bound established by Holmgren and Raz. Our result is established via a reduction to approximate subgroup-type questions from additive combinatorics.

Super-Logarithmic Lower Bounds for Dynamic Graph Problems

Kasper Green Larsen — 2025-02-13T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. In this work, we prove an [math] unconditional lower bound on the maximum of the query time and update time for dynamic data structures supporting reachability queries in [math]-node directed acyclic graphs under edge insertions. This is the first super-logarithmic lower bound for any natural graph problem. In proving the lower bound, we also make novel contributions to the state-of-the-art data structure lower bound techniques that we hope may lead to further progress in proving lower bounds.

ABE for Circuits with poly[math]-Sized Keys from LWE

Valerio Cini — 2025-03-31T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We present a key-policy attribute-based encryption (ABE) scheme for circuits based on the Learning with Errors (LWE) assumption, whose key size is independent of the circuit depth. Our result constitutes the first improvement for ABE for circuits from LWE in almost a decade, given by Gorbunov, Vaikuntanathan, and Wee (Attribute-based encryption for circuits, 2013) and Boneh et al. (Fully key-homomorphic encryption, arithmetic circuit ABE and compact garbled circuits, 2014): We reduce the key size in the latter from [math] to [math]. The starting point of our construction is a recent ABE scheme of Li, Lin, and Luo (ABE for circuits with constant-size secret keys and adaptive security, 2022) which achieves [math] key size but requires pairings and generic bilinear groups in addition to LWE; we introduce new lattice techniques to eliminate the additional requirements.

Attribute-Based Encryption for Circuits of Unbounded Depth from Lattices: Garbled Circuits of Optimal Size, Laconic Functional Evaluation, and More

Yao-Ching Hsieh — 2025-04-17T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Although we have known about fully homomorphic encryption (FHE) from circular security assumptions for over a decade [C. Gentry, STOC ’09, ACM, New York, 2009, pp. 169–178; Z. Brakerski and V. Vaikuntanathan, FOCS ’11, IEEE Computer Society, Los Alamitos, CA, 2011, pp. 97–106], there is still a significant gap in understanding related homomorphic primitives supporting all unrestricted polynomial-size computations. One prominent example is attribute-based encryption (ABE). The state-of-the-art constructions, relying on the hardness of learning with errors (LWE) [S. Gorbunov, V. Vaikuntanathan, and H. Wee, STOC ’13, ACM, New York, 2013, pp. 545–554; D. Boneh et al., Eurocrypt ’14, Springer, Berlin, 2014, pp. 533–556], only accommodate circuits up to a predetermined depth, akin to leveled homomorphic encryption. In addition, their components (master public key, secret keys, and ciphertexts) have sizes polynomial in the maximum circuit depth. Even in the simpler setting where a single key is published (or a single circuit is involved), the depth dependency persists, showing up in constructions of 1-key ABE and related primitives, including laconic function evaluation (LFE), 1-key functional encryption (FE), and reusable garbling schemes. So far, the only approach of eliminating depth dependency relies on indistinguishability obfuscation. An interesting question that has remained open for over a decade is whether the circular security assumptions enabling FHE can similarly benefit ABE. In this work, we introduce new lattice-based techniques to overcome the depth-dependency limitations: relying on a circular security assumption, we construct LFE, 1-key FE, 1-key ABE, and reusable garbling schemes capable of evaluating circuits of unbounded depth and size; based on the evasive circular LWE assumption, a stronger variant of the recently proposed evasive LWE assumption [H. Wee, Eurocrypt ’22, Springer, Cham, Switzerland, 2022, pp. 217–241; R. Tsabary, Crypto ’22, Springer, Cham, Switzerland, 2022, pp. 535–559], we construct full-fledged ABE and predicate encryption (PE) schemes for circuits of unbounded depth and size. Our LFE, 1-key FE, and reusable garbling schemes achieve almost optimal succinctness (up to polynomial factors in the security parameter). Their ciphertexts and input encodings have sizes linear in the input length, while function digest, secret keys, and garbled circuits have constant sizes independent of circuit parameters (for Boolean outputs). In fact, this gives the first constant-size garbled circuits without relying on indistinguishability obfuscation. Our ABE and PE schemes offer short components, with master public key and ciphertext sizes linear in the attribute length and secret key being constant size.

A [math] Monotonicity Tester for Boolean Functions on [math]-Dimensional Hypergrids

Hadley Black — 2025-05-30T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Monotonicity testing of Boolean functions on the hypergrid, [math], is a classic topic in property testing. Determining the nonadaptive complexity of this problem is an important open question. For arbitrary [math], [H. Black, D. Chakrabarty, and C. Seshadhri, Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, 2020, pp. 1975–1994] describes a tester with query complexity [math]. This complexity is independent of [math] but has a suboptimal dependence on [math]. Recently, Braverman et al. [Proceedings of Innovations in Theoretical Computer Science, 2023, pp. 25:1–25:24] and H. Black, D. Chakrabarty, and C. Seshadhri [Proceedings of the 55th Annual ACM Symposium on Theory of Computing, 2023, pp. 233–241] described [math]- and [math]-query testers, respectively. These testers have an almost optimal dependence on [math] but a suboptimal polynomial dependence on [math]. In this paper, we describe a nonadaptive, one-sided monotonicity tester with query complexity [math], independent of [math]. Up to the [math]-factors, our result resolves the nonadaptive complexity of monotonicity testing for Boolean functions on hypergrids. The independence of [math] yields a nonadaptive, one-sided [math]-query monotonicity tester for Boolean functions [math] associated with an arbitrary product measure.

The Full Landscape of Robust Mean Testing: Sharp Separations between Oblivious and Adaptive Contamination

Clément Canonne — 2026-04-15T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We consider the question of Gaussian mean testing, a fundamental task in high-dimensional distribution testing and signal processing, subject to adversarial corruptions of the samples. We focus on the relative power of different adversaries and show that, in contrast to the common wisdom in robust statistics, there exists a strict separation between adaptive adversaries (strong contamination) and oblivious ones (weak contamination) for this task. Specifically, we resolve both the information-theoretic and computational landscapes for robust mean testing. In the exponential-time setting, we establish the tight sample complexity of testing [math] against [math], where [math], with an [math]-fraction of oblivious adversarial corruptions, to be [math], while the complexity against adaptive adversarial corruptions is [math], which is strictly worse for a large range of vanishing [math]. To the best of our knowledge, ours is the first separation in sample complexity between the strong and weak contamination models. In the polynomial-time setting, we close a gap in the literature by providing a polynomial-time algorithm against adaptive adversaries achieving the above sample complexity [math], and a low-degree lower bound (which complements an existing reduction from planted clique) suggesting that all efficient algorithms require this many samples, even in the oblivious-adversary setting.

Removing Additive Structure in 3SUM-Based Reductions

Ce Jin — 2024-09-05T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Our work explores the hardness of 3SUM instances without certain additive structures, and its applications. As our main technical result, we show that solving 3SUM on a size-[math] integer set that avoids solutions to [math] for [math] still requires [math] time, under the 3SUM hypothesis. Such sets are called Sidon sets and are well-studied in the field of additive combinatorics. Combined with previous reductions, this implies that the all-edges sparse triangle problem on [math]-vertex graphs with maximum degree [math] and at most [math] [math]-cycles for every [math] requires [math] time, under the 3SUM hypothesis. This can be used to strengthen the previous conditional lower bounds by Abboud, Bringmann, Khoury, and Zamir [54th Annual ACM SIGACT Symposium on Theory of Computing, 2022] of 4-cycle enumeration, offline approximate distance oracle and approximate dynamic shortest path. In particular, we show that no algorithm for the 4-cycle enumeration problem on [math]-vertex [math]-edge graphs with [math] delays has [math] or [math] preprocessing time for [math]. We also present a matching upper bound via simple modifications of the known algorithms for 4-cycle detection. A slight generalization of the main result also extends the result of Dudek, Gawrychowski, and Starikovskaya [52nd Annual ACM SIGACT Symposium on Theory of Computing, 2020] on the 3SUM-hardness of nontrivial 3-variate linear degeneracy testing (3-LDTs): we show 3SUM-hardness for all nontrivial 4-LDTs. The proof of our main technical result combines a wide range of tools: Balog–Szemerédi–Gowers theorem, sparse convolution algorithm, and a new almost-linear hash function with almost 3-universal guarantee for integers that do not have small-coefficient linear relations.

Resolving Matrix Spencer Conjecture up to Poly-Logarithmic Rank

Nikhil Bansal — 2024-09-12T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric [math] matrices [math] each with [math] and rank at most [math], one can efficiently find [math] signs [math] such that their signed sum has spectral norm [math]. This result also implies a [math] qubit lower bound for quantum random access codes encoding [math] classical bits with advantage [math]. Our proof uses the recent refinement of the noncommutative Khintchine inequality due to Bandeira, Boedihardjo, and van Handel [Invent. Math., 234 (2023), pp. 419–487] for random matrices with correlated Gaussian entries.

Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics

Amir Abboud — 2024-09-13T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The “short cycle removal” technique was recently introduced by Abboud et al. [Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2022, pp. 1487–1500] to prove the fine-grained hardness of approximation. Its main technical result is that listing all triangles in an [math]-regular graph is [math]-hard even when the number of short cycles is small, namely, when the number of [math]-cycles is [math] for [math]. Its corollaries are based on the 3-SUM conjecture and their strength depends on [math], i.e., on how effectively the short cycles are removed. Abboud et al. achieve [math] by applying structure versus randomness arguments on graphs. In this paper, we take a step back and apply conceptually similar arguments on the numbers of the 3-SUM problem, from which the hardness of triangle listing is derived. Consequently, we achieve [math] and the following lower bound corollaries under the 3-SUM conjecture: Approximate distance oracles: The seminal Thorup–Zwick distance oracles achieve stretch [math] after preprocessing a graph in [math] time. For the same stretch, and assuming the query time is [math], Abboud et al. proved an [math] lower bound on the preprocessing time; we improve it to [math], which is only a factor 2 away from the upper bound. Additionally, we obtain tight bounds for stretch [math] and [math] and higher lower bounds for dynamic shortest paths. Listing 4-cycles: Abboud et al. proved the first superlinear lower bound for listing all 4-cycles in a graph, ruling out [math] time algorithms where [math] is the number of 4-cycles. We settle the complexity of this basic problem by showing that the [math] upper bound is tight up to [math] factors. Our results exploit a rich tool set from additive combinatorics, most notably the Balog–Szemerédi–Gowers theorem and Rusza’s covering lemma. A key ingredient that may be of independent interest is a truly subquadratic algorithm for 3-SUM if one of the sets has small doubling.

Generic Reed–Solomon Codes Achieve List-Decoding Capacity

Joshua Brakensiek — 2024-11-04T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. In a recent paper, Brakensiek, Gopi, and Makam [arXiv preprint, https://arxiv.org/abs/2107.10822, 2021] introduced higher-order maximum distance separable (MDS) codes as a generalization of MDS codes. An order-[math] MDS code, denoted by [math], has the property that any [math] subspaces formed from columns of its generator matrix intersect as minimally as possible. An independent work by Roth [arXiv preprint, https://arxiv.org/abs/2111.03210, 2021] defined a different notion of higher-order MDS codes as those achieving a generalized singleton bound for list-decoding. In this work, we show that these two notions of higher-order MDS codes are (nearly) equivalent. We also show that generic Reed–Solomon codes are [math] for all [math], relying crucially on the GM–MDS theorem, which shows that generator matrices of generic Reed–Solomon codes achieve any possible zero pattern. As a corollary, this implies that generic Reed–Solomon codes achieve list-decoding capacity. More concretely, we show that, with high probability, a random Reed–Solomon code of rate [math] over an exponentially large field is list decodable from radius [math] with list size at most [math], resolving a conjecture of Shangguan and Tamo [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, 2020, pp. 538–551].

An Improved Parameterized Algorithm for Treewidth

Tuukka Korhonen — 2024-11-22T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give an algorithm that takes as input an [math]-vertex graph [math] and an integer [math], runs in time [math], and outputs a tree decomposition of [math] of width at most [math] if such a decomposition exists. This resolves the long-standing open problem of whether there is a [math] time algorithm for treewidth. In particular, our algorithm is the first improvement on the dependency on [math] in algorithms for treewidth since the [math] time algorithm given by Bodlaender and Kloks [Proceedings of the 18th International Colloquium of Automata, Languages and Programming (ICALP 1991), Lecture Notes in Comput. Sci. 510, Springer, 1991, pp. 544–555] and Lagergren and Arnborg [Proceedings of the 18th International Colloquium of Automata, Languages and Programming (ICALP 1991), Lecture Notes in Comput. Sci. 510, Springer, 1991, pp. 532–543]. We also give an algorithm that, given an [math]-vertex graph [math], an integer [math], and a rational [math], in time [math] either outputs a tree decomposition of [math] of width at most [math] or determines that the treewidth of [math] is larger than [math]. Prior to our work, no approximation algorithms for treewidth with approximation ratio <2, other than the exact algorithms, were known. Both of our algorithms work in polynomial space.

Almost-Optimal Sublinear Additive Spanners

Zihan Tan — 2024-12-02T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Given an undirected unweighted graph [math] on [math] vertices and [math] edges, a subgraph [math] is a spanner of [math] with stretch function [math], if for every pair [math] of vertices in [math], [math]. When [math], [math] is called a sublinear additive spanner; when [math], [math] is called an additive spanner, and [math] is usually called the additive stretch of [math]. As our primary result, we show that for any constant [math] and constant integer [math], every graph on [math] vertices has a sublinear additive spanner with stretch function [math] and [math] edges. When [math], this improves upon the previous spanner construction with stretch function [math] and [math] edges [S. Chechik, New additive spanners, in Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2013, pp. 498–512]; for any constant integer [math], this improves upon the previous spanner construction with stretch function [math] and [math] edges [S. Pettie, ACM Trans. Algorithms, 6 (2010), 7]. Most importantly, the size of our spanners almost matches the lower bound of [math] [A. Abboud, G. Bodwin, and S. Pettie, SIAM J. Comput., 47 (2018), pp. 2203–2236], which holds for all compression schemes achieving the same stretch function. As our second result, we show a new construction of additive spanners with stretch [math] and [math] edges, which slightly improves upon the previous stretch bound of [math] achieved by linear-size spanners [G. Bodwin and V. V. Williams, ACM Trans. Algorithms, 17 (2021), 36]. An additional advantage of our spanner is that it admits a subquadratic construction runtime of [math], while the previous construction in [G. Bodwin and V. V. Williams, ACM Trans. Algorithms, 17 (2021), 36] requires all-pairs shortest paths computation which takes [math] time.

A Near-Cubic Lower Bound for 3-Query Locally Decodable Codes from Semirandom CSP Refutation

Omar Alrabiah — 2025-01-07T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. A code [math] is a [math]-locally decodable code ([math]-LDC) if one can recover any chosen bit [math] of the message [math] with good confidence by randomly querying the encoding [math] on at most [math] coordinates. Existing constructions of 2-LDCs achieve [math], and lower bounds show that this is in fact tight. However, when [math], far less is known: the best constructions achieve [math], while the best known results only show a quadratic lower bound [math] on the blocklength. In this paper, we prove a near-cubic lower bound of [math] on the blocklength of 3-query LDCs. This improves on the best known prior works by a polynomial factor in [math]. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs (constraint satisfaction problems) developed in [V. Guruswami, P. K. Kothari, and P. Manohar, Algorithms and certificates for Boolean CSP refutation: Smoothed is no harder than random, in Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, 2022, pp, 678–689] and [Y. Ishai and E. Kushilevitz, On the hardness of information-theoretic multiparty computation, in Advances in Cryptology - EUROCRYPT 2004, Springer, 2004, pp. 439–455]. In particular, our work relies on bounding the spectral norm of appropriate Kikuchi matrices.

Faster Isomorphism for [math]-Groups of Class 2 and Exponent [math]

Xiaorui Sun — 2025-04-18T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The group isomorphism problem determines whether two groups, given by their Cayley tables, are isomorphic. For groups with order [math], an algorithm with [math] running time, attributed to Tarjan, was proposed in the 1970s [G. L. Miller, On the [math] isomorphism technique: A preliminary report, in Proceedings of the ACM Symposium on Theory of Computing (STOC), 1978, pp. 51–58]. Despite the extensive study over the past decades, the current best group isomorphism algorithm has an [math] running time [D. J. Rosenbaum, Bidirectional Collision Detection and Faster Deterministic Isomorphism Testing, preprint, 2013]. The isomorphism testing for [math]-groups of (nilpotent) class 2 and exponent [math] has been identified as a major barrier to obtaining an [math] time algorithm for the group isomorphism problem. Although the [math]-groups of class 2 and exponent [math] have much simpler algebraic structures than general groups, the best-known isomorphism testing algorithm for this group class also has an [math] running time. In this paper, we present an isomorphism testing algorithm for [math]-groups of class 2 and exponent [math] with running time [math] for any prime [math]. Our result is based on a novel reduction to the skew-symmetric matrix tuple isometry problem [G. Ivanyos and Y. Qiao, SIAM J. Comput., 48 (2019), pp. 926–963]. To obtain the reduction, we develop several tools for matrix space analysis, including a matrix space individualization-refinement method and a characterization of the low rank matrix spaces.

Doubly Efficient Private Information Retrieval and Fully Homomorphic Ram Computation from Ring LWE

Wei-Kai Lin — 2026-03-05T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. A (single server) private information retrieval (PIR) allows a client to read data from a public database held on a remote server, without revealing to the server which locations she is reading. In a doubly efficient PIR (DEPIR), the database is first preprocessed, but the server can subsequently answer any client’s query in time that is sublinear in the database size. Prior work gave a plausible candidate for a public-key variant of DEPIR, where a trusted party is needed to securely preprocess the database and generate a corresponding public key for the clients; security relied on a new nonstandard code-based assumption and a heuristic use of ideal obfuscation. In this work we construct the stronger unkeyed notion of DEPIR, where the preprocessing is a deterministic procedure that the server can execute on its own. Moreover, we prove security under just the standard ring learning-with-errors (RingLWE) assumption. For a database of size [math] and any constant [math], the preprocessing run-time and size is [math], while the run-time and communication complexity of each PIR query is [math]. We also show how to update the preprocessed database in time [math]. Our approach is to first construct a standard PIR where the server’s computation consists of evaluating a multivariate polynomial; we then convert it to a DEPIR by preprocessing the polynomial to allow for fast evaluation, using the techniques of Kedlaya and Umans [Proceedings of the 49th Annual Symposium on Foundations of Computer Science, 2008, pp. 146–155]. Building on top of our DEPIR, we construct general fully homomorphic encryption for random-access machines (RAM-FHE), which allows a server to homomorphically evaluate an arbitrary RAM program [math] over a client’s encrypted input [math] and the server’s preprocessed plaintext input [math] to derive an encryption of the output [math] in time that scales with the RAM run-time of the computation rather than its circuit size. Prior work only gave a heuristic candidate construction of a restricted notion of RAM-FHE. In this work, we construct RAM-FHE under the RingLWE assumption with circular security. For a RAM program [math] with worst-case run-time [math], the homomorphic evaluation runs in time [math].

A Strong Version of Cobham’s Theorem

Philipp Hieronymi — 2024-01-17T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Let [math] be two multiplicatively independent integers. Cobham’s famous theorem states that a set [math] is both [math]-recognizable and [math]-recognizable if and only if it is definable in Presburger arithmetic. Here we show the following strengthening: let [math] be [math]-recognizable, and let [math] be [math]-recognizable such that both [math] and [math] are not definable in Presburger arithmetic. Then the first-order logical theory of [math] is undecidable. This is in contrast to a well-known theorem of Büchi stating that the first-order logical theory of [math] is decidable.

The Shortest Even Cycle Problem Is Tractable

Andreas Björklund — 2024-02-15T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Given a directed graph as input, we show how to efficiently find a shortest (directed, simple) cycle on an even number of vertices. As far as we know, no polynomial-time algorithm was previously known for this problem. In fact, finding any even cycle in a directed graph in polynomial time was open for more than two decades until Robertson, Seymour, and Thomas (Ann. of Math. (2), 150 (1999), pp. 929–975) and, independently, McCuaig (Electron. J. Combin., 11 (2004), R7900) (announced jointly at STOC 1997) gave an efficiently testable structural characterization of even-cycle-free directed graphs. Methodologically, our algorithm relies on the standard framework of algebraic fingerprinting and randomized polynomial identity testing over a finite field and, in fact, relies on a generating polynomial implicit in a paper of Vazirani and Yannakakis (Discrete Appl. Math., 25 (1989), pp. 179–190) that enumerates weighted cycle covers by the parity of their number of cycles as a difference of a permanent and a determinant polynomial. The need to work with the permanent—known to be #P-hard apart from a very restricted choice of coefficient rings (L. G. Valiant, Theoret. Comput. Sci., 8 (1979), pp. 189–201)—is where our main technical contribution occurs. We design a family of finite commutative rings of characteristic 4 that simultaneously (i) give a nondegenerate representation for the generating polynomial identity via the permanent and the determinant, (ii) support efficient permanent computations by extension of Valiant’s techniques, and (iii) enable emulation of finite-field arithmetic in characteristic 2. Here our work is foreshadowed by that of Björklund and Husfeldt (SIAM J. Comput., 48 (2019), pp. 1698–1710) who used a considerably less efficient commutative ring design—in particular, one lacking finite-field emulation—to obtain a polynomial-time algorithm for the shortest two disjoint paths problem in undirected graphs. Building on work of Gilbert and Tarjan (Numer. Math., 50 (1986), pp. 377–404) as well as Alon and Yuster (J. ACM, 42 (2013), pp. 844–856), we also show how ideas from the nested dissection technique for solving linear equation systems—introduced by George (SIAM J. Numer. Anal., 10 (1973), pp. 345–363) for symmetric positive definite real matrices—leads to faster algorithm designs in our present finite-ring randomized context when we have control of the separator structure of the input graph; for example, this happens when the input has bounded genus.

Clustering Mixtures with Almost Optimal Separation in Polynomial Time

Jerry Li — 2024-02-22T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We consider the problem of clustering mixtures of mean-separated Gaussians in high dimensions. We are given samples from a mixture of [math] identity covariance Gaussians, so that the minimum pairwise distance between any two pairs of means is at least [math], for some parameter [math], and the goal is to recover the ground truth clustering of these samples. It is folklore that separation [math] is both necessary and sufficient to recover a good clustering (say, with constant or [math] error), at least information-theoretically. However, the estimators which achieve this guarantee are inefficient. We give the first algorithm which runs in polynomial time in both [math] and the dimension [math], and which almost matches this guarantee. More precisely, we give an algorithm which takes polynomially many samples and time, and which can successfully recover a good clustering, so long as the separation is [math], for any [math]. Previously, polynomial time algorithms were only known for this problem when the separation was polynomial in [math], and all algorithms which could tolerate [math] separation required quasipolynomial time. We also extend our result to mixtures of translations of a distribution which satisfies the Poincaré inequality, under additional mild assumptions. Our main technical tool, which we believe is of independent interest, is a novel way to implicitly represent and estimate high degree moments of a distribution, which allows us to extract important information about high degree moments without ever writing down the full moment tensors explicitly.

Flow Time Scheduling and Prefix Beck–Fiala

Nikhil Bansal — 2024-02-26T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We relate discrepancy theory with the classic scheduling problems of minimizing max flow time and total flow time on unrelated machines. Specifically, we give a general reduction that allows us to transfer discrepancy bounds in the prefix Beck–Fiala (bounded [math]-norm) setting to bounds on the flow time of an optimal schedule. Combining our reduction with a deep result proved by Banaszczyk via convex geometry gives guarantees of [math] and [math] for max flow time and total flow time, respectively, improving upon the previous best guarantees of [math] and [math]. Apart from the improved guarantees, the reduction motivates seemingly easy versions of prefix discrepancy questions: any constant bound on prefix Beck–Fiala where vectors have sparsity two (sparsity one being trivial) would already yield tight guarantees for both max flow time and total flow time. While known techniques solve this case when the entries take values in [math], we show that they are unlikely to transfer to the more general 2-sparse case of bounded [math]-norm.

Testing Thresholds for High-Dimensional Sparse Random Geometric Graphs

Siqi Liu — 2024-02-27T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The random geometric graph model [math] is a distribution over graphs in which the edges capture a latent geometry. To sample [math], we identify each of our [math] vertices with an independently and uniformly sampled vector from the [math]-dimensional unit sphere [math], and we connect pairs of vertices whose vectors are “sufficiently close,” such that the marginal probability of an edge is [math]. Because of the underlying geometry, this model is natural for applications in data science and beyond. We investigate the problem of testing for this latent geometry, or, in other words, distinguishing an Erdős–Rényi graph [math] from a random geometric graph [math]. It is not too difficult to show that if [math] while [math] is held fixed, the two distributions become indistinguishable; we wish to understand how fast [math] must grow as a function of [math] for indistinguishability to occur. When [math] for constant [math], we prove that if [math], the total variation distance between the two distributions is close to 0; this improves upon the best previous bound of Brennan, Bresler, and Nagaraj (2020), which required [math], and further our result is nearly tight, resolving a conjecture of Bubeck, Ding, Eldan, and Rácz (2016) up to logarithmic factors. We also obtain improved upper bounds on the statistical indistinguishability thresholds in [math] for the full range of [math] satisfying [math], improving upon the previous bounds by polynomial factors. Our analysis uses the belief propagation algorithm to characterize the distributions of (subsets of) the random vectors conditioned on producing a particular graph. In this sense, our analysis is connected to the “cavity method” from statistical physics. To analyze this process, we rely on novel sharp estimates for the area of the intersection of a random sphere cap with an arbitrary subset of [math], which we prove using optimal transport maps and entropy-transport inequalities on the unit sphere. We believe these techniques may be of independent interest.

Complexity Classification of Counting Graph Homomorphisms Modulo a Prime Number

Andrei Bulatov — 2024-07-01T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Counting graph homomorphisms and its generalizations such as the counting constraint satisfaction problem (CSP), variations of the counting CSP, and counting problems in general have been intensively studied since the pioneering work of Valiant. While the complexity of exact counting of graph homomorphisms [M. Dyer and C. Greenhill, Random Structures Algorithms, 17 (2000), pp. 260–289] and the counting CSP [A. A. Bulatov, J. ACM, 60 (2013), pp. 34:1–34:41, and M. E. Dyer and D. Richerby, SIAM J. Comput., 42 (2013), pp. 1245–1274] is well understood, counting modulo some natural number has attracted considerable interest as well. In their 2015 paper, [J. Faben and M. Jerrum, Theory Comput., 11 (2015), pp. 35–57] suggested a conjecture stating that counting homomorphisms to a fixed graph [math] modulo a prime number is hard whenever it is hard to count exactly unless [math] has automorphisms of certain kind. In this paper, we confirm this conjecture. As a part of this investigation, we develop techniques that widen the spectrum of reductions available for modular counting and apply to the general CSP rather than being limited to graph homomorphisms.

Circuits Resilient to Short-Circuit Errors

Klim Efremenko — 2024-07-16T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Given a Boolean circuit [math], we wish to convert it to a circuit [math] that computes the same function as [math], even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [D. J. Kleitman, F. T. Leighton, and Y. Ma, J. Comput. System Sci., 55 (1997), pp. 385–401]. Can we design such a resilient circuit [math] whose size is roughly comparable to that of [math]? Prior work [T. Kalai, A. B. Lewko, and A. Rao, Formulas resilient to short-circuit errors, in Foundations of Computer Science (FOCS), 2012, pp. 490–499; M. Braverman et al., Optimal short-circuit resilient formulas, in Computational Complexity Conference (CCC), Vol. 137, 2019, pp. 10:1–10:22] gave a positive answer for the special case where [math] is a formula. We study the general case and show that any Boolean circuit [math] of size [math] can be converted to a new circuit [math] of quasi-polynomial size [math] that computes the same function as [math], even if a [math] fraction of the gates on any root-to-leaf path in [math] are short circuited. Moreover, if the original circuit [math] is a formula, the resilient circuit [math] is of near-linear size [math]. The construction of our resilient circuits utilizes the connection between circuits and dag-like communication protocols [A. Razborov, Izvestiya of the RAN, 59 (1995), pp. 201–224; P. Pudlák, On extracting computations from propositional proofs (a survey), in Foundations of Software Technology and Theoretical Computer Science (FSTTCS) Vol. 8, 2010, pp. 30–41; D. Sokolov, Dag-like communication and its applications, in Computer Science Symposium in Russia (CSR), Springer, 2017, pp. 294–307], originally introduced in the context of proof complexity.

The Power of Two Choices in Graphical Allocation

Nikhil Bansal — 2024-08-26T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The graphical balls-into-bins process is a generalization of the classical 2-choice balls-into-bins process, where the bins correspond to vertices of an arbitrary underlying graph [math]. At each time step an edge of [math] is chosen uniformly at random, and a ball must be assigned to either of the two endpoints of this edge. The standard 2-choice process corresponds to the case of [math]. For any [math]-edge-connected, [math]-regular graph on [math] vertices, and any number of balls, we give an allocation strategy that, with high probability, ensures a gap of [math] between the load of any two bins. In particular, this implies a polylogarithmic bound for natural graphs such as cycles and tori, for which the classical greedy allocation strategy is conjectured to have a polynomial gap between the bin loads. For every graph [math], we also show an [math] lower bound on the gap achievable by any allocation strategy. This implies that our strategy achieves the optimal gap, up to polylogarithmic factors, for every graph [math]. Our allocation algorithm is simple to implement and requires only [math] time per allocation. It can be viewed as a more global version of the greedy strategy that compares average load on certain fixed sets of vertices, rather than on individual vertices. A key idea is to relate the problem of designing a good allocation strategy to that of finding suitable multicommodity flows. To this end, we consider Räcke’s cut-based decomposition tree and define certain orthogonal flows on it.

Algorithms and Certificates for Boolean CSP Refutation: Smoothed Is No Harder than Random

Venkatesan Guruswami — 2024-09-05T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst- and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an [math]-variable smoothed instance of a [math]-arity CSP, our algorithm runs in [math] time and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from 1, provided that the number of constraints is at least [math]. This matches, up to polylogarithmic factors in [math], the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [P. Raghavendra, S. Rao, and T. Schramm, Strongly refuting random CSPs below the spectral threshold, in STOC’17—Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, 2017, pp. 121–131]. We also make a surprising connection between the analysis of our refutation algorithm in the significantly “randomness starved” setting of semirandom [math]-XOR and the existence of even covers in worst-case hypergraphs. We use this connection to positively resolve Feige’s 2008 conjecture—an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the “spectral threshold” of [math], extending the celebrated result for random 3-SAT of [U. Feige, J. H. Kim, and E. Ofek, Witnesses for non-satisfiability of dense random 3CNF formulas, in Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, 2006, pp. 497–508].

The Optimal Error Resilience of Interactive Communication over Binary Channels

Meghal Gupta — 2024-12-12T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. In interactive coding, Alice and Bob wish to compute some function [math] of their individual private inputs [math] and [math]. They do this by engaging in a nonadaptive (fixed order, fixed length) interactive protocol to jointly compute [math]. The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions to the protocol, both parties still learn [math]. We study the optimal error resilience of such a protocol in the face of adversarial bit flips or erasures. While the optimal error resilience of such a protocol over a large alphabet is well understood, the situation over the binary alphabet has remained open. Over the binary alphabet, there has remained a substantial gap in error resilience between the best protocol construction and the best known upper bound, for both bit flips and erasures. In this work, we construct protocols meeting the known upper bounds for both types of errors, thereby closing this gap and resolving the question of optimal error resilience. Specifically, in the case of erasures, we construct a protocol achieving the optimal [math] erasure resilience, with communication complexity linear in the size of the minimal noiseless protocol computing [math]. In the case of bit flips, we determine the optimal error resilience over the binary bit flip channel for the message exchange problem (where [math]) to be [math]. The communication complexity of our protocol is polynomial in the size of the parties’ inputs. We remark that this implies an interactive coding scheme for any [math] resilient to [math] errors with an exponential blowup in communication complexity.

Hop-Constrained Oblivious Routing

Mohsen Ghaffari — 2023-02-07T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We prove the existence of an oblivious routing scheme that is [math]-competitive in terms of [math], thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize [math], defined as follows: The dilation is the maximum path hop length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have [math] within a [math] factor of the best possible value. More precisely, for any integer hop constraint [math], this oblivious routing scheme selects paths of length at most [math] and is [math]-competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most [math] hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002; Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008], which are [math]-competitive in terms of congestion but are not competitive in terms of dilation.

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches

Federica Cecchetto — 2023-04-12T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We consider the connectivity augmentation problem (CAP), a classical problem in the area of survivable network design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a [math]-edge-connected graph [math] and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to [math] makes the graph [math]-edge-connected. If [math] is odd, the problem is known to reduce to the tree augmentation problem (TAP)—i.e., [math] is a spanning tree—for which significant progress has been achieved recently, leading to approximation factors below 1.5 (the current best factor is 1.458). However, advances on TAP have not carried over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli [Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020, pp. 815–825] managed to obtain the first approximation factor below 2 for CAP by presenting a 1.91-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below 1.5, based on a new analysis technique. Through these ingredients, we obtain a 1.393-approximation algorithm for CAP, and therefore also for TAP. This leads to the current best approximation result for both problems in a unified way, by significantly improving on the abovementioned 1.91-approximation for CAP and also the previously best approximation factor of 1.458 for TAP by Grandoni, Kalaitzis, and Zenklusen [Proceedings of the 50th ACM Symposium on Theory of Computing, 2018, pp. 632–645]. Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio of the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below 1.5.

Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion

Zongchen Chen — 2023-10-20T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We prove an optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari, Liu, and Oveis Gharan [Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 1319–1330] and shows [math] mixing time on any [math]-vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hardcore model on independent sets weighted by a fugacity [math], we establish [math] mixing time for the Glauber dynamics on any [math]-vertex graph of constant maximum degree [math] when [math], where [math] is the critical point for the uniqueness/nonuniqueness phase transition on the [math]-regular tree. More generally, for any antiferromagnetic 2-spin system we prove the [math] mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain [math] mixing for [math]-colorings of triangle-free graphs of maximum degree [math] when the number of colors satisfies [math], where [math], and [math] mixing for generating random matchings of any graph with bounded degree and [math] edges. Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2020, pp. 1198–1211] to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan, Guo, and Mousa, [Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2019, pp. 1358–1370].

Adversarial Laws of Large Numbers and Optimal Regret in Online Classification

Noga Alon — 2023-11-03T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed subpopulation is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by [O. Ben-Eliezer and E. Yogev, The adversarial robustness of sampling, in Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS), 2020, pp. 49–62] and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are online learnable. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of Littlestone’s dimension, thus resolving the main open question from [S. Ben-David, D. Pál, and S. Shalev-Shwartz, Agnostic online learning, in Proceedings of the 22nd Conference on Learning Theory (COLT), 2009], which was also posed by [A. Rakhlin, K. Sridharan, and A. Tewari, J. Mach. Learn. Res., 16 (2015), pp. 155–186].

Discrepancy Minimization via a Self-Balancing Walk

Ryan Alweiss — 2024-01-12T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We study discrepancy minimization for vectors in [math] under various settings. The main result is the analysis of a new simple random process in high dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for online vector balancing against oblivious adversaries, resolving several questions posed by Bansal et al. [STOC, ACM, New York, 2020, pp. 1139–1152], as well as a linear time algorithm for logarithmic bounds for the Komlós conjecture.

Almost Optimal SuperConstant-Pass Streaming Lower Bounds for Reachability

Lijie Chen — 2024-04-09T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give an almost quadratic [math] lower bound on the space usage of any [math]-pass streaming algorithm solving the (directed) [math]-[math] reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including maximum matching, shortest path, matrix rank, and linear programming. Our main technical contribution is the definition and construction of set hiding graphs, that may be of independent interest: we give a general way of encoding a set [math] as a directed graph with [math] vertices, such that deciding whether [math] boils down to deciding if [math] is reachable from [math], for a specific pair of vertices [math] in the graph. Furthermore, we prove that our graph “hides” [math], in the sense that no low-space streaming algorithm with a small number of passes can learn (almost) anything about [math].

Kronecker Products, Low-Depth Circuits, and Matrix Rigidity

Josh Alman — 2025-01-13T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. For a matrix [math] and a positive integer [math], the rank [math] rigidity of [math] is the smallest number of entries of [math] which one must change to make its rank at most [math]. There are many well-known applications of rigidity lower bounds to a variety of areas in complexity theory but fewer applications are known of rigidity upper bounds. In this paper, we use rigidity upper bounds to prove new upper bounds in a few different models of computation. Our results include the following: (1) For any [math], and over any field [math], the [math] Walsh–Hadamard transform has a depth-[math] linear circuit of size [math]. This circumvents a known lower bound of [math] for circuits with bounded coefficients over [math] [P. Pudlák, Inform. Process. Lett., 74 (2000), pp. 197–201] by using coefficients of magnitude polynomial in [math]. Our construction also generalizes to linear transformations given by a Kronecker power of any fixed [math] matrix. (2) The [math] Walsh–Hadamard transform has a linear circuit of size [math], improving on the bound of [math] which one obtains from the standard fast Walsh–Hadamard transform. (3) There is a new rigidity upper bound showing that the following classes of matrices are not rigid enough to prove circuit lower bounds using Valiant’s approach: for any field [math] and any function [math], the matrix [math], given by [math], for any [math] and for any field [math] and any fixed-size matrices [math], the Kronecker product [math]. This generalizes recent results on nonrigidity using a simpler approach which avoids needing the polynomial method. (4) There are new connections between recursive linear transformations, such as Fourier and Walsh–Hadamard transforms, and circuits for matrix multiplication.

Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity

Rahul Santhanam — 2025-06-10T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS (Ideal Proof System) and in the context of propositional proof systems more generally. We give an explicit sequence of formulas in conjuctive normal form (CNF) [math] such that [math] iff there are no polynomial-size IPS proofs for the formulas [math]. This provides a natural equivalence between proof size lower bounds and standard algebraic complexity lower bounds. Our proof of this fact uses the implication from IPS lower bounds to algebraic complexity lower bounds due to Grochow and Pitassi together with a diagonalization argument: the formulas [math] themselves assert the nonexistence of short IPS proofs for formulas encoding [math] at a different input length. Our result also has meta-mathematical implications: it gives evidence for the difficulty of proving strong lower bounds for IPS within IPS. More generally, for any strong enough propositional proof system [math] we propose a new explicit hard candidate, the iterated [math]-lower bound formulas, which inductively asserts the nonexistence of short [math] proofs for formulas encoding this same statement at a different input length. We show that these formulas are unconditionally hard for resolution following recent results of Atserias and Müller and of Garlik. We further give evidence in favor of this hypothesis for other proof systems.

A Polynomial Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection

Deeparnab Chakrabarty — 2023-02-07T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making [math] queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where [math] denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least [math] rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with [math] queries using only polylogarithmic rounds of adaptivity. An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making [math] queries necessarily requires [math] rounds. This left open the possibility of efficient SFM algorithms in polylogarithmic rounds. For matroid intersection, even the possibility of a constant round, [math] query algorithm was not hitherto ruled out. In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making [math] queries requires (Throughout the paper, we use the usual convention of using [math] to denote [math] and using [math] to denote [math] for some unspecified constant [math]) [math] rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most [math] queries for any constant [math]. Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model.

FIXP-Membership via Convex Optimization: Games, Cakes, and Markets

Aris Filos-Ratsikas — 2023-04-04T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We introduce a new technique for proving membership of problems in FIXP: the class capturing the complexity of computing a fixed point of an algebraic circuit. Our technique constructs a “pseudogate,” which can be used as a black box when building FIXP circuits. This pseudogate, which we term the “OPT-gate,” can solve most convex optimization problems. Using the OPT-gate, we prove new FIXP-membership results, and we generalize and simplify several known results from the literature on fair division, game theory, and competitive markets. In particular, we prove complexity results for two classic problems: Computing a market equilibrium in the Arrow–Debreu model with general concave utilities is in FIXP, and computing an envy-free division of a cake with very general valuations is FIXP-complete. We further showcase the wide applicability of our technique by using it to obtain simplified proofs and extensions of known FIXP-membership results for equilibrium computation for various types of strategic games as well as the pseudomarket mechanism of Hylland and Zeckhauser.

Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg–Rao

Yu Gao — 2023-04-26T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give an algorithm for computing exact maximum flows on graphs with [math] edges and integer capacities in the range [math] in [math] time. We use [math] to suppress logarithmic factors in [math]. For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the [math] time bound from Goldberg and Rao [J. ACM, 45 (1998), pp. 783–797]. Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from Ma̧dry [Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, 2016, pp. 593–602]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.

Unambiguous DNFs and Alon–Saks–Seymour

Kaspars Balodis — 2023-10-20T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We exhibit an unambiguous [math]-DNF (disjunctive normal form) formula that requires conjunctive normal form width [math], which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon–Saks–Seymour problem in graph theory (posed in 1991), which asks, How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query/communication complexity, learning theory, and automata theory.

A Single-Exponential Time 2-Approximation Algorithm for Treewidth

Tuukka Korhonen — 2023-11-14T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give an algorithm that, given an [math]-vertex graph [math] and an integer [math], in time [math] either outputs a tree decomposition of [math] of width at most [math] or determines that the treewidth of [math] is larger than [math]. This is the first 2-approximation algorithm for treewidth that is faster than the known exact algorithms, and in particular improves upon the previous best approximation ratio of 5 in time [math] given by Bodlaender et al. [SIAM J. Comput., 45 (2016), pp. 317–378]. Our algorithm works by applying incremental improvement operations to a tree decomposition, using an approach inspired by a proof of Bellenbaum and Diestel [Combin. Probab. Comput., 11 (2002), pp. 541–547].

Breaking the Cubic Barrier for (Unweighted) Tree Edit Distance

Xiao Mao — 2023-11-16T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. The (unweighted) tree edit distance problem for [math] node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in [math] time [Demaine et al., Automata, Languages and Programming, Springer, Berlin, 2007, pp. 146–157]. The same paper also showed that [math] is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly subcubic time under the All-Pairs Shortest Paths conjecture [Bringmann et al., ACM Trans. Algorithms, 16 (2020), pp. 48:1–48:22]. In this paper, we break the cubic barrier by showing an [math] time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly subcubic time [Bringmann et al., Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, 2016, pp. 375–384].

Rapid Mixing of Glauber Dynamics via Spectral Independence for All Degrees

Xiaoyu Chen — 2024-06-13T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We prove an optimal [math] lower bound on a spectral gap of the Glauber dynamics for antiferromagnetic two-spin systems with [math] vertices in the tree uniqueness regime. This spectral gap holds for any, including unbounded, maximum degree [math]. Consequently, we have the following mixing time bounds for the models satisfying the uniqueness condition with a slack [math]: [math] mixing time for the hardcore model with fugacity [math] and [math] mixing time for the Ising model with edge activity [math], where the maximum degree [math] may depend on the number of vertices [math] and [math] depends only on [math]. Our proof is built on the recently developed connections between the Glauber dynamics for spin systems and high-dimensional expander walks. In particular, we prove a stronger notion of spectral independence, called complete spectral independence, and use a novel Markov chain, called field dynamics, to connect this stronger spectral independence to the rapid mixing of Glauber dynamics for all degrees.

Approximating Maximum Independent Set for Rectangles in the Plane

Joseph Mitchell — 2024-07-23T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We give a polynomial-time constant-factor approximation algorithm for the maximum independent set of (axis-aligned) rectangles problem in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is [math]. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.

Hardness vs. Randomness, Revised: Uniform, Non-Black-Box, and Instance-wise

Lijie Chen — 2024-12-04T08:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. We propose a new approach to the hardness-to-randomness framework and to the [math] conjecture. Classical results rely on nonuniform hardness assumptions to construct derandomization algorithms that work in the worst case, or rely on uniform hardness assumptions to construct derandomization algorithms that work only in the average case. In both types of results, the derandomization algorithm is “black-box” and uses the standard approach based on pseudorandom generators (PRGs). In this work we present results that closely relate new and natural uniform hardness assumptions to worst-case derandomization of [math], where the algorithms underlying the latter derandomization are non-black-box. In our main result, we show that [math] if the following holds: There exists a multi-output function computable by logspace-uniform circuits of polynomial size and depth [math] that cannot be computed by uniform probabilistic algorithms in time [math], for some universal constant [math], on almost all inputs. The required failure on “almost all inputs” is stronger than the standard requirement of failing on one input of each length; however, the same assumption without the depth restriction on [math] is necessary for the conclusion. This suggests a potential equivalence between worst-case derandomization of [math] of any form (i.e., not necessarily by a black-box algorithm) and the existence of efficiently computable functions that are hard for probabilistic algorithms on almost all inputs. In our second result, we introduce a new and uniform hardness-to-randomness tradeoff for the setting of superfast average-case derandomization; prior to this work, superfast average-case derandomization was known only under nonuniform hardness assumptions. In an extreme instantiation of our new tradeoff, under appealing uniform hardness assumptions and if one-way functions exist, we show that for every polynomial [math] and constant [math] it holds that [math], where the “[math]” prefix means that no polynomial-time algorithm can find, with nonnegligible probability, an input on which the deterministic simulation errs. Technically, our approach is to design targeted PRGs and hitting-set generators (HSGs), as introduced by Goldreich [In a world of [math], in Studies in Complexity and Cryptography, Lecture Notes in Comput. Sci. 6650, Springer, 2011, pp. 191–232]. Our targeted PRGs/HSGs “produce randomness from the input,” as suggested by Goldreich and Wigderson [Proceedings of the 6th International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM), 2002, pp. 209–223], and our analysis of these targeted PRGs/HSGs relies on non-black-box versions of the reconstruction procedure of Impagliazzo and Wigderson [Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 1998, pp. 734–743]. Our main reconstruction procedure crucially relies on the ideas underlying the proof system of Goldwasser, Kalai, and Rothblum [J. ACM, 62 (2015), 27].

The Minimum Formula Size Problem is (ETH) Hard

Rahul Ilango — 2025-05-22T07:00:00Z

SIAM Journal on Computing, Ahead of Print.
Abstract. A long-standing open question is whether the minimum circuit size problem (MCSP) is NP-complete. In fact, even determining whether MCSP has a search-to-decision reduction has been open for more than 20 years. We show that under the exponential time hypothesis, the minimum (De Morgan) formula size problem (MFSP) is not in P. Building on this, we show that MFSP has a polynomial-time (exact) search-to-decision reduction, a result that does not relativize. Our main lemma relates the formula complexity of a partial function with the formula complexity of an associated total function and is proved using the “leaf weighting” technique of Buchfuhrer and Umans [J. Comput. System. Sci., 77 (2011), pp. 142–153].