SIAM Journal on Computing, Volume 53, Issue 5, Page 1354-1380, October 2024. Abstract. A resizable array is an array that can grow and shrink by the addition or removal of items from its end, or both its ends, while still supporting constant-time access to each item stored in the array given its index. Since the size of an array, i.e., the number of items in it, varies over time, space-efficient maintenance of a resizable array requires dynamic memory management. A standard doubling technique allows the maintenance of an array of size [math] using only [math] space, with [math] amortized time, or even [math] worst-case time, per operation. Sitarski, and (apparently independently) Brodnik, Carlsson, Demaine, Munro, and Sedgewick describe much better solutions that maintain a resizable array of size [math] using only [math] space, still with [math] time per operation. Brodnik et al. give a simple proof that this is best possible. We distinguish between the space needed for storing a resizable array, and accessing its items, and the temporary space that may be needed while growing or shrinking the array. For every integer [math], we show that [math] space is sufficient for storing and accessing an array of size [math], if [math] space can be used briefly during grow and shrink operations. Accessing an item by index takes [math] worst-case time, while grow and shrink operations take [math] amortized time. Using an exact analysis of a growth game, we show that for any data structure from a wide class of data structures that uses only [math] space to store the array, the amortized cost of grow is [math], even if only grow and access operations are allowed. The time for grow and shrink operations cannot be made worst-case unless [math].
{"title":"Optimal Resizable Arrays","authors":"Robert E. Tarjan, Uri Zwick","doi":"10.1137/23m1575792","DOIUrl":"https://doi.org/10.1137/23m1575792","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 5, Page 1354-1380, October 2024. <br/> Abstract. A resizable array is an array that can grow and shrink by the addition or removal of items from its end, or both its ends, while still supporting constant-time access to each item stored in the array given its index. Since the size of an array, i.e., the number of items in it, varies over time, space-efficient maintenance of a resizable array requires dynamic memory management. A standard doubling technique allows the maintenance of an array of size [math] using only [math] space, with [math] amortized time, or even [math] worst-case time, per operation. Sitarski, and (apparently independently) Brodnik, Carlsson, Demaine, Munro, and Sedgewick describe much better solutions that maintain a resizable array of size [math] using only [math] space, still with [math] time per operation. Brodnik et al. give a simple proof that this is best possible. We distinguish between the space needed for storing a resizable array, and accessing its items, and the temporary space that may be needed while growing or shrinking the array. For every integer [math], we show that [math] space is sufficient for storing and accessing an array of size [math], if [math] space can be used briefly during grow and shrink operations. Accessing an item by index takes [math] worst-case time, while grow and shrink operations take [math] amortized time. Using an exact analysis of a growth game, we show that for any data structure from a wide class of data structures that uses only [math] space to store the array, the amortized cost of grow is [math], even if only grow and access operations are allowed. The time for grow and shrink operations cannot be made worst-case unless [math].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"206 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
SIAM 计算期刊》，提前印刷。 摘要Abboud 等人最近提出了 "短周期去除 "技术[Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2022, pp.它的主要技术结果是，即使短循环的数量很少，即当[math]循环的数量为[math]时，列出[math]规则图中的所有三角形也是[math]难的。它的推论基于 3-SUM 猜想，其强度取决于 [math]，即短周期被移除的有效程度。Abboud 等人通过在图上应用结构与随机性论证实现了 [math]。在本文中，我们退一步，将概念上类似的论证应用于 3-SUM 问题的数字上，并由此推导出三角形列表的难度。因此，我们实现了 [math] 和 3-SUM 猜想下的以下下界推论：近似距离神谕：开创性的 Thorup-Zwick 距离算子在[math]时间内对图进行预处理后实现了拉伸[math]。对于同样的拉伸，假设查询时间为[math]，阿布德等人证明了预处理时间的[math]下界；我们将其改进为[math]，与上界只相差 2 倍。此外，我们还获得了拉伸 [math] 和 [math] 的紧约束，以及动态最短路径的更高下限。列出 4 循环：Abboud 等人首次证明了列出图中所有 4 循环的超线性下界，排除了[math]时间算法，其中[math]是 4 循环的数量。我们通过证明[math]上界在[math]因子以内都很紧凑，解决了这一基本问题的复杂性。我们的结果利用了加法组合论中丰富的工具集，其中最著名的是巴洛格-塞梅雷迪-高尔斯定理和鲁萨覆盖稃。如果其中一个集合的倍率较小，3-SUM 的真正亚二次方算法就是一个可能引起独立兴趣的关键要素。
{"title":"Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics","authors":"Amir Abboud, Karl Bringmann, Nick Fischer","doi":"10.1137/23m1611348","DOIUrl":"https://doi.org/10.1137/23m1611348","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> 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.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"40 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142258571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
SIAM 计算期刊》，提前印刷。 摘要我们给出了矩阵斯宾塞猜想到多对数秩的一个简单证明：给定对称[数学]矩阵[数学]，每个矩阵[数学]的秩最多[数学]，我们可以有效地找到[数学]符号[数学]，使得它们的符号和具有谱规范[数学]。这一结果也意味着以[数学]优势编码[数学]经典比特的量子随机存取码的[数学]比特下限。我们的证明使用了班代拉、布埃迪哈卓和范-汉德尔（Bandeira, Boedihardjo, and van Handel）[《发明数学》，234 (2023)，第 419-487 页]最近对具有相关高斯条目的随机矩阵的非交换 Khintchine 不等式的改进。
{"title":"Resolving Matrix Spencer Conjecture up to Poly-Logarithmic Rank","authors":"Nikhil Bansal, Haotian Jiang, Raghu Meka","doi":"10.1137/23m1592201","DOIUrl":"https://doi.org/10.1137/23m1592201","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> 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.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198438","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Manuel Bodirsky, Peter Jonsson, Barnaby Martin, Antoine Mottet, Žaneta Semanišinová
SIAM Journal on Computing, Volume 53, Issue 5, Page 1293-1353, October 2024. Abstract. We study the complexity of infinite-domain constraint satisfaction problems (CSPs): our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure [math] can be transferred to a classification of the CSPs of first-order expansions of another structure [math]. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the [math]-fold algebraic power of [math]. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of [math] and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen’s Interval Algebra, the [math]-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyze with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the artificial intelligence (AI) literature.
{"title":"Complexity Classification Transfer for CSPs via Algebraic Products","authors":"Manuel Bodirsky, Peter Jonsson, Barnaby Martin, Antoine Mottet, Žaneta Semanišinová","doi":"10.1137/22m1534304","DOIUrl":"https://doi.org/10.1137/22m1534304","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 5, Page 1293-1353, October 2024. <br/> Abstract. We study the complexity of infinite-domain constraint satisfaction problems (CSPs): our basic setting is that a complexity classification for the CSPs of first-order expansions of a structure [math] can be transferred to a classification of the CSPs of first-order expansions of another structure [math]. We exploit a product of structures (the algebraic product) that corresponds to the product of the respective polymorphism clones and present a complete complexity classification of the CSPs for first-order expansions of the [math]-fold algebraic power of [math]. This is proved by various algebraic and logical methods in combination with knowledge of the polymorphisms of the tractable first-order expansions of [math] and explicit descriptions of the expressible relations in terms of syntactically restricted first-order formulas. By combining our classification result with general classification transfer techniques, we obtain surprisingly strong new classification results for highly relevant formalisms such as Allen’s Interval Algebra, the [math]-dimensional Block Algebra, and the Cardinal Direction Calculus, even if higher-arity relations are allowed. Our results confirm the infinite-domain tractability conjecture for classes of structures that have been difficult to analyze with older methods. For the special case of structures with binary signatures, the results can be substantially strengthened and tightly connected to Ord-Horn formulas; this solves several longstanding open problems from the artificial intelligence (AI) literature.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"448 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225312","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Volume 53, Issue 5, Page 1257-1292, October 2024. Abstract. In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets, and polytopes. Our method is based on a simple and versatile algorithm for computing a Hamilton path on the skeleton of a 0/1-polytope [math], where [math]. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem [math], and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is larger than the running time of the optimization algorithm only by a factor of [math]. When [math] encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope [math] along a Hamilton path corresponds to listing the combinatorial objects by local change operations; i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all ([math]-optimal) bases and independent sets in a matroid; ([math]-optimal) spanning trees, forests, matchings, maximum matchings, and [math]-optimal matchings in a graph; vertex covers, minimum vertex covers, [math]-optimal vertex covers, stable sets, maximum stable sets, and [math]-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, [math]-optimal antichains, and [math]-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope, and order polytope, respectively. As another corollary from our framework, we obtain an [math] delay algorithm for the vertex enumeration problem on 0/1-polytopes [math], where [math] and [math], and [math] is the time needed to solve the linear program [math]. This improves upon the 25-year-old [math] delay algorithm due to Bussieck and Lübbecke.
{"title":"Traversing Combinatorial 0/1-Polytopes via Optimization","authors":"Arturo Merino, Torsten Mütze","doi":"10.1137/23m1612019","DOIUrl":"https://doi.org/10.1137/23m1612019","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 5, Page 1257-1292, October 2024. <br/> Abstract. In this paper, we present a new framework that exploits combinatorial optimization for efficiently generating a large variety of combinatorial objects based on graphs, matroids, posets, and polytopes. Our method is based on a simple and versatile algorithm for computing a Hamilton path on the skeleton of a 0/1-polytope [math], where [math]. The algorithm uses as a black box any algorithm that solves a variant of the classical linear optimization problem [math], and the resulting delay, i.e., the running time per visited vertex on the Hamilton path, is larger than the running time of the optimization algorithm only by a factor of [math]. When [math] encodes a particular class of combinatorial objects, then traversing the skeleton of the polytope [math] along a Hamilton path corresponds to listing the combinatorial objects by local change operations; i.e., we obtain Gray code listings. As concrete results of our general framework, we obtain efficient algorithms for generating all ([math]-optimal) bases and independent sets in a matroid; ([math]-optimal) spanning trees, forests, matchings, maximum matchings, and [math]-optimal matchings in a graph; vertex covers, minimum vertex covers, [math]-optimal vertex covers, stable sets, maximum stable sets, and [math]-optimal stable sets in a bipartite graph; as well as antichains, maximum antichains, [math]-optimal antichains, and [math]-optimal ideals of a poset. Specifically, the delay and space required by these algorithms are polynomial in the size of the matroid ground set, graph, or poset, respectively. Furthermore, all of these listings correspond to Hamilton paths on the corresponding combinatorial polytopes, namely the base polytope, matching polytope, vertex cover polytope, stable set polytope, chain polytope, and order polytope, respectively. As another corollary from our framework, we obtain an [math] delay algorithm for the vertex enumeration problem on 0/1-polytopes [math], where [math] and [math], and [math] is the time needed to solve the linear program [math]. This improves upon the 25-year-old [math] delay algorithm due to Bussieck and Lübbecke.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"10 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases random spanning tree distributions and determinantal point processes. For a graph [math], we show how to approximately sample uniformly random spanning trees from [math] in [math] (Throughout, [math] hides polylogarithmic factors in [math].) time per sample after an initial [math] time preprocessing. This is the first nearly linear runtime in the output size, which is clearly optimal. For a determinantal point process on [math]-sized subsets of a ground set of [math] elements, defined via an [math] kernel matrix, we show how to approximately sample in [math] time after an initial [math] time preprocessing, where [math] is the matrix multiplication exponent. The time to compute just the weight of the output set is simply [math], a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of [math] to [math]. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution [math] on [math] is reduced to sampling from related distributions on [math] for [math]. We show that for strongly Rayleigh distributions, the domain size can be reduced to nearly linear in the output size [math], improving the state of the art from [math] for general strongly Rayleigh distributions and the more specialized [math] for spanning tree distributions. Our reduction involves sampling from [math] domain-sparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of [math] are known and stored in a convenient data structure. Having access to marginals is the discrete analogue of having access to the mean and covariance of a continuous distribution, or equivalently knowing “isotropy” for the distribution, the key behind optimal samplers in the continuous setting based on the famous Kannan–Lovász–Simonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures.
{"title":"Optimal Sublinear Sampling of Spanning Trees and Determinantal Point Processes via Average-Case Entropic Independence","authors":"Nima Anari, Yang P. Liu, Thuy-Duong Vuong","doi":"10.1137/22m1524321","DOIUrl":"https://doi.org/10.1137/22m1524321","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We design fast algorithms for repeatedly sampling from strongly Rayleigh distributions, which include as special cases random spanning tree distributions and determinantal point processes. For a graph [math], we show how to approximately sample uniformly random spanning trees from [math] in [math] (Throughout, [math] hides polylogarithmic factors in [math].) time per sample after an initial [math] time preprocessing. This is the first nearly linear runtime in the output size, which is clearly optimal. For a determinantal point process on [math]-sized subsets of a ground set of [math] elements, defined via an [math] kernel matrix, we show how to approximately sample in [math] time after an initial [math] time preprocessing, where [math] is the matrix multiplication exponent. The time to compute just the weight of the output set is simply [math], a natural barrier that suggests our runtime might be optimal for determinantal point processes as well. As a corollary, we even improve the state of the art for obtaining a single sample from a determinantal point process, from the prior runtime of [math] to [math]. In our main technical result, we achieve the optimal limit on domain sparsification for strongly Rayleigh distributions. In domain sparsification, sampling from a distribution [math] on [math] is reduced to sampling from related distributions on [math] for [math]. We show that for strongly Rayleigh distributions, the domain size can be reduced to nearly linear in the output size [math], improving the state of the art from [math] for general strongly Rayleigh distributions and the more specialized [math] for spanning tree distributions. Our reduction involves sampling from [math] domain-sparsified distributions, all of which can be produced efficiently assuming approximate overestimates for marginals of [math] are known and stored in a convenient data structure. Having access to marginals is the discrete analogue of having access to the mean and covariance of a continuous distribution, or equivalently knowing “isotropy” for the distribution, the key behind optimal samplers in the continuous setting based on the famous Kannan–Lovász–Simonovits (KLS) conjecture. We view our result as analogous in spirit to the KLS conjecture and its consequences for sampling, but rather for discrete strongly Rayleigh measures.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"673 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198404","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar
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].
SIAM 计算期刊》，提前印刷。 摘要我们提出了一种强反驳所有布尔 CSP 平滑实例的算法。平滑模型是最坏情况输入模型和平均情况输入模型的混合体，其中输入是 CSP 的任意实例，只有字面的否定模式以某种小概率重新随机化。对于一个[数学]变量平滑的[数学]稀有度 CSP 实例，我们的算法可以在[数学]时间内运行，并且只要约束的数量至少为[数学]，就能以很高的概率成功地将可满足约束的最佳分数限定在 1 以外。这与用于驳斥 CSP 完全随机实例的最先进算法在运行时间和约束数量之间的权衡[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.］我们还在半随机[math]-XOR 的显著 "随机性饥饿 "设置中对我们的驳斥算法的分析与最坏情况超图中偶数覆盖的存在之间建立了令人惊讶的联系。我们利用这种联系正面解决了费吉 2008 年的猜想--一个关于在足够密集的超图中是否存在偶数盖的极端组合学猜想，它概括了众所周知的图周长摩尔约束。作为推论，我们证明了任意平滑 CSP 实例都存在多项式大小的驳斥见证，其约束数比 [math] 的 "谱阈值 "低一个多项式因子，从而扩展了 [U. Feige, J. H. Kim.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.］
{"title":"Algorithms and Certificates for Boolean CSP Refutation: Smoothed Is No Harder than Random","authors":"Venkatesan Guruswami, Pravesh K. Kothari, Peter Manohar","doi":"10.1137/22m1537771","DOIUrl":"https://doi.org/10.1137/22m1537771","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> 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].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"2 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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.
{"title":"Removing Additive Structure in 3SUM-Based Reductions","authors":"Ce Jin, Yinzhan Xu","doi":"10.1137/23m1589967","DOIUrl":"https://doi.org/10.1137/23m1589967","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> 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.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"5087 3 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Volume 53, Issue 5, Page 1217-1256, October 2024. Abstract. Mehta and Panigrahi (FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) introduce the problem of online matching with stochastic rewards, where edges are associated with success probabilities and a match succeeds with the probability of the corresponding edge. It is one of the few online matching problems that have defied the randomized online primal dual framework by Devanur, Jain, and Kleinberg (SODA 2013, SIAM, Philadelphia, 2013, pp. 101–107) thus far. This paper unlocks the power of randomized online primal dual in online matching with stochastic rewards by employing the configuration linear program rather than the standard matching linear program used in previous works. Our main result is a 0.572 competitive algorithm for the case of vanishing and unequal probabilities, improving the best previous bound of 0.534 by Mehta, Waggoner, and Zadimoghaddam (SODA 2015, SIAM, Philadelphia, 2015, pp. 1388–1404) and, in fact, is even better than the best previous bound of 0.567 by Mehta and Panigrahi (FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) for the more restricted case of vanishing and equal probabilities. For vanishing and equal probabilities, we get a better competitive ratio of 0.576. Our results further generalize to the vertex-weighted case due to the intrinsic robustness of the randomized online primal dual analysis.
{"title":"Online Primal Dual Meets Online Matching with Stochastic Rewards: Configuration LP to the Rescue","authors":"Zhiyi Huang, Qiankun Zhang","doi":"10.1137/21m1454705","DOIUrl":"https://doi.org/10.1137/21m1454705","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 5, Page 1217-1256, October 2024. <br/> Abstract. Mehta and Panigrahi (FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) introduce the problem of online matching with stochastic rewards, where edges are associated with success probabilities and a match succeeds with the probability of the corresponding edge. It is one of the few online matching problems that have defied the randomized online primal dual framework by Devanur, Jain, and Kleinberg (SODA 2013, SIAM, Philadelphia, 2013, pp. 101–107) thus far. This paper unlocks the power of randomized online primal dual in online matching with stochastic rewards by employing the configuration linear program rather than the standard matching linear program used in previous works. Our main result is a 0.572 competitive algorithm for the case of vanishing and unequal probabilities, improving the best previous bound of 0.534 by Mehta, Waggoner, and Zadimoghaddam (SODA 2015, SIAM, Philadelphia, 2015, pp. 1388–1404) and, in fact, is even better than the best previous bound of 0.567 by Mehta and Panigrahi (FOCS 2012, IEEE, Piscataway, NJ, 2012, pp. 728–737) for the more restricted case of vanishing and equal probabilities. For vanishing and equal probabilities, we get a better competitive ratio of 0.576. Our results further generalize to the vertex-weighted case due to the intrinsic robustness of the randomized online primal dual analysis.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"74 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Computing, Ahead of Print. Abstract. For any finite group [math], we give an algebraic algorithm to compute the generalized discrete Fourier transform with respect to [math], using [math] operations, for any [math]. Here, [math] is the exponent of matrix multiplication.
{"title":"Fast Generalized DFTs for All Finite Groups","authors":"Chris Umans","doi":"10.1137/20m1316342","DOIUrl":"https://doi.org/10.1137/20m1316342","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. For any finite group [math], we give an algebraic algorithm to compute the generalized discrete Fourier transform with respect to [math], using [math] operations, for any [math]. Here, [math] is the exponent of matrix multiplication.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"7 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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