The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We further show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary, we get that, for the graph isomorphism problem, the Lasserre/sums-of-squares semidefinite programming hierarchy of relaxations collapses to the Sherali–Adams linear programming hierarchy, up to a small loss in the degree.
{"title":"Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Graph Isomorphism Problem","authors":"Albert Atserias, Joanna Fijalkow","doi":"10.1137/20m1338435","DOIUrl":"https://doi.org/10.1137/20m1338435","url":null,"abstract":"The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We further show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary, we get that, for the graph isomorphism problem, the Lasserre/sums-of-squares semidefinite programming hierarchy of relaxations collapses to the Sherali–Adams linear programming hierarchy, up to a small loss in the degree.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1131 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136112396","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}
In the dynamic minimum set cover problem, the challenge is to minimize the update time while guaranteeing a close-to-optimal approximation factor. (Throughout, , , , and are parameters denoting the maximum number of elements, the number of sets, the frequency, and the cost range.) In the high-frequency range, when , this was achieved by a deterministic -approximation algorithm with amortized update time by Gupta et al. [Online and dynamic algorithms for set cover, in Proceedings STOC 2017, ACM, pp. 537–550]. In this paper we consider the low-frequency range, when , and obtain deterministic algorithms with a -approximation ratio and the following guarantees on the update time. (1) amortized update time: Prior to our work, the best approximation ratio guaranteed by deterministic algorithms was of Bhattacharya, Henzinger, and Italiano [Design of dynamic algorithms via primal-dual method, in Proceedings ICALP 2015, Springer, pp. 206–218]. In contrast, the only result with -approximation was that of Abboud et al. [Dynamic set cover: Improved algorithms and lower bounds, in Proceedings STOC 2019, ACM, pp. 114–125], who designed a randomized -approximation algorithm with amortized update time. (2) amortized update time: This result improves the above update time bound for most values of in the low-frequency range, i.e., . It is also the first result that is independent of and . It subsumes the constant amortized update time of Bhattacharya and Kulkarni [Deterministically maintaining a -approximate minimum vertex cover in amortized update time, in Proceedings SODA 2019, SIAM, pp. 1872–1885] for unweighted dynamic vertex cover (i.e., when and ). (3) worst-case update time: No nontrivial worst-case update time was previously known for the dynamic set cover problem. Our bound subsumes and improves by a logarithmic factor the worst-case update time for the unweighted dynamic vertex cover problem (i.e., when and ) of Bhattacharya, Henzinger, and Nanongkai [Fully dynamic approximate maximum matching and minimum vertex cover in worst case update time, in Proceedings SODA 2017, SIAM, pp. 470–489]. We achieve our results via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. Prior work in dynamic algorithms that employs the primal-dual approach uses a local update scheme that maintains relaxed complementary slackness conditions for every set. For our first result we use instead a global update scheme that does not always maintain complementary slackness conditions. For our second result we combine the global and the local update schema. To achieve our third result we use a hierarchy of background schedulers. It is an interesting open question whether this background scheduler technique can also be used to transform algorithms with amortized running time bounds into algorithms with worst-case running time bounds.
在动态最小集覆盖问题中，挑战是在保证近似因子接近最优的情况下最小化更新时间。(在整个，，，，中，和是参数，表示元素的最大数量、集合的数量、频率和成本范围。)在高频范围内，当，这是通过Gupta等人的具有平摊更新时间的确定性逼近算法实现的[集合覆盖的在线和动态算法，在Proceedings STOC 2017, ACM, pp. 537-550]。在本文中，我们考虑了低频范围，当，并获得了具有-近似比的确定性算法和更新时间的以下保证。(1)平摊更新时间:在我们的工作之前，确定性算法所保证的最佳近似比是Bhattacharya, Henzinger和Italiano [Design of dynamic algorithms via prial -对偶方法，in Proceedings ICALP 2015, Springer, pp. 206-218]。相比之下，使用-逼近的唯一结果是Abboud等人的结果。[动态集覆盖:改进的算法和下界，在Proceedings STOC 2019, ACM, pp. 114-125]，他们设计了一个具有平摊更新时间的随机逼近算法。(2)平摊更新时间:该结果改进了上述低频范围内大多数值的更新时间界限，即。它也是第一个独立于和的结果。它包含了Bhattacharya和Kulkarni[在平摊更新时间内确定性地保持-近似最小顶点覆盖，Proceedings SODA 2019, SIAM, pp. 1872-1885]的恒定平摊更新时间，用于非加权动态顶点覆盖(即当和)。(3)最坏情况更新时间:动态集覆盖问题没有已知的非平凡最坏情况更新时间。Bhattacharya、Henzinger和Nanongkai [Proceedings SODA 2017, SIAM, pp. 470-489]的非加权动态顶点覆盖问题的最坏更新时间(即当和)的对数因子纳入并改进了我们的边界。我们通过原始对偶方法实现我们的结果，通过将分数包装解决方案维护为双重证书。在采用原始对偶方法的动态算法中，先前的工作使用了一种局部更新方案，该方案为每个集保持宽松的互补松弛条件。对于我们的第一个结果，我们使用不总是保持互补松弛条件的全局更新方案。对于第二个结果，我们将全局更新模式和本地更新模式结合起来。为了实现第三个结果，我们使用了一个后台调度器层次结构。这种后台调度技术是否也可以用于将具有平摊运行时间界限的算法转换为具有最坏情况运行时间界限的算法，这是一个有趣的开放性问题。
{"title":"Deterministic Near-Optimal Approximation Algorithms for Dynamic Set Cover","authors":"Sayan Bhattacharya, Monika Henzinger, Danupon Nanongkai, Xiaowei Wu","doi":"10.1137/21m1428649","DOIUrl":"https://doi.org/10.1137/21m1428649","url":null,"abstract":"In the dynamic minimum set cover problem, the challenge is to minimize the update time while guaranteeing a close-to-optimal approximation factor. (Throughout, , , , and are parameters denoting the maximum number of elements, the number of sets, the frequency, and the cost range.) In the high-frequency range, when , this was achieved by a deterministic -approximation algorithm with amortized update time by Gupta et al. [Online and dynamic algorithms for set cover, in Proceedings STOC 2017, ACM, pp. 537–550]. In this paper we consider the low-frequency range, when , and obtain deterministic algorithms with a -approximation ratio and the following guarantees on the update time. (1) amortized update time: Prior to our work, the best approximation ratio guaranteed by deterministic algorithms was of Bhattacharya, Henzinger, and Italiano [Design of dynamic algorithms via primal-dual method, in Proceedings ICALP 2015, Springer, pp. 206–218]. In contrast, the only result with -approximation was that of Abboud et al. [Dynamic set cover: Improved algorithms and lower bounds, in Proceedings STOC 2019, ACM, pp. 114–125], who designed a randomized -approximation algorithm with amortized update time. (2) amortized update time: This result improves the above update time bound for most values of in the low-frequency range, i.e., . It is also the first result that is independent of and . It subsumes the constant amortized update time of Bhattacharya and Kulkarni [Deterministically maintaining a -approximate minimum vertex cover in amortized update time, in Proceedings SODA 2019, SIAM, pp. 1872–1885] for unweighted dynamic vertex cover (i.e., when and ). (3) worst-case update time: No nontrivial worst-case update time was previously known for the dynamic set cover problem. Our bound subsumes and improves by a logarithmic factor the worst-case update time for the unweighted dynamic vertex cover problem (i.e., when and ) of Bhattacharya, Henzinger, and Nanongkai [Fully dynamic approximate maximum matching and minimum vertex cover in worst case update time, in Proceedings SODA 2017, SIAM, pp. 470–489]. We achieve our results via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. Prior work in dynamic algorithms that employs the primal-dual approach uses a local update scheme that maintains relaxed complementary slackness conditions for every set. For our first result we use instead a global update scheme that does not always maintain complementary slackness conditions. For our second result we combine the global and the local update schema. To achieve our third result we use a hierarchy of background schedulers. It is an interesting open question whether this background scheduler technique can also be used to transform algorithms with amortized running time bounds into algorithms with worst-case running time bounds.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135351377","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}
Konstantinos Panagiotou, Leon Ramzews, Benedikt Stufler
We create a novel connection between Boltzmann sampling methods and Devroye’s algorithm to develop highly efficient sampling procedures that generate objects from important combinatorial classes with a given size in expected time . This performance is best possible and significantly improves the state of the art for samplers of subcritical graph classes (such as cactus graphs, outerplanar graphs, and series-parallel graphs), subcritical substitution-closed classes of permutations, Bienaymé–Galton–Watson trees conditioned on their number of leaves, and several further examples. Our approach allows for this high level of universality, as it applies in general to classes admitting bijective encodings by so-called enriched trees, which are rooted trees with additional structures on the offspring of each node.
{"title":"Exact-Size Sampling of Enriched Trees in Linear Time","authors":"Konstantinos Panagiotou, Leon Ramzews, Benedikt Stufler","doi":"10.1137/21m1459733","DOIUrl":"https://doi.org/10.1137/21m1459733","url":null,"abstract":"We create a novel connection between Boltzmann sampling methods and Devroye’s algorithm to develop highly efficient sampling procedures that generate objects from important combinatorial classes with a given size in expected time . This performance is best possible and significantly improves the state of the art for samplers of subcritical graph classes (such as cactus graphs, outerplanar graphs, and series-parallel graphs), subcritical substitution-closed classes of permutations, Bienaymé–Galton–Watson trees conditioned on their number of leaves, and several further examples. Our approach allows for this high level of universality, as it applies in general to classes admitting bijective encodings by so-called enriched trees, which are rooted trees with additional structures on the offspring of each node.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135547518","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}
Modern applications of graph algorithms often involve the use of the output sets (usually, a subset of edges or vertices of the input graph) as inputs to other algorithms. Since the input graphs of interest are large and dynamic, it is desirable for an algorithm’s output to not change drastically when a few random edges are removed from the input graph, so as to prevent issues in postprocessing. Alternately, having such a guarantee also means that one can revise the solution obtained by running the algorithm on the original graph in just a few places in order to obtain a solution for the new graph. We formalize this feature by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover’s distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity of graph algorithms. After deriving basic properties of average sensitivity such as composition, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the minimum - cut problem, and the maximum matching problem. In addition, we prove that the average sensitivity of our global minimum cut algorithm is almost optimal, by showing a nearly matching lower bound. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. One of the main ideas involved in designing our algorithms with low average sensitivity is the following fact: if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms and local computation algorithms. Using this fact in conjunction with our average sensitivity lower bound for 2-coloring, we show that every local computation algorithm for 2-coloring has query complexity linear in the number of vertices, thereby answering an open question.
{"title":"Average Sensitivity of Graph Algorithms","authors":"Nithin Varma, Yuichi Yoshida","doi":"10.1137/21m1399592","DOIUrl":"https://doi.org/10.1137/21m1399592","url":null,"abstract":"Modern applications of graph algorithms often involve the use of the output sets (usually, a subset of edges or vertices of the input graph) as inputs to other algorithms. Since the input graphs of interest are large and dynamic, it is desirable for an algorithm’s output to not change drastically when a few random edges are removed from the input graph, so as to prevent issues in postprocessing. Alternately, having such a guarantee also means that one can revise the solution obtained by running the algorithm on the original graph in just a few places in order to obtain a solution for the new graph. We formalize this feature by introducing the notion of average sensitivity of graph algorithms, which is the average earth mover’s distance between the output distributions of an algorithm on a graph and its subgraph obtained by removing an edge, where the average is over the edges removed and the distance between two outputs is the Hamming distance. In this work, we initiate a systematic study of average sensitivity of graph algorithms. After deriving basic properties of average sensitivity such as composition, we provide efficient approximation algorithms with low average sensitivities for concrete graph problems, including the minimum spanning forest problem, the global minimum cut problem, the minimum - cut problem, and the maximum matching problem. In addition, we prove that the average sensitivity of our global minimum cut algorithm is almost optimal, by showing a nearly matching lower bound. We also show that every algorithm for the 2-coloring problem has average sensitivity linear in the number of vertices. One of the main ideas involved in designing our algorithms with low average sensitivity is the following fact: if the presence of a vertex or an edge in the solution output by an algorithm can be decided locally, then the algorithm has a low average sensitivity, allowing us to reuse the analyses of known sublinear-time algorithms and local computation algorithms. Using this fact in conjunction with our average sensitivity lower bound for 2-coloring, we show that every local computation algorithm for 2-coloring has query complexity linear in the number of vertices, thereby answering an open question.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135215291","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}
The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the hardness of classically simulating the QSVT. A recent result by Chia et al. [Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), 2020, pp. 387–400] showed that the QSVT can be efficiently “dequantized” for low-rank matrices, and discussed its implication to quantum machine learning. In this work, motivated by establishing the superiority of quantum algorithms for quantum chemistry and making progress on the quantum PCP conjecture, we focus on the other main class of matrices considered in applications of the QSVT, sparse matrices. We first show how to efficiently “dequantize”, with arbitrarily small constant precision, the QSVT associated with a low-degree polynomial. We apply this technique to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix. We show, in particular, that a central computational problem considered by quantum algorithms for quantum chemistry (estimating the ground state energy of a local Hamiltonian when given, as an additional input, a state sufficiently close to the ground state) can be solved efficiently with constant precision on a classical computer. As a complementary result, we prove that with inverse-polynomial precision, the same problem becomes -complete. This gives theoretical evidence for the superiority of quantum algorithms for chemistry, and strongly suggests that said superiority stems from the improved precision achievable in the quantum setting. We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.
量子奇异值变换(QSVT)是一种最新的技术，它提供了一个统一的框架来描述迄今为止发现的大多数量子算法，并可能导致新的量子算法的发展。本文研究了经典模拟QSVT的难度。Chia等人最近的一项研究结果[基于采样的亚线性低秩矩阵算法框架用于去量化量子机器学习，发表在第52届ACM SIGACT计算理论研讨会(STOC 2020)， 2020, pp. 387-400]表明，QSVT可以有效地“去量化”低秩矩阵，并讨论了其对量子机器学习的意义。在这项工作中，受量子化学中量子算法的优越性和量子PCP猜想的进展的激励，我们将重点放在QSVT应用中考虑的另一类主要矩阵——稀疏矩阵上。我们首先展示了如何以任意小的常数精度有效地“去量化”与低次多项式相关的QSVT。我们将此技术应用于设计经典算法，以恒定的精度估计稀疏矩阵的奇异值。我们特别表明，量子化学的量子算法所考虑的一个中心计算问题(当给定一个足够接近基态的状态作为额外输入时，估计局部哈密顿量的基态能量)可以在经典计算机上以恒定的精度有效地解决。作为补充结果，我们证明了在逆多项式精度下，同样的问题变为-完备。这为量子算法在化学中的优越性提供了理论证据，并强烈表明，这种优越性源于量子设置中可实现的精度的提高。我们还讨论了这种去量子化技术如何有助于在中心量子PCP猜想上取得进展。
{"title":"Dequantizing the Quantum Singular Value Transformation: Hardness and Applications to Quantum Chemistry and the Quantum PCP Conjecture","authors":"Sevag Gharibian, François Le Gall","doi":"10.1137/22m1513721","DOIUrl":"https://doi.org/10.1137/22m1513721","url":null,"abstract":"The Quantum Singular Value Transformation (QSVT) is a recent technique that gives a unified framework to describe most quantum algorithms discovered so far, and may lead to the development of novel quantum algorithms. In this paper we investigate the hardness of classically simulating the QSVT. A recent result by Chia et al. [Sampling-based sublinear low-rank matrix arithmetic framework for dequantizing quantum machine learning, in Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC 2020), 2020, pp. 387–400] showed that the QSVT can be efficiently “dequantized” for low-rank matrices, and discussed its implication to quantum machine learning. In this work, motivated by establishing the superiority of quantum algorithms for quantum chemistry and making progress on the quantum PCP conjecture, we focus on the other main class of matrices considered in applications of the QSVT, sparse matrices. We first show how to efficiently “dequantize”, with arbitrarily small constant precision, the QSVT associated with a low-degree polynomial. We apply this technique to design classical algorithms that estimate, with constant precision, the singular values of a sparse matrix. We show, in particular, that a central computational problem considered by quantum algorithms for quantum chemistry (estimating the ground state energy of a local Hamiltonian when given, as an additional input, a state sufficiently close to the ground state) can be solved efficiently with constant precision on a classical computer. As a complementary result, we prove that with inverse-polynomial precision, the same problem becomes -complete. This gives theoretical evidence for the superiority of quantum algorithms for chemistry, and strongly suggests that said superiority stems from the improved precision achievable in the quantum setting. We also discuss how this dequantization technique may help make progress on the central quantum PCP conjecture.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135553853","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}
Dan Alistarh, James Aspnes, Faith Ellen, Rati Gelashvili, Leqi Zhu
SIAM Journal on Computing, Volume 52, Issue 4, Page 913-944, August 2023. Abstract. We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve [math]-set agreement among [math] processes or approximate agreement on a cycle of length 4 among [math] processes in a wait-free manner in asynchronous models where processes communicate using objects that can be constructed from shared registers. However, it was unknown whether proofs based on simpler techniques were possible. We show that these impossibility results cannot be obtained by extension-based proofs in the iterated snapshot model and, hence, extension-based proofs are limited in power.
{"title":"Why Extension-Based Proofs Fail","authors":"Dan Alistarh, James Aspnes, Faith Ellen, Rati Gelashvili, Leqi Zhu","doi":"10.1137/20m1375851","DOIUrl":"https://doi.org/10.1137/20m1375851","url":null,"abstract":"SIAM Journal on Computing, Volume 52, Issue 4, Page 913-944, August 2023. <br/> Abstract. We introduce extension-based proofs, a class of impossibility proofs that includes valency arguments. They are modelled as an interaction between a prover and a protocol. Using proofs based on combinatorial topology, it has been shown that it is impossible to deterministically solve [math]-set agreement among [math] processes or approximate agreement on a cycle of length 4 among [math] processes in a wait-free manner in asynchronous models where processes communicate using objects that can be constructed from shared registers. However, it was unknown whether proofs based on simpler techniques were possible. We show that these impossibility results cannot be obtained by extension-based proofs in the iterated snapshot model and, hence, extension-based proofs are limited in power.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"3 4","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520740","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}
Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Sudeshna Kolay
We study exact algorithms for Metric TSP in . In the early 1990s, algorithms with running time were presented for the planar case, and some years later an algorithm with running time was presented for any . Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Metric TSP, except for a lower bound stating that the problem admits no algorithm unless ETH fails. In this paper we settle the complexity of Metric TSP, up to constant factors in the exponent and under ETH, by giving an algorithm with running time .
{"title":"An ETH-Tight Exact Algorithm for Euclidean TSP","authors":"Mark de Berg, Hans L. Bodlaender, Sándor Kisfaludi-Bak, Sudeshna Kolay","doi":"10.1137/22m1469122","DOIUrl":"https://doi.org/10.1137/22m1469122","url":null,"abstract":"We study exact algorithms for Metric TSP in . In the early 1990s, algorithms with running time were presented for the planar case, and some years later an algorithm with running time was presented for any . Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Metric TSP, except for a lower bound stating that the problem admits no algorithm unless ETH fails. In this paper we settle the complexity of Metric TSP, up to constant factors in the exponent and under ETH, by giving an algorithm with running time .","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135657468","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}
The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem is arguably one of the most basic problems in this area: given a (-edge)-connected graph and a set of extra edges (links), select a minimum cardinality subset of links such that adding to increases its edge connectivity to . Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and Jájá, SIAM J. Comput., 10 (1981), pp. 270–283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290–306] that can be reduced to the case , also known as the tree augmentation problem for odd , and to the case , also known as the cactus augmentation problem for even . Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC’20, ACM, New York, 2020, pp. 815–825], several better than 2 approximation algorithms were known for , culminating with a recent approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC’18, ACM, New York, 1918, pp. 632–645]. However, for the best known approximation was 2. In this paper we breach the 2 approximation barrier for , hence, for , by presenting a polynomial-time approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP ’14, Springer, Berlin, 2014, pp. 800–811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we “open the box” and exploit the specific properties of the instances of a Steiner tree arising from . In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area.
生存性网络设计的基本目标是构建一个廉价的网络，该网络在给定的节点集之间保持连通性，尽管有一些边/节点出现故障。连接性增强问题可以说是这个领域中最基本的问题之一:给定一个(边)连接的图和一组额外的边(链接)，选择一个最小基数的链接子集，这样添加到增加其边的连接性。直觉上，人们希望通过增加额外的边缘来增强现有网络的可靠性。对于这个np困难问题，最著名的近似因子是2，这可以通过多种方法来实现(第一个这样的结果是在[G。N. Frederickson和Jájá， SIAM J. Comput。， 10 (1981)， pp 270-283]。它是已知的[E]。A. Dinitz, A. V. Karzanov和M. V. Lomonosov，《离散优化研究》，Nauka, Moscow, 1976, pp. 290-306]，可以简化为奇数的树增强问题，也称为奇数的树增强问题，以及偶数的仙人掌增强问题。会议前本论文的版本[J]。Byrka, F. Grandoni和a . Jabal Ameli, STOC ' 20, ACM, New York, 2020, pp. 815-825]，几个比2更好的近似算法已知，以最近的近似达到最终[F]。Grandoni, C. Kalaitzis和R. Zenklusen, STOC ' 18, ACM，纽约，1918,pp. 632-645]。然而，最著名的近似是2。在本文中，我们突破了2个近似障碍，因此，通过提出一个多项式时间近似。从技术的角度来看，我们的方法与以前的工作有很大的不同。特别是，优于2的近似算法要么利用贪婪式算法，要么基于四舍五入精心设计的lp。相反，我们使用了先前在参数化算法中使用的Steiner树问题的简化[Basavaraju等人，ICALP ' 14, Springer, Berlin, 2014, pp. 800-811]。这种减少并不是近似保留，并且使用Steiner树的当前最佳近似因子[Byrka et al.， J. ACM, 60(2013)， 6]作为黑盒将不足以改进2。为了实现后一个目标，我们“打开盒子”并利用斯坦纳树实例的特定属性。在我们看来，可生存网络设计的近似算法和steiner类型问题之间的这种联系是有趣的，并且可能导致该领域的其他结果。
{"title":"Breaching the 2-Approximation Barrier for Connectivity Augmentation: A Reduction to Steiner Tree","authors":"Jarosław Byrka, Fabrizio Grandoni, Afrouz Jabal Ameli","doi":"10.1137/21m1421143","DOIUrl":"https://doi.org/10.1137/21m1421143","url":null,"abstract":"The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The connectivity augmentation problem is arguably one of the most basic problems in this area: given a (-edge)-connected graph and a set of extra edges (links), select a minimum cardinality subset of links such that adding to increases its edge connectivity to . Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is 2, and this can be achieved with multiple approaches (the first such result is in [G. N. Frederickson and Jájá, SIAM J. Comput., 10 (1981), pp. 270–283]. It is known [E. A. Dinitz, A. V. Karzanov, and M. V. Lomonosov, Studies in Discrete Optimization, Nauka, Moscow, 1976, pp. 290–306] that can be reduced to the case , also known as the tree augmentation problem for odd , and to the case , also known as the cactus augmentation problem for even . Prior to the conference version of this paper [J. Byrka, F. Grandoni, and A. Jabal Ameli, STOC’20, ACM, New York, 2020, pp. 815–825], several better than 2 approximation algorithms were known for , culminating with a recent approximation [F. Grandoni, C. Kalaitzis, and R. Zenklusen, STOC’18, ACM, New York, 1918, pp. 632–645]. However, for the best known approximation was 2. In this paper we breach the 2 approximation barrier for , hence, for , by presenting a polynomial-time approximation. From a technical point of view, our approach deviates quite substantially from previous work. In particular, the better-than-2 approximation algorithms for either exploit greedy-style algorithms or are based on rounding carefully designed LPs. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al., ICALP ’14, Springer, Berlin, 2014, pp. 800–811]. This reduction is not approximation preserving, and using the current best approximation factor for a Steiner tree [Byrka et al., J. ACM, 60 (2013), 6] as a black box would not be good enough to improve on 2. To achieve the latter goal, we “open the box” and exploit the specific properties of the instances of a Steiner tree arising from . In our opinion this connection between approximation algorithms for survivable network design and Steiner-type problems is interesting, and might lead to other results in the area.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135140634","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}
{"title":"Special Section on the Fifty-First Annual ACM Sympositum on the Theory of Computing (STOC 2019)","authors":"Dana Moshkovitz, Sushant Sachdeva","doi":"10.1137/23n975661","DOIUrl":"https://doi.org/10.1137/23n975661","url":null,"abstract":"","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81424973","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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