SIAM Journal on Computing最新文献

{"title":"Finding Maximum Edge-Disjoint Paths Between Multiple Terminals","authors":"Satoru Iwata, Yu Yokoi","doi":"10.1137/22m1494804","DOIUrl":"https://doi.org/10.1137/22m1494804","url":null,"abstract":"Let $G=(V,E)$ be a multigraph with a set $Tsubseteq V$ of terminals. A path in $G$ is called a $T$-path if its ends are distinct vertices in $T$ and no internal vertices belong to $T$. In 1978, Mader showed a characterization of the maximum number of edge-disjoint $T$-paths. In this paper, we provide a combinatorial, deterministic algorithm for finding the maximum number of edge-disjoint $T$-paths. The algorithm adopts an augmenting path approach. More specifically, we utilize a new concept of short augmenting walks in auxiliary labeled graphs to capture a possible augmentation of the number of edge-disjoint $T$-paths. To design a search procedure for a short augmenting walk, we introduce blossoms analogously to the matching algorithm of Edmonds (1965). When the search procedure terminates without finding a short augmenting walk, the algorithm provides a certificate for the optimality of the current edge-disjoint $T$-paths. From this certificate, one can obtain the Edmonds--Gallai type decomposition introduced by SebH{o} and SzegH{o} (2004). The algorithm runs in $O(|E|^2)$ time, which is much faster than the best known deterministic algorithm based on a reduction to linear matroid parity. We also present a strongly polynomial algorithm for the maximum integer free multiflow problem, which asks for a nonnegative integer combination of $T$-paths maximizing the sum of the coefficients subject to capacity constraints on the edges.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136033342","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":"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"title":"A Polynomial-Time Algorithm for 1/2-Well-Supported Nash Equilibria in Bimatrix Games","authors":"Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis","doi":"10.1137/23m1549237","DOIUrl":"https://doi.org/10.1137/23m1549237","url":null,"abstract":"","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135878177","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":"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}

{"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":null,"pages":null},"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}