SIAM Journal on Computing最新文献

{"title":"Approximation Algorithms for LCS and LIS with Truly Improved Running Times","authors":"Aviad Rubinstein, Saeed Seddighin, Zhao Song, Xiaorui Sun","doi":"10.1137/20m1316500","DOIUrl":"https://doi.org/10.1137/20m1316500","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence (LIS) problem which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size to the input size. We denote this ratio by [math] and obtain the following results for LCS and LIS without any prior knowledge of [math]: a truly subquadratic time algorithm for LCS with approximation factor [math] and a truly sublinear time algorithm for LIS with approximation factor [math]. The triangle inequality was recently used by M. Boroujeni, S. Ehsani, M. Ghodsi, M. HajiAghayi, and S. Seddingham [Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 1170–1189] and D. Chakraborty, D. Das, E. Goldenberg, M. Koucky, and M. Saks [Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science, 2018, pp. 979–990] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of the triangle inequality to nonmetric settings.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"33 3","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520735","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":"Breaking the Cubic Barrier for (Unweighted) Tree Edit Distance","authors":"Xiao Mao","doi":"10.1137/22m1480719","DOIUrl":"https://doi.org/10.1137/22m1480719","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. The (unweighted) tree edit distance problem for [math] node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in [math] time [Demaine et al., Automata, Languages and Programming, Springer, Berlin, 2007, pp. 146–157]. The same paper also showed that [math] is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly subcubic time under the All-Pairs Shortest Paths conjecture [Bringmann et al., ACM Trans. Algorithms, 16 (2020), pp. 48:1–48:22]. In this paper, we break the cubic barrier by showing an [math] time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly subcubic time [Bringmann et al., Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, 2016, pp. 375–384].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"59 5","pages":""},"PeriodicalIF":1.6,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520772","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":"Sampling Graphs without Forbidden Subgraphs and Unbalanced Expanders with Negligible Error","authors":"Benny Applebaum, Eliran Kachlon","doi":"10.1137/22m1484134","DOIUrl":"https://doi.org/10.1137/22m1484134","url":null,"abstract":"Suppose that you wish to sample a random graph over vertices and edges conditioned on the event that does not contain a “small” -size graph (e.g., clique) as a subgraph. Assuming that most such graphs are -free, the problem can be solved by a simple rejected-sampling algorithm (that tests for -cliques) with an expected running time of . Is it possible to solve the problem in a running time that does not grow polynomially with ? In this paper, we introduce the general problem of sampling a “random looking” graph with a given edge density that avoids some arbitrary predefined -size subgraph . As our main result, we show that the problem is solvable with respect to some specially crafted -wise independent distribution over graphs. That is, we design a sampling algorithm for -wise independent graphs that supports efficient testing for subgraph-freeness in time , where is a function of and the constant in the exponent is independent of . Our solution extends to the case where both and are -uniform hypergraphs. We use these algorithms to obtain the first probabilistic construction of constant-degree polynomially unbalanced expander graphs whose failure probability is negligible in (i.e., ). In particular, given constants , we output a bipartite graph that has left nodes and right nodes with right-degree of so that any right set of size at most expands by factor of . This result is extended to the setting of unique expansion as well. We observe that such a negligible-error construction can be employed in many useful settings and present applications in coding theory (batch codes and low-density parity-check codes), pseudorandomness (low-bias generators and randomness extractors), and cryptography. Notably, we show that our constructions yield a collection of polynomial-stretch locally computable cryptographic pseudorandom generators based on Goldreich’s one-wayness assumption resolving a central open problem in the area of parallel-time cryptography (e.g., Applebaum, Ishai, and Kushilevitz [SIAM J. Comput., 36 (2006), pp. 845–888] and Ishai et al. [Proceedings of the 40th Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 433–442]).","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"29 46","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134953498","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 Single-Exponential Time 2-Approximation Algorithm for Treewidth","authors":"Tuukka Korhonen","doi":"10.1137/22m147551x","DOIUrl":"https://doi.org/10.1137/22m147551x","url":null,"abstract":"We give an algorithm that, given an -vertex graph and an integer , in time either outputs a tree decomposition of of width at most or determines that the treewidth of is larger than . This is the first 2-approximation algorithm for treewidth that is faster than the known exact algorithms, and in particular improves upon the previous best approximation ratio of 5 in time given by Bodlaender et al. [SIAM J. Comput., 45 (2016), pp. 317–378]. Our algorithm works by applying incremental improvement operations to a tree decomposition, using an approach inspired by a proof of Bellenbaum and Diestel [Combin. Probab. Comput., 11 (2002), pp. 541–547].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"5 22","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229838","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":"Adversarial Laws of Large Numbers and Optimal Regret in Online Classification","authors":"Noga Alon, Omri Ben-Eliezer, Yuval Dagan, Shay Moran, Moni Naor, Eylon Yogev","doi":"10.1137/21m1441924","DOIUrl":"https://doi.org/10.1137/21m1441924","url":null,"abstract":"Laws of large numbers guarantee that given a large enough sample from some population, the measure of any fixed subpopulation is well-estimated by its frequency in the sample. We study laws of large numbers in sampling processes that can affect the environment they are acting upon and interact with it. Specifically, we consider the sequential sampling model proposed by [O. Ben-Eliezer and E. Yogev, The adversarial robustness of sampling, in Proceedings of the 39th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems (PODS), 2020, pp. 49–62] and characterize the classes which admit a uniform law of large numbers in this model: these are exactly the classes that are online learnable. Our characterization may be interpreted as an online analogue to the equivalence between learnability and uniform convergence in statistical (PAC) learning. The sample-complexity bounds we obtain are tight for many parameter regimes, and as an application, we determine the optimal regret bounds in online learning, stated in terms of Littlestone’s dimension, thus resolving the main open question from [S. Ben-David, D. Pál, and S. Shalev-Shwartz, Agnostic online learning, in Proceedings of the 22nd Conference on Learning Theory (COLT), 2009], which was also posed by [A. Rakhlin, K. Sridharan, and A. Tewari, J. Mach. Learn. Res., 16 (2015), pp. 155–186].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"243 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775116","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 Massively Parallel Connectivity","authors":"Sam Coy, Artur Czumaj","doi":"10.1137/22m1520177","DOIUrl":"https://doi.org/10.1137/22m1520177","url":null,"abstract":"We consider the problem of designing fundamental graph algorithms on the model of massively parallel computation (MPC). The input to the problem is an undirected graph with vertices and edges and with being the maximum diameter of any connected component in . We consider the MPC with low local space, allowing each machine to store only words for an arbitrary constant and with linear global space (which is the number of machines times the local space available), that is, with optimal utilization. In a recent breakthrough, Andoni et al. [Parallel graph connectivity in log diameter rounds, 2018] and Behnezhad, Hajiaghayi, and Harris [Exponentially faster massively parallel maximal matching, 2019] designed parallel randomized algorithms that in rounds on an MPC with low local space determine all connected components of a graph, improving on the classic bound of derived from earlier works on PRAM algorithms. In this paper, we show that asymptotically identical bounds can be also achieved for deterministic algorithms: We present a deterministic MPC low local space algorithm that in rounds determines connected components of the input graph. Our result matches the complexity of state-of-the-art randomized algorithms for this task. We complement our upper bounds by extending a recent lower bound for the connectivity on an MPC conditioned on the 1-vs-2-cycles conjecture (which requires ) by showing a related conditional hardness of MPC rounds for the entire spectrum of , covering a particularly interesting range when .","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136262768","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":"Corrigendum: Metric Embedding via Shortest Path Decompositions","authors":"Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman","doi":"10.1137/23m1546245","DOIUrl":"https://doi.org/10.1137/23m1546245","url":null,"abstract":"This note points out an error in the proof of Theorem 4 in the article “Metric Embedding via Shortest Path Decompositions,” SIAM J. Comput., 51 (2022), pp. 290–314, by the authors, and withdraws the associated claim of Theorem 4.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"40 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136261743","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":"Optimal Mixing of Glauber Dynamics: Entropy Factorization via High-Dimensional Expansion","authors":"Zongchen Chen, Kuikui Liu, Eric Vigoda","doi":"10.1137/21m1443340","DOIUrl":"https://doi.org/10.1137/21m1443340","url":null,"abstract":"We prove an optimal mixing time bound for the single-site update Markov chain known as the Glauber dynamics or Gibbs sampling in a variety of settings. Our work presents an improved version of the spectral independence approach of Anari, Liu, and Oveis Gharan [Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 1319–1330] and shows mixing time on any -vertex graph of bounded degree when the maximum eigenvalue of an associated influence matrix is bounded. As an application of our results, for the hardcore model on independent sets weighted by a fugacity , we establish mixing time for the Glauber dynamics on any -vertex graph of constant maximum degree when , where is the critical point for the uniqueness/nonuniqueness phase transition on the -regular tree. More generally, for any antiferromagnetic 2-spin system we prove the mixing time of the Glauber dynamics on any bounded degree graph in the corresponding tree uniqueness region. Our results apply more broadly; for example, we also obtain mixing for -colorings of triangle-free graphs of maximum degree when the number of colors satisfies , where , and mixing for generating random matchings of any graph with bounded degree and edges. Our approach is based on two steps. First, we show that the approximate tensorization of entropy (i.e., factorizing entropy into single vertices), which is a key step for establishing the modified log-Sobolev inequality in many previous works, can be deduced from entropy factorization into blocks of fixed linear size. Second, we adapt the local-to-global scheme of Alev and Lau [Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing (STOC), 2020, pp. 1198–1211] to establish such block factorization of entropy in a more general setting of pure weighted simplicial complexes satisfying local spectral expansion; this also substantially generalizes the result of Cryan, Guo, and Mousa, [Proceedings of the 60th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2019, pp. 1358–1370].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"77 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135616351","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":"Unambiguous DNFs and Alon–Saks–Seymour","authors":"Kaspars Balodis, Shalev Ben-David, Mika Göös, Siddhartha Jain, Robin Kothari","doi":"10.1137/22m1480616","DOIUrl":"https://doi.org/10.1137/22m1480616","url":null,"abstract":"We exhibit an unambiguous -DNF (disjunctive normal form) formula that requires conjunctive normal form width , which is optimal up to logarithmic factors. As a consequence, we get a near-optimal solution to the Alon–Saks–Seymour problem in graph theory (posed in 1991), which asks, How large a gap can there be between the chromatic number of a graph and its biclique partition number? Our result is also known to imply several other improved separations in query/communication complexity, learning theory, and automata theory.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135513930","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":"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":"1 1","pages":"0"},"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}