Aviad Rubinstein, Saeed Seddighin, Zhao Song, Xiaorui Sun
SIAM Journal on Computing, Ahead of Print. 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.
SIAM计算机杂志，出版前。摘要。最长公共子序列(LCS)是组合优化中的一个经典和核心问题。虽然LCS承认一个二次时间解，但最近的证据表明，在真正的次二次时间内解决问题可能是不可能的。LCS中每个字符在每个字符串中最多出现一次的一种特殊情况相当于最长递增子序列(LIS)问题，该问题可以在拟线性时间内解决。在这项工作中，我们提出了在真正次二次时间内逼近LCS和在真正次线性时间内逼近LCS的新算法。我们的近似因子取决于最优解大小与输入大小的比值。我们用[math]表示这个比值，在不事先知道[math]的情况下，得到LCS和LIS的以下结果:一个具有近似因子[math]的LCS真正的次二次时间算法和一个具有近似因子[math]的LIS真正的次线性时间算法。最近，M. Boroujeni, S. Ehsani, M. Ghodsi, M. HajiAghayi和S. Seddingham[第29届ACM-SIAM离散算法研讨会论文集，2018，第1170-1189页]和D. Chakraborty, D. Das, E. Goldenberg, M. Koucky和M. Saks[第59届IEEE计算机科学基础研讨会论文集，2018，第979-990页]使用三角形不等式来提出新的编辑距离近似算法。我们的LCS技术将三角不等式的概念扩展到非度量设置。
{"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}
SIAM Journal on Computing, Ahead of Print. Abstract. The (unweighted) tree edit distance problem for [math] node trees asks to compute a measure of dissimilarity between two rooted trees with node labels. The current best algorithm from more than a decade ago runs in [math] time [Demaine et al., Automata, Languages and Programming, Springer, Berlin, 2007, pp. 146–157]. The same paper also showed that [math] is the best possible running time for any algorithm using the so-called decomposition strategy, which underlies almost all the known algorithms for this problem. These algorithms would also work for the weighted tree edit distance problem, which cannot be solved in truly subcubic time under the All-Pairs Shortest Paths conjecture [Bringmann et al., ACM Trans. Algorithms, 16 (2020), pp. 48:1–48:22]. In this paper, we break the cubic barrier by showing an [math] time algorithm for the unweighted tree edit distance problem. We consider an equivalent maximization problem and use a dynamic programming scheme involving matrices with many special properties. By using a decomposition scheme as well as several combinatorial techniques, we reduce tree edit distance to the max-plus product of bounded-difference matrices, which can be solved in truly subcubic time [Bringmann et al., Proceedings of the IEEE 57th Annual Symposium on Foundations of Computer Science, 2016, pp. 375–384].
SIAM计算机杂志，出版前。摘要。[math]节点树的(未加权的)树编辑距离问题要求计算两个带有节点标签的根树之间的不相似性度量。目前最好的算法是十多年前的[数学]时间[Demaine et al.， Automata, Languages and Programming, Springer, Berlin, 2007, pp. 146-157]。同一篇论文还表明，对于使用所谓的分解策略的任何算法来说，[math]是可能的最佳运行时间，这是几乎所有已知算法解决该问题的基础。这些算法也适用于加权树编辑距离问题，该问题在全对最短路径猜想下无法在真正的次立方时间内解决[Bringmann等人，ACM Trans.]。算法，16 (2020)，pp. 48:1-48:22]。在本文中，我们通过展示一种[数学]时间算法来解决未加权树编辑距离问题，从而打破了三次障碍。我们考虑了一个等价最大化问题，并使用了一个包含许多特殊性质矩阵的动态规划方案。通过使用分解方案和几种组合技术，我们将树编辑距离减少到有界差分矩阵的最大+积，可以在真正的次立方时间内解决[Bringmann等人，Proceedings of the IEEE第57届计算机科学基础年度研讨会，2016,pp. 375-384]。
{"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}
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]).
假设您希望对一个随机图进行采样，该图的顶点和边取决于不包含“小”尺寸图(例如，clique)作为子图的事件。假设大多数这样的图是自由的，这个问题可以通过一个简单的拒绝抽样算法(测试-cliques)来解决，其预期运行时间为。是否有可能在一个运行时间内解决这个问题，而不是多项式地增长?在本文中，我们介绍了一个具有给定边缘密度的“随机看起来”图的一般采样问题，该问题避免了一些任意预定义大小的子图。作为我们的主要结果，我们证明了这个问题是可解决的，相对于一些特别设计的-智能独立分布在图上。也就是说，我们为-wise独立图设计了一种采样算法，该算法支持子图在时间上的自由度的有效测试，其中是的函数，指数中的常数是独立的。我们的解决方案扩展到两者都是-一致超图的情况。我们使用这些算法获得了失效概率在(即)中可以忽略的常次多项式不平衡展开图的第一个概率构造。特别地，在给定常数的情况下，我们输出一个二部图，其中左节点和右节点的右度为，使得任何大小的右集最多扩展为。这一结果也推广到唯一性展开的情况下。我们观察到，这种可忽略的误差结构可以在许多有用的设置中使用，并在编码理论(批码和低密度奇偶校验码)，伪随机(低偏差生成器和随机提取器)和密码学中应用。值得注意的是，我们表明我们的构造产生了一组基于Goldreich的单向假设的多项式伸缩局部可计算密码伪随机生成器，解决了并行时间密码学领域的一个中心开放问题(例如，Applebaum, Ishai和Kushilevitz [SIAM J. Comput])。[第40届ACM计算理论研讨会论文集，ACM, 2008, pp. 433-442])。
{"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}
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].
我们给出了一个算法，给定一个无顶点图和一个整数，在时间上要么输出一个最宽的树分解，要么确定树的宽度大于。这是第一个比已知精确算法更快的树宽2逼近算法，特别是在Bodlaender等人给出的5的最佳逼近比的基础上得到了改进。， 45 (2016)， pp. 317-378]。我们的算法通过将增量改进操作应用于树分解，使用一种受Bellenbaum和Diestel [Combin]证明启发的方法。Probab。第一版。， 11(2002)，第541-547页。
{"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}
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].
大数定律保证，给定来自某个群体的足够大的样本，任何固定子群体的测量都可以通过样本中的频率得到很好的估计。我们研究采样过程中的大数定律，这些定律可以影响它们所作用的环境并与之相互作用。具体来说，我们考虑了由[O。Ben-Eliezer和E. Yogev，《抽样的对抗性稳稳性》，发表于第39届ACM SIGMOD-SIGACT-SIGAI数据库系统原理研讨会(PODS)， 2020年，第49-62页)，并描述了在该模型中承认大数统一定律的类:这些正是在线可学习的类。我们的表征可以被解释为统计(PAC)学习中可学习性和均匀收敛之间等价的在线模拟。我们获得的样本复杂度界限对于许多参数体系都是严格的，并且作为一个应用，我们确定了在线学习中的最优后悔界限，用Littlestone维表示，从而解决了[S]中的主要开放问题。Ben-David, D. Pál, S. shalov - shwartz，不可知论在线学习，第22届学习理论会议论文集(COLT)， 2009)。Rakhlin, K. Sridharan和A. Tewari, J. Mach。学习。Res.， 16 (2015)， pp. 155-186]。
{"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}
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 .
研究了基于大规模并行计算(MPC)模型的基本图算法设计问题。问题的输入是一个无向图，有顶点和边，并且是中任何连接分量的最大直径。我们考虑具有低本地空间的MPC，允许每台机器仅存储任意常数的单词，并且具有线性全局空间(即机器数量乘以可用的本地空间)，即具有最佳利用率。在最近的一项突破中，Andoni等人[对数直径轮的并行图连通性，2018]和Behnezhad, Hajiaghayi和Harris[指数更快的大规模并行最大匹配，2019]设计了并行随机算法，该算法在低局部空间的MPC轮上确定图的所有连接组件，改进了从早期PRAM算法中衍生的经典界。在本文中，我们证明了确定性算法也可以实现渐近同界:我们提出了一个确定性MPC低局部空间算法，它以轮为单位确定输入图的连通分量。我们的结果与最先进的随机算法的复杂性相匹配。我们通过扩展最近的MPC连通性的下界来补充我们的上界，该下界以1 vs 2周期猜想为条件(这需要)，通过展示整个谱的MPC轮的相关条件硬度，覆盖了一个特别有趣的范围。
{"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}
Ittai Abraham, Arnold Filtser, Anupam Gupta, Ofer Neiman
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.
本文指出了SIAM J. Comput文章“通过最短路径分解的度量嵌入”中定理4证明中的一个错误。， 51 (2022)， pp. 290-314，并撤销定理4的相关主张。
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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].
我们证明了单点更新马尔可夫链的最优混合时间界限，即各种设置下的格劳伯动力学或吉布斯采样。我们的工作提出了Anari, Liu和Oveis Gharan的谱独立方法的改进版本[第61届IEEE计算机科学基础研讨会(FOCS)， 2020, pp. 1319-1330]，并显示了当相关影响矩阵的最大特征值有界时，任何有界度的-顶点图上的混合时间。作为我们研究结果的应用，对于由逸度加权的独立集上的核心模型，我们建立了在任意最大度不变的-顶点图上的Glauber动力学的混合时间，其中为规则树上惟一/非惟一相变的临界点。更一般地，对于任意反铁磁2自旋系统，我们证明了在相应的树唯一性区域内任意有界度图上的Glauber动力学的混合时间。我们的研究结果适用范围更广;例如，当颜色个数满足时，我们还得到了最大度无三角形图的-着色的混合，其中，以及生成任意有界度和边的图的随机匹配的混合。我们的方法基于两个步骤。首先，我们证明了熵的近似张紧化(即将熵分解成单个顶点)可以从熵分解成固定线性大小的块中推导出来，这是先前许多工作中建立修正log-Sobolev不等式的关键步骤。其次，我们采用Alev和Lau的局部到全局方案[第52届ACM SIGACT计算理论研讨会(STOC)， 2020, pp. 1198-1211]在满足局部谱展开的纯加权简单复合体的更一般设置中建立这样的熵块分解;这也从本质上概括了Cryan, Guo和Mousa的结果，[第60届IEEE计算机科学基础研讨会(FOCS)， 2019, pp. 1358-1370]。
{"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}
Kaspars Balodis, Shalev Ben-David, Mika Göös, Siddhartha Jain, Robin Kothari
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.
{"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}
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.
{"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}
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