We give an algorithm for computing exact maximum flows on graphs with edges and integer capacities in the range in time. We use to suppress logarithmic factors in . For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the time bound from Goldberg and Rao [J. ACM, 45 (1998), pp. 783–797]. Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from Ma̧dry [Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, 2016, pp. 593–602]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.
给出了在时间范围内具有边和整数容量的图上精确计算最大流的算法。我们用来抑制对数因子。对于具有多项式有界整数容量的稀疏图，这是Goldberg和Rao [J]在时间限制上的第一个改进。ACM, 45 (1998)， pp. 783-797。我们的算法围绕动态维持基于Ma ' o dry的内点法算法核心的增强电流展开[第57届IEEE计算机科学基础年度研讨会论文集，2016,pp. 593-602]。这需要设计数据结构，在有限的设置下，在进行阻力更新的图中返回具有大电能的边。
{"title":"Fully Dynamic Electrical Flows: Sparse Maxflow Faster Than Goldberg–Rao","authors":"Yu Gao, Yang Liu, Richard Peng","doi":"10.1137/22m1476666","DOIUrl":"https://doi.org/10.1137/22m1476666","url":null,"abstract":"We give an algorithm for computing exact maximum flows on graphs with edges and integer capacities in the range in time. We use to suppress logarithmic factors in . For sparse graphs with polynomially bounded integer capacities, this is the first improvement over the time bound from Goldberg and Rao [J. ACM, 45 (1998), pp. 783–797]. Our algorithm revolves around dynamically maintaining the augmenting electrical flows at the core of the interior point method based algorithm from Ma̧dry [Proceedings of the 57th IEEE Annual Symposium on Foundations of Computer Science, 2016, pp. 593–602]. This entails designing data structures that, in limited settings, return edges with large electric energy in a graph undergoing resistance updates.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136381017","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 determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density and provide (i) for randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most in -vertex graphs of maximum degree , and (ii) a proof that unless NP = RP, no such algorithms exist for . The critical density is the occupancy fraction of the hard-core model on the complete graph at the uniqueness threshold on the infinite -regular tree, giving as . Our methods apply more generally to antiferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.
{"title":"Approximately Counting Independent Sets of a Given Size in Bounded-Degree Graphs","authors":"Ewan Davies, Will Perkins","doi":"10.1137/21m1466220","DOIUrl":"https://doi.org/10.1137/21m1466220","url":null,"abstract":"We determine the computational complexity of approximately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density and provide (i) for randomized polynomial-time algorithms for approximately sampling and counting independent sets of given size at most in -vertex graphs of maximum degree , and (ii) a proof that unless NP = RP, no such algorithms exist for . The critical density is the occupancy fraction of the hard-core model on the complete graph at the uniqueness threshold on the infinite -regular tree, giving as . Our methods apply more generally to antiferromagnetic 2-spin systems and motivate new questions in extremal combinatorics.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136319418","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}
Alexander A. Sherstov, Andrey A. Storozhenko, Pei Wu
We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order sum to at most , where is the number of variables, is the tree depth, and is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with , becoming trivial already at . As an application, we obtain, for every integer , a partial Boolean function on bits that has bounded-error quantum query complexity at most and randomized query complexity . This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: versus for any [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of versus for bounded-error quantum versus randomized communication complexity for any . The best previous separation was polynomially weaker: versus (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).
我们证明了对于每个决策树，给定阶和的傅里叶系数的绝对值不超过，其中是变量的数量，是树的深度，并且是一个绝对常数。这个界限本质上是紧密的，并且解决了一个猜想，因为Tal[关于量子和随机查询复杂性之间的最佳分离，在第61届IEEE计算机科学基础研讨会(FOCS)， 2020年，第228-239页]。我们工作之前的界限迅速退化，已经变得微不足道。作为一种应用，我们得到了对于每一个整数，在比特上的一个部分布尔函数，该函数最大具有有界误差量子查询复杂度和随机查询复杂度。Aaronson和Ambainis [SIAM J. Comput]的结果表明，这种有界错误量子与随机查询复杂性的分离是最好的。和Bravyi等。[经典的关联算法，arXiv预印本，2021]。在我们的工作之前，最著名的分离是多项式弱的:相对于任何[A]。Tal，量子和随机查询复杂性之间的最佳分离，第61届IEEE计算机科学基础研讨会论文集(FOCS)， 2020, pp. 228-239]。作为另一个应用，我们获得了对有界误差量子和随机通信复杂性的本质上最优的分离。最好的先前分离是多项式弱的:相对于(这隐含在[A]。Tal，量子和随机查询复杂性之间的最佳分离，第61届IEEE计算机科学基础研讨会论文集(FOCS)， 2020, pp. 228-239])。
{"title":"An Optimal Separation of Randomized and Quantum Query Complexity","authors":"Alexander A. Sherstov, Andrey A. Storozhenko, Pei Wu","doi":"10.1137/22m1468943","DOIUrl":"https://doi.org/10.1137/22m1468943","url":null,"abstract":"We prove that for every decision tree, the absolute values of the Fourier coefficients of a given order sum to at most , where is the number of variables, is the tree depth, and is an absolute constant. This bound is essentially tight and settles a conjecture due to Tal [Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. The bounds prior to our work degraded rapidly with , becoming trivial already at . As an application, we obtain, for every integer , a partial Boolean function on bits that has bounded-error quantum query complexity at most and randomized query complexity . This separation of bounded-error quantum versus randomized query complexity is best possible, by the results of Aaronson and Ambainis [SIAM J. Comput., 47 (2018), pp. 982–1038] and Bravyi et al. [Classical Algorithms for Forrelation, arXiv preprint, 2021]. Prior to our work, the best known separation was polynomially weaker: versus for any [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]. As another application, we obtain an essentially optimal separation of versus for bounded-error quantum versus randomized communication complexity for any . The best previous separation was polynomially weaker: versus (this is implicit in [A. Tal, Towards optimal separations between quantum and randomized query complexities, in Proceedings of the Sixty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2020, pp. 228–239]).","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136267126","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":"On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness","authors":"Joshua Grochow, Youming Qiao","doi":"10.1137/21m1441110","DOIUrl":"https://doi.org/10.1137/21m1441110","url":null,"abstract":"","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136319109","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 connectivity augmentation problem (CAP), a classical problem in the area of survivable network design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a -edge-connected graph and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to makes the graph -edge-connected. If is odd, the problem is known to reduce to the tree augmentation problem (TAP)—i.e., is a spanning tree—for which significant progress has been achieved recently, leading to approximation factors below 1.5 (the current best factor is 1.458). However, advances on TAP have not carried over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli [Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020, pp. 815–825] managed to obtain the first approximation factor below 2 for CAP by presenting a 1.91-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below 1.5, based on a new analysis technique. Through these ingredients, we obtain a 1.393-approximation algorithm for CAP, and therefore also for TAP. This leads to the current best approximation result for both problems in a unified way, by significantly improving on the abovementioned 1.91-approximation for CAP and also the previously best approximation factor of 1.458 for TAP by Grandoni, Kalaitzis, and Zenklusen [Proceedings of the 50th ACM Symposium on Theory of Computing, 2018, pp. 632–645]. Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio of the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below 1.5.
{"title":"Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches","authors":"Federica Cecchetto, Vera Traub, Rico Zenklusen","doi":"10.1137/21m1430601","DOIUrl":"https://doi.org/10.1137/21m1430601","url":null,"abstract":"We consider the connectivity augmentation problem (CAP), a classical problem in the area of survivable network design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a -edge-connected graph and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to makes the graph -edge-connected. If is odd, the problem is known to reduce to the tree augmentation problem (TAP)—i.e., is a spanning tree—for which significant progress has been achieved recently, leading to approximation factors below 1.5 (the current best factor is 1.458). However, advances on TAP have not carried over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli [Proceedings of the 52nd ACM Symposium on Theory of Computing, 2020, pp. 815–825] managed to obtain the first approximation factor below 2 for CAP by presenting a 1.91-approximation algorithm based on a method that is disjoint from recent advances for TAP. We first bridge the gap between TAP and CAP by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below 1.5, based on a new analysis technique. Through these ingredients, we obtain a 1.393-approximation algorithm for CAP, and therefore also for TAP. This leads to the current best approximation result for both problems in a unified way, by significantly improving on the abovementioned 1.91-approximation for CAP and also the previously best approximation factor of 1.458 for TAP by Grandoni, Kalaitzis, and Zenklusen [Proceedings of the 50th ACM Symposium on Theory of Computing, 2018, pp. 632–645]. Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio of the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below 1.5.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135289251","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 $varepsilon$-Consensus-Halving problem, a fundamental problem in fair division, there are $n$ agents with valuations over the interval $[0,1]$, and the goal is to divide the interval into pieces and assign a label "$+$" or "$-$" to each piece, such that every agent values the total amount of "$+$" and the total amount of "$-$" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first "natural" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving $varepsilon$-Consensus-Halving for any $varepsilon$, as well as a polynomial-time algorithm for $varepsilon=1/2$. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-$1/k$-Division problem [Simmons and Su, 2003]. In particular, we prove that $varepsilon$-Consensus-$1/3$-Division is PPAD-hard.
在公平分割的基本问题——varepsilon - consensus - halving问题中，有n个智能体在区间$[0,1]$上赋值，目标是将区间划分为若干块，并为每个块分配一个“$+$”或“$-$”的标签，使得每个智能体对“$+$”和“$-$”的总值几乎相等。最近，Filos-Ratsikas和Goldberg[2019]证明了这个问题是计算类PPA的第一个“自然”完整问题，回答了一个十年来的开放问题。在本文中，我们研究了在多大程度上，如果一个人限制了估值函数的类别，这个问题变得容易解决。为此，我们提供以下贡献。首先，我们得到了[Filos-Ratsikas and Goldberg, 2019]的ppa -硬度结果的强化，以处理只有两个区块的代理具有分段统一估值的情况。我们通过新的约简得到了这个结果，实际上在概念上比[Filos-Ratsikas and Goldberg, 2019]中的相应结果简单得多。然后，我们考虑了单块(统一)估值的情况，并提供了一个参数化的多项式时间算法来求解任意$varepsilon$-Consensus-Halving，以及$varepsilon=1/2$的多项式时间算法。最后，我们新技术的一个重要应用是Consensus- halving的推广的第一个硬度结果，即Consensus- 1/k$- division问题[Simmons and Su, 2003]。特别是，我们证明了$varepsilon$- consensus -$1/3$- division是PPAD-hard的。
{"title":"Consensus-Halving: Does It Ever Get Easier?","authors":"Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, Manolis Zampetakis","doi":"10.1137/20m1387493","DOIUrl":"https://doi.org/10.1137/20m1387493","url":null,"abstract":"In the $varepsilon$-Consensus-Halving problem, a fundamental problem in fair division, there are $n$ agents with valuations over the interval $[0,1]$, and the goal is to divide the interval into pieces and assign a label \"$+$\" or \"$-$\" to each piece, such that every agent values the total amount of \"$+$\" and the total amount of \"$-$\" almost equally. The problem was recently proven by Filos-Ratsikas and Goldberg [2019] to be the first \"natural\" complete problem for the computational class PPA, answering a decade-old open question. In this paper, we examine the extent to which the problem becomes easy to solve, if one restricts the class of valuation functions. To this end, we provide the following contributions. First, we obtain a strengthening of the PPA-hardness result of [Filos-Ratsikas and Goldberg, 2019], to the case when agents have piecewise uniform valuations with only two blocks. We obtain this result via a new reduction, which is in fact conceptually much simpler than the corresponding one in [Filos-Ratsikas and Goldberg, 2019]. Then, we consider the case of single-block (uniform) valuations and provide a parameterized polynomial time algorithm for solving $varepsilon$-Consensus-Halving for any $varepsilon$, as well as a polynomial-time algorithm for $varepsilon=1/2$. Finally, an important application of our new techniques is the first hardness result for a generalization of Consensus-Halving, the Consensus-$1/k$-Division problem [Simmons and Su, 2003]. In particular, we prove that $varepsilon$-Consensus-$1/3$-Division is PPAD-hard.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136049107","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}
A tensor network is a diagram that specifies a way to “multiply” a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.
{"title":"Spectral Methods from Tensor Networks","authors":"Ankur Moitra, Alexander S. Wein","doi":"10.1137/20m1311661","DOIUrl":"https://doi.org/10.1137/20m1311661","url":null,"abstract":"A tensor network is a diagram that specifies a way to “multiply” a collection of tensors together to produce another tensor (or matrix). Many existing algorithms for tensor problems (such as tensor decomposition and tensor PCA), although they are not presented this way, can be viewed as spectral methods on matrices built from simple tensor networks. In this work we leverage the full power of this abstraction to design new algorithms for certain continuous tensor decomposition problems. An important and challenging family of tensor problems comes from orbit recovery, a class of inference problems involving group actions (inspired by applications such as cryo-electron microscopy). Orbit recovery problems over finite groups can often be solved via standard tensor methods. However, for infinite groups, no general algorithms are known. We give a new spectral algorithm based on tensor networks for one such problem: continuous multi-reference alignment over the infinite group SO(2). Our algorithm extends to the more general heterogeneous case.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136047507","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}
A team of mobile agents with different labels, starting from different nodes of an unknown anonymous network, must meet at the same node and declare that they all met. This task of gathering was traditionally considered assuming that agents at the same node can exchange information. We ask if this ability of talking is needed. The answer turns out to be no. We design two deterministic algorithms that accomplish gathering in a much weaker model. We only assume that each agent knows how many agents are at the node that it currently occupies. Our first algorithm assumes that agents know some upper bound N on the size of the network, and works in time polynomial in N and in the length of the smallest label. Our second algorithm does not assume any knowledge about the network, but its complexity is at least exponential in the size of the network and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product we solve, in the same weak model, the fundamental problem of leader election among agents. As an application we solve the gossiping problem in this model: if each agent has a message, all agents can learn all messages. This is perhaps our most surprising finding: agents without any transmitting devices can solve the most general information exchange problem if they can count the number of agents present at visited nodes.
{"title":"Want to Gather? No Need to Chatter!","authors":"Sébastien Bouchard, Yoann Dieudonné, Andrzej Pelc","doi":"10.1137/20m1362899","DOIUrl":"https://doi.org/10.1137/20m1362899","url":null,"abstract":"A team of mobile agents with different labels, starting from different nodes of an unknown anonymous network, must meet at the same node and declare that they all met. This task of gathering was traditionally considered assuming that agents at the same node can exchange information. We ask if this ability of talking is needed. The answer turns out to be no. We design two deterministic algorithms that accomplish gathering in a much weaker model. We only assume that each agent knows how many agents are at the node that it currently occupies. Our first algorithm assumes that agents know some upper bound N on the size of the network, and works in time polynomial in N and in the length of the smallest label. Our second algorithm does not assume any knowledge about the network, but its complexity is at least exponential in the size of the network and in the labels of agents. Its purpose is to show feasibility of gathering under this harsher scenario. As a by-product we solve, in the same weak model, the fundamental problem of leader election among agents. As an application we solve the gossiping problem in this model: if each agent has a message, all agents can learn all messages. This is perhaps our most surprising finding: agents without any transmitting devices can solve the most general information exchange problem if they can count the number of agents present at visited nodes.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"85 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135598817","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 prove that there is a graph isomorphism test running in time $n^{operatorname{polylog}(h)}$ on $n$-vertex graphs excluding some $h$-vertex graph as a minor. Previously known bounds were $n^{operatorname{poly}(h)}$ (Ponomarenko, 1988) and $n^{operatorname{polylog}(n)}$ (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments.
我们证明了在$n^{operatorname{polylog}(h)}$ n -顶点图上存在一个图同构测试，该测试在$n^{operatorname{polylog}}$上运行，不包括$h$顶点图作为次要图。以前已知的边界是$n^{operatorname{poly}(h)}$ (Ponomarenko, 1988)和$n^{operatorname{polylog}(n)}$ (Babai, STOC 2016)。对于该算法，我们将群论图同构机制的最新进展与新的图论论证结合起来。
{"title":"Isomorphism Testing for Graphs Excluding Small Minors","authors":"Martin Grohe, Daniel Neuen, Daniel Wiebking","doi":"10.1137/21m1401930","DOIUrl":"https://doi.org/10.1137/21m1401930","url":null,"abstract":"We prove that there is a graph isomorphism test running in time $n^{operatorname{polylog}(h)}$ on $n$-vertex graphs excluding some $h$-vertex graph as a minor. Previously known bounds were $n^{operatorname{poly}(h)}$ (Ponomarenko, 1988) and $n^{operatorname{polylog}(n)}$ (Babai, STOC 2016). For the algorithm we combine recent advances in the group-theoretic graph isomorphism machinery with new graph-theoretic arguments.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"209 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135582522","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}
For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an for the partition function on graphs of maximum degree when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064, 2021]. Moreover, in the tree nonuniqueness region, Sly in [Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no to estimate the partition function unless . The algorithmic results follow from the correlation decay approach due to Weitz [Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant . For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex on other vertices, as opposed to the total influence of other vertices on , thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.
{"title":"Rapid Mixing of Glauber Dynamics up to Uniqueness via Contraction","authors":"Zongchen Chen, Kuikui Liu, Eric Vigoda","doi":"10.1137/20m136685x","DOIUrl":"https://doi.org/10.1137/20m136685x","url":null,"abstract":"For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an for the partition function on graphs of maximum degree when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064, 2021]. Moreover, in the tree nonuniqueness region, Sly in [Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no to estimate the partition function unless . The algorithmic results follow from the correlation decay approach due to Weitz [Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant . For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex on other vertices, as opposed to the total influence of other vertices on , thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135532958","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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