SIAM Journal on Computing, Volume 53, Issue 2, Page 221-246, April 2024. Abstract. We define and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph [math] and a set of demand pairs [math], a reachability preserver is a sparse subgraph [math] that preserves reachability between all demand pairs Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an [math]-node graph and demand pairs of the form [math] for a small node subset [math], there is always a reachability preserver on [math] edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which [math] size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the [math] of Chlamtáč et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534–553] to [math].
{"title":"Reachability Preservers: New Extremal Bounds and Approximation Algorithms","authors":"Amir Abboud, Greg Bodwin","doi":"10.1137/21m1442176","DOIUrl":"https://doi.org/10.1137/21m1442176","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 221-246, April 2024. <br/> Abstract. We define and study reachability preservers, a graph-theoretic primitive that has been implicit in prior work on network design. Given a directed graph [math] and a set of demand pairs [math], a reachability preserver is a sparse subgraph [math] that preserves reachability between all demand pairs Our first contribution is a series of extremal bounds on the size of reachability preservers. Our main result states that, for an [math]-node graph and demand pairs of the form [math] for a small node subset [math], there is always a reachability preserver on [math] edges. We additionally give a lower bound construction demonstrating that this upper bound characterizes the settings in which [math] size reachability preservers are generally possible, in a large range of parameters. The second contribution of this paper is a new connection between extremal graph sparsification results and classical Steiner Network Design problems. Surprisingly, prior to this work, the osmosis of techniques between these two fields had been superficial. This allows us to improve the state of the art approximation algorithms for the most basic Steiner-type problem in directed graphs from the [math] of Chlamtáč et al. [Approximating spanners and directed steiner forest: Upper and lower bounds, in Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, 2017, pp. 534–553] to [math].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"33 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126992","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}
Ruben Becker, Yuval Emek, Mohsen Ghaffari, Christoph Lenzen
SIAM Journal on Computing, Volume 53, Issue 2, Page 247-286, April 2024. Abstract. We study the problem of approximating the distances in an undirected weighted graph [math] by the distances in trees based on the notion of stretch. Focusing on decentralized models of computation such as the [math], [math], and semi-streaming models, our main results are as follows: (1) We develop a simple randomized algorithm that constructs a spanning tree such that the expected stretch of every edge is [math], where [math] is the number of nodes in [math]. If [math] is unweighted, then this algorithm can be implemented to run in [math] rounds in the [math] model, where [math] is the hop-diameter of [math]; thus our algorithm is asymptotically optimal in this case. In the weighted case, the run-time of the algorithm matches the currently best known bound for exact single source shortest path (SSSP) computations, which despite recent progress is still separated from the lower bound of [math] by polynomial factors. A naive attempt to replace exact SSSP computations with approximate ones in order to improve the complexity in the weighted case encounters a fundamental challenge, as the underlying decomposition technique fails to work under distance approximation. (2) We overcome this obstacle by developing a technique termed blurry ball growing. This technique, in combination with a clever algorithmic idea of Miller, Peng, and Xu (SPAA 2013), allows us to obtain low diameter graph decompositions with small edge cutting probabilities based solely on approximate SSSP computations. (3) Using these decompositions, we in turn obtain metric tree embedding algorithms in the vein of the celebrated work of Bartal (FOCS 1996), whose computational complexity is optimal up to polylogarithmic factors not only in the [math] model but also in the [math] and semi-streaming models. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is “used” only logarithmically many times. This property is of interest for capacitated problems and for simulating [math] algorithms on the tree into which the graph is embedded.
{"title":"Decentralized Low-Stretch Trees via Low Diameter Graph Decompositions","authors":"Ruben Becker, Yuval Emek, Mohsen Ghaffari, Christoph Lenzen","doi":"10.1137/22m1489034","DOIUrl":"https://doi.org/10.1137/22m1489034","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 247-286, April 2024. <br/> Abstract. We study the problem of approximating the distances in an undirected weighted graph [math] by the distances in trees based on the notion of stretch. Focusing on decentralized models of computation such as the [math], [math], and semi-streaming models, our main results are as follows: (1) We develop a simple randomized algorithm that constructs a spanning tree such that the expected stretch of every edge is [math], where [math] is the number of nodes in [math]. If [math] is unweighted, then this algorithm can be implemented to run in [math] rounds in the [math] model, where [math] is the hop-diameter of [math]; thus our algorithm is asymptotically optimal in this case. In the weighted case, the run-time of the algorithm matches the currently best known bound for exact single source shortest path (SSSP) computations, which despite recent progress is still separated from the lower bound of [math] by polynomial factors. A naive attempt to replace exact SSSP computations with approximate ones in order to improve the complexity in the weighted case encounters a fundamental challenge, as the underlying decomposition technique fails to work under distance approximation. (2) We overcome this obstacle by developing a technique termed blurry ball growing. This technique, in combination with a clever algorithmic idea of Miller, Peng, and Xu (SPAA 2013), allows us to obtain low diameter graph decompositions with small edge cutting probabilities based solely on approximate SSSP computations. (3) Using these decompositions, we in turn obtain metric tree embedding algorithms in the vein of the celebrated work of Bartal (FOCS 1996), whose computational complexity is optimal up to polylogarithmic factors not only in the [math] model but also in the [math] and semi-streaming models. Our embeddings have the additional useful property that the tree can be mapped back to the original graph such that each edge is “used” only logarithmically many times. This property is of interest for capacitated problems and for simulating [math] algorithms on the tree into which the graph is embedded.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"12 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140127117","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, Volume 53, Issue 2, Page 189-220, April 2024. Abstract. We study the computational complexity of the problem [math] of counting [math]-vertex induced subgraphs of a graph [math] that satisfy a graph property [math]. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): If a hereditary property [math] is true for all graphs, or if it is true only for finitely many graphs, then [math] is solvable in polynomial time. Otherwise, [math] is [math]-complete when parameterized by [math], and, assuming ETH, it cannot be solved in time [math] for any function [math]. This classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [Theoret. Comput. Sci., 289 (2002), pp. 997–1008]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem. As an additional result, we also present an exhaustive and explicit parameterized complexity classification for all properties that are invariant under homomorphic equivalence. By covering one of the most natural and general notions of closure, namely, closure under vertex-deletion (hereditary), we generalize some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of [math] for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [SIAM J. Comput., (2022), pp. FOCS20-139–FOCS20-174].
{"title":"Counting Small Induced Subgraphs with Hereditary Properties","authors":"Jacob Focke, Marc Roth","doi":"10.1137/22m1512211","DOIUrl":"https://doi.org/10.1137/22m1512211","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 2, Page 189-220, April 2024. <br/> Abstract. We study the computational complexity of the problem [math] of counting [math]-vertex induced subgraphs of a graph [math] that satisfy a graph property [math]. Our main result establishes an exhaustive and explicit classification for all hereditary properties, including tight conditional lower bounds under the Exponential Time Hypothesis (ETH): If a hereditary property [math] is true for all graphs, or if it is true only for finitely many graphs, then [math] is solvable in polynomial time. Otherwise, [math] is [math]-complete when parameterized by [math], and, assuming ETH, it cannot be solved in time [math] for any function [math]. This classification features a wide range of properties for which the corresponding detection problem (as classified by Khot and Raman [Theoret. Comput. Sci., 289 (2002), pp. 997–1008]) is tractable but counting is hard. Moreover, even for properties which are already intractable in their decision version, our results yield significantly stronger lower bounds for the counting problem. As an additional result, we also present an exhaustive and explicit parameterized complexity classification for all properties that are invariant under homomorphic equivalence. By covering one of the most natural and general notions of closure, namely, closure under vertex-deletion (hereditary), we generalize some of the earlier results on this problem. For instance, our results fully subsume and strengthen the existing classification of [math] for monotone (subgraph-closed) properties due to Roth, Schmitt, and Wellnitz [SIAM J. Comput., (2022), pp. FOCS20-139–FOCS20-174].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"16 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-03-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126911","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, Volume 53, Issue 1, Page 146-187, February 2024. Abstract. This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem.
{"title":"Four-Coloring [math]-Free Graphs. II. Finding an Excellent Precoloring","authors":"Maria Chudnovsky, Sophie Spirkl, Mingxian Zhong","doi":"10.1137/18m1234849","DOIUrl":"https://doi.org/10.1137/18m1234849","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 1, Page 146-187, February 2024. <br/> Abstract. This is the second paper in a series of two. The goal of the series is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial time-algorithm that starts with a 4-precoloring of a graph with no induced six-vertex path and outputs a polynomial-sized collection of so-called excellent precolorings. Excellent precolorings are easier to handle than general ones, and, in addition, in order to determine whether the initial precoloring can be extended to the whole graph, it is enough to answer the same question for each of the excellent precolorings in the collection. The first paper in the series deals with excellent precolorings, thus providing a complete solution to the problem.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"76 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002305","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, Volume 53, Issue 1, Page 111-145, February 2024. Abstract. This is the first paper in a series whose goal is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial-time algorithm that determines if a special kind of precoloring of a [math]-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem.
{"title":"Four-Coloring [math]-Free Graphs. I. Extending an Excellent Precoloring","authors":"Maria Chudnovsky, Sophie Spirkl, Mingxian Zhong","doi":"10.1137/18m1234837","DOIUrl":"https://doi.org/10.1137/18m1234837","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 1, Page 111-145, February 2024. <br/> Abstract. This is the first paper in a series whose goal is to give a polynomial-time algorithm for the 4-coloring problem and the 4-precoloring extension problem restricted to the class of graphs with no induced six-vertex path, thus proving a conjecture of Huang. Combined with previously known results, this completes the classification of the complexity of the 4-coloring problem for graphs with a connected forbidden induced subgraph. In this paper we give a polynomial-time algorithm that determines if a special kind of precoloring of a [math]-free graph has a precoloring extension, and constructs such an extension if one exists. Combined with the main result of the second paper of the series, this gives a complete solution to the problem.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002727","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}
Siqi Liu, Sidhanth Mohanty, Tselil Schramm, Elizabeth Yang
SIAM Journal on Computing, Ahead of Print. Abstract. The random geometric graph model [math] is a distribution over graphs in which the edges capture a latent geometry. To sample [math], we identify each of our [math] vertices with an independently and uniformly sampled vector from the [math]-dimensional unit sphere [math], and we connect pairs of vertices whose vectors are “sufficiently close,” such that the marginal probability of an edge is [math]. Because of the underlying geometry, this model is natural for applications in data science and beyond. We investigate the problem of testing for this latent geometry, or, in other words, distinguishing an Erdős–Rényi graph [math] from a random geometric graph [math]. It is not too difficult to show that if [math] while [math] is held fixed, the two distributions become indistinguishable; we wish to understand how fast [math] must grow as a function of [math] for indistinguishability to occur. When [math] for constant [math], we prove that if [math], the total variation distance between the two distributions is close to 0; this improves upon the best previous bound of Brennan, Bresler, and Nagaraj (2020), which required [math], and further our result is nearly tight, resolving a conjecture of Bubeck, Ding, Eldan, and Rácz (2016) up to logarithmic factors. We also obtain improved upper bounds on the statistical indistinguishability thresholds in [math] for the full range of [math] satisfying [math], improving upon the previous bounds by polynomial factors. Our analysis uses the belief propagation algorithm to characterize the distributions of (subsets of) the random vectors conditioned on producing a particular graph. In this sense, our analysis is connected to the “cavity method” from statistical physics. To analyze this process, we rely on novel sharp estimates for the area of the intersection of a random sphere cap with an arbitrary subset of [math], which we prove using optimal transport maps and entropy-transport inequalities on the unit sphere. We believe these techniques may be of independent interest.
{"title":"Testing Thresholds for High-Dimensional Sparse Random Geometric Graphs","authors":"Siqi Liu, Sidhanth Mohanty, Tselil Schramm, Elizabeth Yang","doi":"10.1137/23m1545203","DOIUrl":"https://doi.org/10.1137/23m1545203","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. The random geometric graph model [math] is a distribution over graphs in which the edges capture a latent geometry. To sample [math], we identify each of our [math] vertices with an independently and uniformly sampled vector from the [math]-dimensional unit sphere [math], and we connect pairs of vertices whose vectors are “sufficiently close,” such that the marginal probability of an edge is [math]. Because of the underlying geometry, this model is natural for applications in data science and beyond. We investigate the problem of testing for this latent geometry, or, in other words, distinguishing an Erdős–Rényi graph [math] from a random geometric graph [math]. It is not too difficult to show that if [math] while [math] is held fixed, the two distributions become indistinguishable; we wish to understand how fast [math] must grow as a function of [math] for indistinguishability to occur. When [math] for constant [math], we prove that if [math], the total variation distance between the two distributions is close to 0; this improves upon the best previous bound of Brennan, Bresler, and Nagaraj (2020), which required [math], and further our result is nearly tight, resolving a conjecture of Bubeck, Ding, Eldan, and Rácz (2016) up to logarithmic factors. We also obtain improved upper bounds on the statistical indistinguishability thresholds in [math] for the full range of [math] satisfying [math], improving upon the previous bounds by polynomial factors. Our analysis uses the belief propagation algorithm to characterize the distributions of (subsets of) the random vectors conditioned on producing a particular graph. In this sense, our analysis is connected to the “cavity method” from statistical physics. To analyze this process, we rely on novel sharp estimates for the area of the intersection of a random sphere cap with an arbitrary subset of [math], which we prove using optimal transport maps and entropy-transport inequalities on the unit sphere. We believe these techniques may be of independent interest.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"25 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002382","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. We show that lower bounds on the border rank of matrix multiplication can be used to nontrivially derandomize polynomial identity testing for small algebraic circuits. Letting [math] denote the border rank of [math] matrix multiplication, we construct a hitting set generator with seed length [math] that hits [math]-variate circuits of multiplicative complexity [math]. If the matrix multiplication exponent [math] is not 2, our generator has seed length [math] and hits circuits of size [math] for sufficiently small [math]. Surprisingly, the fact that [math] already yields new, nontrivial hitting set generators for circuits of sublinear multiplicative complexity.
{"title":"On Matrix Multiplication and Polynomial Identity Testing","authors":"Robert Andrews","doi":"10.1137/22m1536169","DOIUrl":"https://doi.org/10.1137/22m1536169","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/>Abstract. We show that lower bounds on the border rank of matrix multiplication can be used to nontrivially derandomize polynomial identity testing for small algebraic circuits. Letting [math] denote the border rank of [math] matrix multiplication, we construct a hitting set generator with seed length [math] that hits [math]-variate circuits of multiplicative complexity [math]. If the matrix multiplication exponent [math] is not 2, our generator has seed length [math] and hits circuits of size [math] for sufficiently small [math]. Surprisingly, the fact that [math] already yields new, nontrivial hitting set generators for circuits of sublinear multiplicative complexity.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"25 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140002729","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. We relate discrepancy theory with the classic scheduling problems of minimizing max flow time and total flow time on unrelated machines. Specifically, we give a general reduction that allows us to transfer discrepancy bounds in the prefix Beck–Fiala (bounded [math]-norm) setting to bounds on the flow time of an optimal schedule. Combining our reduction with a deep result proved by Banaszczyk via convex geometry gives guarantees of [math] and [math] for max flow time and total flow time, respectively, improving upon the previous best guarantees of [math] and [math]. Apart from the improved guarantees, the reduction motivates seemingly easy versions of prefix discrepancy questions: any constant bound on prefix Beck–Fiala where vectors have sparsity two (sparsity one being trivial) would already yield tight guarantees for both max flow time and total flow time. While known techniques solve this case when the entries take values in [math], we show that they are unlikely to transfer to the more general 2-sparse case of bounded [math]-norm.
{"title":"Flow Time Scheduling and Prefix Beck–Fiala","authors":"Nikhil Bansal, Lars Rohwedder, Ola Svensson","doi":"10.1137/22m1541010","DOIUrl":"https://doi.org/10.1137/22m1541010","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We relate discrepancy theory with the classic scheduling problems of minimizing max flow time and total flow time on unrelated machines. Specifically, we give a general reduction that allows us to transfer discrepancy bounds in the prefix Beck–Fiala (bounded [math]-norm) setting to bounds on the flow time of an optimal schedule. Combining our reduction with a deep result proved by Banaszczyk via convex geometry gives guarantees of [math] and [math] for max flow time and total flow time, respectively, improving upon the previous best guarantees of [math] and [math]. Apart from the improved guarantees, the reduction motivates seemingly easy versions of prefix discrepancy questions: any constant bound on prefix Beck–Fiala where vectors have sparsity two (sparsity one being trivial) would already yield tight guarantees for both max flow time and total flow time. While known techniques solve this case when the entries take values in [math], we show that they are unlikely to transfer to the more general 2-sparse case of bounded [math]-norm.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"17 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978288","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}
Janardhan Kulkarni, Yang P. Liu, Ashwin Sah, Mehtaab S. Sawhney, Jakub Tarnawski
SIAM Journal on Computing, Volume 53, Issue 1, Page 87-110, February 2024. Abstract. We give an online algorithm that with high probability computes a [math] edge coloring on a graph [math] with maximum degree [math] under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrence based approach for arguing correlation decay.
{"title":"Online Edge Coloring via Tree Recurrences and Correlation Decay","authors":"Janardhan Kulkarni, Yang P. Liu, Ashwin Sah, Mehtaab S. Sawhney, Jakub Tarnawski","doi":"10.1137/22m152431x","DOIUrl":"https://doi.org/10.1137/22m152431x","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 1, Page 87-110, February 2024. <br/> Abstract. We give an online algorithm that with high probability computes a [math] edge coloring on a graph [math] with maximum degree [math] under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrence based approach for arguing correlation decay.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"36 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139978290","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. Following the seminal work of [R. R. Williams, J. ACM, 61 (2014)], in a recent breakthrough, [C. D. Murray and R. R. Williams, STOC 2018] proved that [math] (nondeterministic quasi-polynomial time) does not have polynomial-size [math] circuits (constant depth circuits consisting of [math]/[math]/[math] gates for a fixed constant [math], a frontier class in circuit complexity). We strengthen the above lower bound to an average-case one, by proving that for all constants [math], there is a language in [math] that cannot be [math]-approximated by polynomial-size [math] circuits. Our work also improves the average-case lower bound for [math] against polynomial-size [math] circuits by [R. Chen, I. C. Oliveira, and R. Santhanam, LATIN 2018, pp. 317–330]. Our new lower bound builds on several interesting components, including the following: 1. Barrington’s theorem and the existence of an [math]-complete language that is random self-reducible. 2. The subexponential witness-size lower bound for [math] against [math] and the conditional nondeterministic pseudorandom generator (PRG) construction in [R. R. Williams, SIAM J. Comput., 45 (2016), pp. 497–529]. 3. An “almost” almost-everywhere [math] average-case lower bound (which strengthens the corresponding worst-case lower bound in [C. D. Murray and R. R. Williams, STOC 2018]). 4. A [math]-complete language that is downward self-reducible, same-length checkable, error-correctable, and paddable. Moreover, all its reducibility properties have corresponding low-depth nonadaptive oracle circuits. Our construction builds on [L. Trevisan and S. P. Vadhan, Comput. Complexity, 16 (2007), pp. 331–364]. Like other lower bounds proved via the “algorithmic approach,” the only property of [math] exploited by us is the existence of a nontrivial [math] algorithm for [math] [R. R. Williams, J. ACM, 61 (2014)]. Therefore, for any typical circuit class [math], our results apply to [math] as well if a nontrivial [math] (in fact, [math]) algorithm for [math] is discovered.
SIAM 计算期刊》，提前印刷。 摘要继[R. R. Williams, J. ACM, 61 (2014)]的开创性工作之后，[C. D. Murray and R. R. Williams, STOC 2018]最近又有突破性进展，证明了[math]（非确定性准多项式时间）不存在多项式大小的[math]电路（由固定常数[math]的[math]/[math]/[math]门组成的恒定深度电路，是电路复杂性的前沿类）。我们通过证明对于所有常数[math]，[math]中存在一种语言无法用多项式大小的[math]电路来[math]逼近，从而将上述下界强化为平均情况下的下界。我们的工作还改进了 [R. Chen, I. C. Oliveira, and R. Santhanam, LATIN 2018, pp.我们的新下界建立在几个有趣的组成部分之上，包括以下内容：1.巴林顿定理和随机自还原的[数学]完全语言的存在。2.R. R. Williams, SIAM J. Comput., 45 (2016), pp.3.一个 "几乎 "几乎无处不在的[数学]平均情况下界（它加强了[C. D. Murray and R. R. Williams, STOC 2018]中相应的最坏情况下界）。4.一种[math]完备语言，它是向下自可还原的、同长可检查的、可纠错的和可填充的。此外，它的所有可还原性都有相应的低深度非适应性甲骨文电路。我们的构造建立在 [L. Trevisan 和 S. P. Vadhan, Comput. Complexity, 16 (2007), pp.］与其他通过 "算法方法 "证明的下界一样，我们所利用的 [math] 的唯一属性是 [math] 存在一个非难 [math] 算法 [R. R. Williams, J. ACM, 61 (2014)]。因此，对于任何典型的电路类[math]，如果发现了[math]的非难[math]（事实上是[math]）算法，我们的结果也适用于[math]。
{"title":"Nondeterministic Quasi-Polynomial Time is Average-Case Hard for [math] Circuits","authors":"Lijie Chen","doi":"10.1137/20m1321231","DOIUrl":"https://doi.org/10.1137/20m1321231","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. Following the seminal work of [R. R. Williams, J. ACM, 61 (2014)], in a recent breakthrough, [C. D. Murray and R. R. Williams, STOC 2018] proved that [math] (nondeterministic quasi-polynomial time) does not have polynomial-size [math] circuits (constant depth circuits consisting of [math]/[math]/[math] gates for a fixed constant [math], a frontier class in circuit complexity). We strengthen the above lower bound to an average-case one, by proving that for all constants [math], there is a language in [math] that cannot be [math]-approximated by polynomial-size [math] circuits. Our work also improves the average-case lower bound for [math] against polynomial-size [math] circuits by [R. Chen, I. C. Oliveira, and R. Santhanam, LATIN 2018, pp. 317–330]. Our new lower bound builds on several interesting components, including the following: 1. Barrington’s theorem and the existence of an [math]-complete language that is random self-reducible. 2. The subexponential witness-size lower bound for [math] against [math] and the conditional nondeterministic pseudorandom generator (PRG) construction in [R. R. Williams, SIAM J. Comput., 45 (2016), pp. 497–529]. 3. An “almost” almost-everywhere [math] average-case lower bound (which strengthens the corresponding worst-case lower bound in [C. D. Murray and R. R. Williams, STOC 2018]). 4. A [math]-complete language that is downward self-reducible, same-length checkable, error-correctable, and paddable. Moreover, all its reducibility properties have corresponding low-depth nonadaptive oracle circuits. Our construction builds on [L. Trevisan and S. P. Vadhan, Comput. Complexity, 16 (2007), pp. 331–364]. Like other lower bounds proved via the “algorithmic approach,” the only property of [math] exploited by us is the existence of a nontrivial [math] algorithm for [math] [R. R. Williams, J. ACM, 61 (2014)]. Therefore, for any typical circuit class [math], our results apply to [math] as well if a nontrivial [math] (in fact, [math]) algorithm for [math] is discovered.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"20 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139949926","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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