SIAM Journal on Computing, Ahead of Print. Abstract. The graphical balls-into-bins process is a generalization of the classical 2-choice balls-into-bins process, where the bins correspond to vertices of an arbitrary underlying graph [math]. At each time step an edge of [math] is chosen uniformly at random, and a ball must be assigned to either of the two endpoints of this edge. The standard 2-choice process corresponds to the case of [math]. For any [math]-edge-connected, [math]-regular graph on [math] vertices, and any number of balls, we give an allocation strategy that, with high probability, ensures a gap of [math] between the load of any two bins. In particular, this implies a polylogarithmic bound for natural graphs such as cycles and tori, for which the classical greedy allocation strategy is conjectured to have a polynomial gap between the bin loads. For every graph [math], we also show an [math] lower bound on the gap achievable by any allocation strategy. This implies that our strategy achieves the optimal gap, up to polylogarithmic factors, for every graph [math]. Our allocation algorithm is simple to implement and requires only [math] time per allocation. It can be viewed as a more global version of the greedy strategy that compares average load on certain fixed sets of vertices, rather than on individual vertices. A key idea is to relate the problem of designing a good allocation strategy to that of finding suitable multicommodity flows. To this end, we consider Räcke’s cut-based decomposition tree and define certain orthogonal flows on it.
{"title":"The Power of Two Choices in Graphical Allocation","authors":"Nikhil Bansal, Ohad Feldheim","doi":"10.1137/22m1541800","DOIUrl":"https://doi.org/10.1137/22m1541800","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. The graphical balls-into-bins process is a generalization of the classical 2-choice balls-into-bins process, where the bins correspond to vertices of an arbitrary underlying graph [math]. At each time step an edge of [math] is chosen uniformly at random, and a ball must be assigned to either of the two endpoints of this edge. The standard 2-choice process corresponds to the case of [math]. For any [math]-edge-connected, [math]-regular graph on [math] vertices, and any number of balls, we give an allocation strategy that, with high probability, ensures a gap of [math] between the load of any two bins. In particular, this implies a polylogarithmic bound for natural graphs such as cycles and tori, for which the classical greedy allocation strategy is conjectured to have a polynomial gap between the bin loads. For every graph [math], we also show an [math] lower bound on the gap achievable by any allocation strategy. This implies that our strategy achieves the optimal gap, up to polylogarithmic factors, for every graph [math]. Our allocation algorithm is simple to implement and requires only [math] time per allocation. It can be viewed as a more global version of the greedy strategy that compares average load on certain fixed sets of vertices, rather than on individual vertices. A key idea is to relate the problem of designing a good allocation strategy to that of finding suitable multicommodity flows. To this end, we consider Räcke’s cut-based decomposition tree and define certain orthogonal flows on it.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"69 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198437","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 4, Page 1181-1215, August 2024. Abstract. In this paper, we study the following Minimum Cost Multicovering (MCMC) problem: Given a set of [math] client points [math] and a set of [math] server points [math] in a fixed dimensional [math] space, determine a set of disks centered at these server points so that each client point [math] is covered by at least [math] disks and the total cost of these disks is minimized, where [math] is a function that maps every client point to some nonnegative integer no more than [math] and the cost of each disk is measured by the [math]th power of its radius for some constant [math]. MCMC is a fundamental optimization problem with applications in many areas such as wireless/sensor networking. Despite extensive research on this problem for about two decades, only constant approximations were known for general [math]. It has been a long standing open problem to determine whether a PTAS is possible. In this paper, we give an affirmative answer to this question by presenting the first PTAS for it. Our approach is based on a number of novel techniques, such as balanced recursive realization and bubble charging, and new counterintuitive insights to the problem. Particularly, we approximate each disk with a set of sub-boxes and optimize them at the subdisk level. This allows us to first compute an approximate disk cover through dynamic programming, and then obtain the desired disk cover through a balanced recursive realization procedure.
{"title":"PTAS for Minimum Cost MultiCovering with Disks","authors":"Ziyun Huang, Qilong Feng, Jianxin Wang, Jinhui Xu","doi":"10.1137/22m1523352","DOIUrl":"https://doi.org/10.1137/22m1523352","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1181-1215, August 2024. <br/> Abstract. In this paper, we study the following Minimum Cost Multicovering (MCMC) problem: Given a set of [math] client points [math] and a set of [math] server points [math] in a fixed dimensional [math] space, determine a set of disks centered at these server points so that each client point [math] is covered by at least [math] disks and the total cost of these disks is minimized, where [math] is a function that maps every client point to some nonnegative integer no more than [math] and the cost of each disk is measured by the [math]th power of its radius for some constant [math]. MCMC is a fundamental optimization problem with applications in many areas such as wireless/sensor networking. Despite extensive research on this problem for about two decades, only constant approximations were known for general [math]. It has been a long standing open problem to determine whether a PTAS is possible. In this paper, we give an affirmative answer to this question by presenting the first PTAS for it. Our approach is based on a number of novel techniques, such as balanced recursive realization and bubble charging, and new counterintuitive insights to the problem. Particularly, we approximate each disk with a set of sub-boxes and optimize them at the subdisk level. This allows us to first compute an approximate disk cover through dynamic programming, and then obtain the desired disk cover through a balanced recursive realization procedure.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"14 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198439","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 4, Page 1162-1180, August 2024. Abstract. The Gomory–Hu tree or cut tree [R. E. Gomory and T. C. Hu, J. Soc. Indust. Appl. Math., 9 (1961), pp. 551–570] is a classic data structure for reporting [math]-mincuts (and by duality, the values of [math]-maxflows) for all-pairs of vertices [math] and [math] in an undirected graph. Gomory and Hu showed that it can be computed using [math] exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory–Hu trees more than 50 years later, even for approximate mincuts. In this paper, we break this longstanding barrier and give an algorithm for computing a [math]-approximate Gomory–Hu tree using [math] maxflow computations. Specifically, we obtain the running time bounds we describe below. We obtain a randomized (Monte Carlo) algorithm for undirected, weighted graphs that runs in [math] time and returns a [math]-approximate Gomory–Hu tree with high probability (w.h.p.). Previously, the best running time known was [math], which is obtained by running Gomory and Hu’s original algorithm on a cut sparsifier of the graph. Next, we obtain a randomized (Monte Carlo) algorithm for undirected, unweighted graphs that runs in [math] time and returns a [math]-approximate Gomory–Hu tree w.h.p. This improves on our first result for sparse graphs, namely [math]. Previously, the best running time known for unweighted graphs was [math] for an exact Gomory–Hu tree [A. Bhalgat et al., Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, 2007, pp. 605–614]; no better result is known if approximations are allowed. As a consequence of our Gomory–Hu tree algorithms, we also solve the [math]-approximate all-pairs mincut (APMC) and single-source mincut (SSMC) problems in the same time bounds. (These problems are simpler in that the goal is to only return the [math]-mincut values, and not the mincuts.) This improves on the recent algorithm for these problems in [math] time due to Abboud, Krauthgamer, and Trabelsi [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 2020, pp. 105–118].
SIAM 计算期刊》，第 53 卷第 4 期，第 1162-1180 页，2024 年 8 月。 摘要。Gomory-Hu 树或切割树 [R. E. Gomory and T. C. Hu, J. Soc.E. Gomory and T. C. Hu, J. Soc. Indust.应用数学》，9 (1961)，第 551-570 页]是一种经典的数据结构，用于报告无向图中所有成对顶点[math]和[math]的[math]-mincuts（以及对偶性，[math]-maxflows 的值）。Gomory 和 Hu 证明，可以用 [math] 精确最大流计算来计算它。令人惊讶的是，50 多年后的今天，这仍然是 Gomory-Hu 树的最佳算法，甚至对于近似最小切分也是如此。在本文中，我们打破了这一长期存在的障碍，给出了一种使用[math] maxflow 计算来计算[math]近似 Gomory-Hu 树的算法。具体来说，我们获得了下面描述的运行时间边界。我们获得了一种针对无向加权图的随机（蒙特卡洛）算法，它能在[math]时间内运行，并以高概率（w.h.p.）返回一棵[math]近似的 Gomory-Hu 树。在此之前，已知的最佳运行时间是[math]，它是通过在图的剪切疏解器上运行 Gomory 和 Hu 的原始算法得到的。接下来，我们得到了一种针对无向、无权重图的随机（蒙特卡罗）算法，该算法的运行时间为[math]，并能返回一个[math]近似的 Gomory-Hu 树w.h.p。在此之前，已知无权重图的最佳运行时间是精确 Gomory-Hu 树的 [math] [A. Bhalgat et al.Bhalgat et al., Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, 2007, pp.由于我们的 Gomory-Hu 树算法，我们还能在相同的时间界限内解决[math]近似全对最小裁剪（APMC）和单源最小裁剪（SSMC）问题。(这些问题比较简单，因为我们的目标只是返回[math]-mincut 值，而不是 mincut）。这改进了 Abboud、Krauthgamer 和 Trabelsi 最近提出的在[math]时间内解决这些问题的算法[2020 IEEE 第 61 届计算机科学基础年度研讨会，IEEE 计算机学会，2020 年，第 105-118 页]。
{"title":"Approximate Gomory–Hu Tree is Faster than [math] Maximum Flows","authors":"Jason Li, Debmalya Panigrahi","doi":"10.1137/21m1463379","DOIUrl":"https://doi.org/10.1137/21m1463379","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1162-1180, August 2024. <br/> Abstract. The Gomory–Hu tree or cut tree [R. E. Gomory and T. C. Hu, J. Soc. Indust. Appl. Math., 9 (1961), pp. 551–570] is a classic data structure for reporting [math]-mincuts (and by duality, the values of [math]-maxflows) for all-pairs of vertices [math] and [math] in an undirected graph. Gomory and Hu showed that it can be computed using [math] exact maxflow computations. Surprisingly, this remains the best algorithm for Gomory–Hu trees more than 50 years later, even for approximate mincuts. In this paper, we break this longstanding barrier and give an algorithm for computing a [math]-approximate Gomory–Hu tree using [math] maxflow computations. Specifically, we obtain the running time bounds we describe below. We obtain a randomized (Monte Carlo) algorithm for undirected, weighted graphs that runs in [math] time and returns a [math]-approximate Gomory–Hu tree with high probability (w.h.p.). Previously, the best running time known was [math], which is obtained by running Gomory and Hu’s original algorithm on a cut sparsifier of the graph. Next, we obtain a randomized (Monte Carlo) algorithm for undirected, unweighted graphs that runs in [math] time and returns a [math]-approximate Gomory–Hu tree w.h.p. This improves on our first result for sparse graphs, namely [math]. Previously, the best running time known for unweighted graphs was [math] for an exact Gomory–Hu tree [A. Bhalgat et al., Proceedings of the 39th Annual ACM Symposium on Theory of Computing, San Diego, CA, 2007, pp. 605–614]; no better result is known if approximations are allowed. As a consequence of our Gomory–Hu tree algorithms, we also solve the [math]-approximate all-pairs mincut (APMC) and single-source mincut (SSMC) problems in the same time bounds. (These problems are simpler in that the goal is to only return the [math]-mincut values, and not the mincuts.) This improves on the recent algorithm for these problems in [math] time due to Abboud, Krauthgamer, and Trabelsi [2020 IEEE 61st Annual Symposium on Foundations of Computer Science, IEEE Computer Society, 2020, pp. 105–118].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"69 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142198440","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 4, Page 1132-1161, August 2024. Abstract. In this paper, we first consider the subpath convex hull query problem: Given a simple path [math] of [math] vertices, preprocess it so that the convex hull of any query subpath of [math] can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [Int. J. Comput. Geom. Appl., 1 (1991), pp. 1–22; first appeared in SODA 1990] proposed a data structure of [math] space and [math] query time; they also reduced the query time to [math] by increasing the space to [math]. We present an improved result that uses [math] space while achieving [math] query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in [math] time each. The preprocessing time of our data structure is [math] after the vertices of [math] are sorted by [math]-coordinate. As the subpath convex hull query problem has many applications, our new result leads to improvements for several other problems. In particular, with the help of the above result, along with other techniques, we present new algorithms for the ray-shooting problem among segments. Given a set of [math] (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of [math] space that can answer each query in [math] time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to [math]. As a by-product, given a set of [math] (possibly intersecting) segments in the plane, we build a data structure of [math] space that can determine whether a query line intersects a segment in [math] time. The preprocessing time is [math] for all four problems, which can be reduced to [math] time by a randomized algorithm so that the query time is bounded by [math] with high probability. All these are classical problems that have been studied extensively. Previously data structures of [math] query time were known in the early 1990s (the notation [math] suppresses a polylogarithmic factor); nearly no progress has been made for more than two decades. For all these problems, our new results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better.
{"title":"Algorithms for Subpath Convex Hull Queries and Ray-Shooting among Segments","authors":"Haitao Wang","doi":"10.1137/21m145118x","DOIUrl":"https://doi.org/10.1137/21m145118x","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1132-1161, August 2024. <br/> Abstract. In this paper, we first consider the subpath convex hull query problem: Given a simple path [math] of [math] vertices, preprocess it so that the convex hull of any query subpath of [math] can be quickly obtained. Previously, Guibas, Hershberger, and Snoeyink [Int. J. Comput. Geom. Appl., 1 (1991), pp. 1–22; first appeared in SODA 1990] proposed a data structure of [math] space and [math] query time; they also reduced the query time to [math] by increasing the space to [math]. We present an improved result that uses [math] space while achieving [math] query time. Like the previous work, our query algorithm returns a compact interval tree representing the convex hull so that standard binary-search-based queries on the hull can be performed in [math] time each. The preprocessing time of our data structure is [math] after the vertices of [math] are sorted by [math]-coordinate. As the subpath convex hull query problem has many applications, our new result leads to improvements for several other problems. In particular, with the help of the above result, along with other techniques, we present new algorithms for the ray-shooting problem among segments. Given a set of [math] (possibly intersecting) line segments in the plane, preprocess it so that the first segment hit by a query ray can be quickly found. We give a data structure of [math] space that can answer each query in [math] time. If the segments are nonintersecting or if the segments are lines, then the space can be reduced to [math]. As a by-product, given a set of [math] (possibly intersecting) segments in the plane, we build a data structure of [math] space that can determine whether a query line intersects a segment in [math] time. The preprocessing time is [math] for all four problems, which can be reduced to [math] time by a randomized algorithm so that the query time is bounded by [math] with high probability. All these are classical problems that have been studied extensively. Previously data structures of [math] query time were known in the early 1990s (the notation [math] suppresses a polylogarithmic factor); nearly no progress has been made for more than two decades. For all these problems, our new results provide improvements by reducing the space of the data structures by at least a logarithmic factor while the preprocessing and query times are the same as before or even better.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"60 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142225314","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 4, Page 1085-1131, August 2024. Abstract. Branchwidth determines how graphs and, more generally, arbitrary connectivity (symmetric and submodular) functions can be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations can decrease the width of a branch decomposition or the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework. The first is an algorithm that, for a given [math]-vertex graph [math] and integer [math], in time [math] either constructs a rank decomposition of [math] of width at most [math] or concludes that the rankwidth of [math] is more than [math]. It also yields a [math]-approximation algorithm for cliquewidth within the same time complexity, which in turn improves to [math] the running times of various algorithms on graphs of cliquewidth [math]. Breaking the “cubic barrier” for rankwidth and cliquewidth was an open problem in the area. The second application is an algorithm that, for a given [math]-vertex graph [math] and integer [math], in time [math] either constructs a branch decomposition of [math] of width at most [math] or concludes that the branchwidth of [math] is more than [math]. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].
{"title":"Fast FPT-Approximation of Branchwidth","authors":"Fedor V. Fomin, Tuukka Korhonen","doi":"10.1137/22m153937x","DOIUrl":"https://doi.org/10.1137/22m153937x","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1085-1131, August 2024. <br/> Abstract. Branchwidth determines how graphs and, more generally, arbitrary connectivity (symmetric and submodular) functions can be decomposed into a tree-like structure by specific cuts. We develop a general framework for designing fixed-parameter tractable 2-approximation algorithms for branchwidth of connectivity functions. The first ingredient of our framework is combinatorial. We prove a structural theorem establishing that either a sequence of particular refinement operations can decrease the width of a branch decomposition or the width of the decomposition is already within a factor of 2 from the optimum. The second ingredient is an efficient implementation of the refinement operations for branch decompositions that support efficient dynamic programming. We present two concrete applications of our general framework. The first is an algorithm that, for a given [math]-vertex graph [math] and integer [math], in time [math] either constructs a rank decomposition of [math] of width at most [math] or concludes that the rankwidth of [math] is more than [math]. It also yields a [math]-approximation algorithm for cliquewidth within the same time complexity, which in turn improves to [math] the running times of various algorithms on graphs of cliquewidth [math]. Breaking the “cubic barrier” for rankwidth and cliquewidth was an open problem in the area. The second application is an algorithm that, for a given [math]-vertex graph [math] and integer [math], in time [math] either constructs a branch decomposition of [math] of width at most [math] or concludes that the branchwidth of [math] is more than [math]. This improves over the 3-approximation that follows from the recent treewidth 2-approximation of Korhonen [FOCS 2021].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"163 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141937388","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 4, Page 1039-1084, August 2024. Abstract. Determining the number of registers required for solving obstruction-free (or randomized wait-free) [math]-set agreement is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. The best known upper bound on the number of registers needed to solve this problem among [math] processes is [math] registers. No general lower bound better than 2 was known. We prove that any obstruction-free protocol solving [math]-set agreement among [math] processes must use at least [math] registers. In particular, we get a tight lower bound of exactly [math] registers for solving obstruction-free and randomized wait-free consensus. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free [math]-set agreement. In particular, we show that if an obstruction-free protocol for [math]-set agreement uses fewer registers, then it is possible for [math] processes to simulate the protocol and deterministically solve [math]-set agreement in a wait-free manner, which is impossible. An important aspect of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce an augmented snapshot object, which facilitates this. More generally, our simulation applies to the broad class of colorless tasks. We can use it to prove, for example, a lower bound on the number of registers needed to solve obstruction-free [math]-approximate agreement, which matches the best known upper bound to within a factor of 2 when [math] is sufficiently small. No general lower bound for this problem was known. Finally, we prove that any lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo-termination. Hence, our lower bounds for obstruction-free protocols also hold for randomized wait-free protocols.
{"title":"Revisionist Simulations: A New Approach to Proving Space Lower Bounds","authors":"Faith Ellen, Rati Gelashvili, Leqi Zhu","doi":"10.1137/20m1322923","DOIUrl":"https://doi.org/10.1137/20m1322923","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1039-1084, August 2024. <br/> Abstract. Determining the number of registers required for solving obstruction-free (or randomized wait-free) [math]-set agreement is an open problem that highlights important gaps in our understanding of the space complexity of synchronization. The best known upper bound on the number of registers needed to solve this problem among [math] processes is [math] registers. No general lower bound better than 2 was known. We prove that any obstruction-free protocol solving [math]-set agreement among [math] processes must use at least [math] registers. In particular, we get a tight lower bound of exactly [math] registers for solving obstruction-free and randomized wait-free consensus. Our main tool is a simulation that serves as a reduction from the impossibility of deterministic wait-free [math]-set agreement. In particular, we show that if an obstruction-free protocol for [math]-set agreement uses fewer registers, then it is possible for [math] processes to simulate the protocol and deterministically solve [math]-set agreement in a wait-free manner, which is impossible. An important aspect of the simulation is the ability of simulating processes to revise the past of simulated processes. We introduce an augmented snapshot object, which facilitates this. More generally, our simulation applies to the broad class of colorless tasks. We can use it to prove, for example, a lower bound on the number of registers needed to solve obstruction-free [math]-approximate agreement, which matches the best known upper bound to within a factor of 2 when [math] is sufficiently small. No general lower bound for this problem was known. Finally, we prove that any lower bound on the number of registers used by obstruction-free protocols applies to protocols that satisfy nondeterministic solo-termination. Hence, our lower bounds for obstruction-free protocols also hold for randomized wait-free protocols.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"12 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882486","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}
Sunil Arya, Guilherme D. da Fonseca, David M. Mount
SIAM Journal on Computing, Volume 53, Issue 4, Page 1002-1038, August 2024. Abstract. Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body [math] and [math], a covering is a collection of convex bodies whose union covers [math] such that a constant factor expansion of each body lies within an [math] expansion of [math]. Coverings have been employed in many applications, such as approximations for diameter, width, and [math]-kernels of point sets, approximate nearest neighbor searching, polytope approximations with low combinatorial complexity, and approximations to the closest vector problem (CVP). It is known how to construct coverings of size [math] for general convex bodies in [math]. In special cases, such as when the convex body is the [math] unit ball, this bound has been improved to [math]. This raises the question of whether such a bound generally holds. In this paper we answer the question in the affirmative. We demonstrate the power and versatility of our coverings by applying them to the problem of approximating a convex body by a polytope, where the error is measured through the Banach–Mazur metric. Given a well-centered convex body [math] and an approximation parameter [math], we show that there exists a polytope [math] consisting of [math] vertices (facets) such that [math]. This bound is optimal in the worst case up to factors of [math]. (This bound has been established recently using different techniques, but our approach is arguably simpler and more elegant.) As an additional consequence, we obtain the fastest [math]-approximate CVP algorithm that works in any norm, with a running time of [math] up to polynomial factors in the input size, and we obtain the fastest [math]-approximation algorithm for integer programming. We also present a framework for constructing coverings of optimal size for any convex body (up to factors of [math]).
{"title":"Economical Convex Coverings and Applications","authors":"Sunil Arya, Guilherme D. da Fonseca, David M. Mount","doi":"10.1137/23m1568351","DOIUrl":"https://doi.org/10.1137/23m1568351","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 1002-1038, August 2024. <br/> Abstract. Coverings of convex bodies have emerged as a central component in the design of efficient solutions to approximation problems involving convex bodies. Intuitively, given a convex body [math] and [math], a covering is a collection of convex bodies whose union covers [math] such that a constant factor expansion of each body lies within an [math] expansion of [math]. Coverings have been employed in many applications, such as approximations for diameter, width, and [math]-kernels of point sets, approximate nearest neighbor searching, polytope approximations with low combinatorial complexity, and approximations to the closest vector problem (CVP). It is known how to construct coverings of size [math] for general convex bodies in [math]. In special cases, such as when the convex body is the [math] unit ball, this bound has been improved to [math]. This raises the question of whether such a bound generally holds. In this paper we answer the question in the affirmative. We demonstrate the power and versatility of our coverings by applying them to the problem of approximating a convex body by a polytope, where the error is measured through the Banach–Mazur metric. Given a well-centered convex body [math] and an approximation parameter [math], we show that there exists a polytope [math] consisting of [math] vertices (facets) such that [math]. This bound is optimal in the worst case up to factors of [math]. (This bound has been established recently using different techniques, but our approach is arguably simpler and more elegant.) As an additional consequence, we obtain the fastest [math]-approximate CVP algorithm that works in any norm, with a running time of [math] up to polynomial factors in the input size, and we obtain the fastest [math]-approximation algorithm for integer programming. We also present a framework for constructing coverings of optimal size for any convex body (up to factors of [math]).","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"36 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141882623","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 give a polynomial-time constant-factor approximation algorithm for the maximum independent set of (axis-aligned) rectangles problem in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is [math]. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.
{"title":"Approximating Maximum Independent Set for Rectangles in the Plane","authors":"Joseph Mitchell","doi":"10.1137/22m1475521","DOIUrl":"https://doi.org/10.1137/22m1475521","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. We give a polynomial-time constant-factor approximation algorithm for the maximum independent set of (axis-aligned) rectangles problem in the plane. Using a polynomial-time algorithm, the best approximation factor previously known is [math]. The results are based on a new form of recursive partitioning in the plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"71 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141782513","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 4, Page 969-1001, August 2024. Abstract. A central issue in machine learning is how to train models on sensitive user data. Industry has widely adopted a simple algorithm: Stochastic Gradient Descent (SGD) with noise (a.k.a. Stochastic Gradient Langevin Dynamics). However, foundational theoretical questions about this algorithm’s privacy loss remain open—even in the seemingly simple setting of smooth convex losses over a bounded domain. Our main result resolves these questions: for a large range of parameters, we characterize the differential privacy up to a constant factor. This result reveals that all previous analyses for this setting have the wrong qualitative behavior. Specifically, while previous privacy analyses increase ad infinitum in the number of iterations, we show that after a small burn-in period, running SGD longer leaks no further privacy. Our analysis departs from previous approaches based on fast mixing, instead using techniques based on optimal transport (namely, Privacy Amplification by Iteration) and the Sampled Gaussian Mechanism (namely, Privacy Amplification by Sampling). Our techniques readily extend to other settings, e.g., strongly convex losses, nonuniform stepsizes, arbitrary batch sizes, and random or cyclic choice of batches.
{"title":"On the Privacy of Noisy Stochastic Gradient Descent for Convex Optimization","authors":"Jason M. Altschuler, Jinho Bok, Kunal Talwar","doi":"10.1137/23m1556538","DOIUrl":"https://doi.org/10.1137/23m1556538","url":null,"abstract":"SIAM Journal on Computing, Volume 53, Issue 4, Page 969-1001, August 2024. <br/> Abstract. A central issue in machine learning is how to train models on sensitive user data. Industry has widely adopted a simple algorithm: Stochastic Gradient Descent (SGD) with noise (a.k.a. Stochastic Gradient Langevin Dynamics). However, foundational theoretical questions about this algorithm’s privacy loss remain open—even in the seemingly simple setting of smooth convex losses over a bounded domain. Our main result resolves these questions: for a large range of parameters, we characterize the differential privacy up to a constant factor. This result reveals that all previous analyses for this setting have the wrong qualitative behavior. Specifically, while previous privacy analyses increase ad infinitum in the number of iterations, we show that after a small burn-in period, running SGD longer leaks no further privacy. Our analysis departs from previous approaches based on fast mixing, instead using techniques based on optimal transport (namely, Privacy Amplification by Iteration) and the Sampled Gaussian Mechanism (namely, Privacy Amplification by Sampling). Our techniques readily extend to other settings, e.g., strongly convex losses, nonuniform stepsizes, arbitrary batch sizes, and random or cyclic choice of batches.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"18 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141737512","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}
Klim Efremenko, Bernhard Haeupler, Yael Tauman Kalai, Pritish Kamath, Gillat Kol, Nicolas Resch, Raghuvansh R. Saxena
SIAM Journal on Computing, Ahead of Print. Abstract. Given a Boolean circuit [math], we wish to convert it to a circuit [math] that computes the same function as [math], even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [D. J. Kleitman, F. T. Leighton, and Y. Ma, J. Comput. System Sci., 55 (1997), pp. 385–401]. Can we design such a resilient circuit [math] whose size is roughly comparable to that of [math]? Prior work [T. Kalai, A. B. Lewko, and A. Rao, Formulas resilient to short-circuit errors, in Foundations of Computer Science (FOCS), 2012, pp. 490–499; M. Braverman et al., Optimal short-circuit resilient formulas, in Computational Complexity Conference (CCC), Vol. 137, 2019, pp. 10:1–10:22] gave a positive answer for the special case where [math] is a formula. We study the general case and show that any Boolean circuit [math] of size [math] can be converted to a new circuit [math] of quasi-polynomial size [math] that computes the same function as [math], even if a [math] fraction of the gates on any root-to-leaf path in [math] are short circuited. Moreover, if the original circuit [math] is a formula, the resilient circuit [math] is of near-linear size [math]. The construction of our resilient circuits utilizes the connection between circuits and dag-like communication protocols [A. Razborov, Izvestiya of the RAN, 59 (1995), pp. 201–224; P. Pudlák, On extracting computations from propositional proofs (a survey), in Foundations of Software Technology and Theoretical Computer Science (FSTTCS) Vol. 8, 2010, pp. 30–41; D. Sokolov, Dag-like communication and its applications, in Computer Science Symposium in Russia (CSR), Springer, 2017, pp. 294–307], originally introduced in the context of proof complexity.
SIAM 计算期刊》，提前印刷。 摘要给定一个布尔电路 [math]，我们希望将它转换成一个电路 [math]，即使其中一些门出现对抗性短路错误，即它们的输出被其中一个输入的值所取代，它仍能计算与 [math] 相同的函数 [D. J. Kleitman, F. T. Leighton, and Y. Ma, J. Computing.J. Kleitman, F. T. Leighton, and Y. Ma, J. Comput.系统科学》，55 (1997)，第 385-401 页]。我们能否设计出这样一种弹性电路[math]，其大小与[math]大致相当呢？先前的工作 [T. Kalai, A. B. Lew.Kalai, A. B. Lewko, and A. Rao, Formulas resilient to short-circuit errors, in Foundations of Computer Science (FOCS), 2012, pp.我们研究了一般情况，并证明任何大小为 [math] 的布尔电路 [math] 都能转换成一个准多项式大小为 [math] 的新电路 [math]，即使 [math] 中任何根到叶路径上有 [math] 部分的门被短路，它也能计算与 [math] 相同的函数。此外，如果原始电路[math]是一个公式，弹性电路[math]的大小[math]也接近线性。我们的弹性电路的构造利用了电路与类似达格的通信协议之间的联系 [A. Razborov, Izvests.Razborov, Izvestiya of the RAN, 59 (1995), pp. 201-224; P. Pudlák, On extracting computations from propositional proofs (a survey), in Foundations of Software Technology and Theoretical Computer Science (FSTTCS) Vol. 8, 2010, pp.
{"title":"Circuits Resilient to Short-Circuit Errors","authors":"Klim Efremenko, Bernhard Haeupler, Yael Tauman Kalai, Pritish Kamath, Gillat Kol, Nicolas Resch, Raghuvansh R. Saxena","doi":"10.1137/22m1520578","DOIUrl":"https://doi.org/10.1137/22m1520578","url":null,"abstract":"SIAM Journal on Computing, Ahead of Print. <br/> Abstract. Given a Boolean circuit [math], we wish to convert it to a circuit [math] that computes the same function as [math], even if some of its gates suffer from adversarial short circuit errors, i.e., their output is replaced by the value of one of their inputs [D. J. Kleitman, F. T. Leighton, and Y. Ma, J. Comput. System Sci., 55 (1997), pp. 385–401]. Can we design such a resilient circuit [math] whose size is roughly comparable to that of [math]? Prior work [T. Kalai, A. B. Lewko, and A. Rao, Formulas resilient to short-circuit errors, in Foundations of Computer Science (FOCS), 2012, pp. 490–499; M. Braverman et al., Optimal short-circuit resilient formulas, in Computational Complexity Conference (CCC), Vol. 137, 2019, pp. 10:1–10:22] gave a positive answer for the special case where [math] is a formula. We study the general case and show that any Boolean circuit [math] of size [math] can be converted to a new circuit [math] of quasi-polynomial size [math] that computes the same function as [math], even if a [math] fraction of the gates on any root-to-leaf path in [math] are short circuited. Moreover, if the original circuit [math] is a formula, the resilient circuit [math] is of near-linear size [math]. The construction of our resilient circuits utilizes the connection between circuits and dag-like communication protocols [A. Razborov, Izvestiya of the RAN, 59 (1995), pp. 201–224; P. Pudlák, On extracting computations from propositional proofs (a survey), in Foundations of Software Technology and Theoretical Computer Science (FSTTCS) Vol. 8, 2010, pp. 30–41; D. Sokolov, Dag-like communication and its applications, in Computer Science Symposium in Russia (CSR), Springer, 2017, pp. 294–307], originally introduced in the context of proof complexity.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"30 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141717902","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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