We introduce synchronization strings, which provide a novel way to efficiently deal with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to cope with than more commonly considered Hamming-type errors, i.e., symbol substitutions and erasures. For every ε > 0, synchronization strings allow us to index a sequence with an ε-O(1)-size alphabet, such that one can efficiently transform k synchronization errors into (1 + ε)k Hamming-type errors. This powerful new technique has many applications. In this article, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion-deletion channels. While ECCs for both Hamming-type errors and synchronization errors have been intensely studied, the latter has largely resisted progress. As Mitzenmacher puts it in his 2009 survey [30]: “Channels with synchronization errors...are simply not adequately understood by current theory. Given the near-complete knowledge, we have for channels with erasures and errors...our lack of understanding about channels with synchronization errors is truly remarkable.” Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate insdel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regimes, these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straightforward application of our synchronization strings-based indexing method gives a simple black-box construction that transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs into the realm of insdel codes. Most notably, for the complete noise spectrum, we obtain efficient “near-MDS” insdel codes, which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and ε > 0, we give a family of insdel codes achieving a rate of 1 - δ - ε over a constant-size alphabet that efficiently corrects a δ fraction of insertions or deletions.
{"title":"Synchronization Strings: Codes for Insertions and Deletions Approaching the Singleton Bound","authors":"Bernhard Haeupler, Amirbehshad Shahrasbi","doi":"10.1145/3468265","DOIUrl":"https://doi.org/10.1145/3468265","url":null,"abstract":"We introduce synchronization strings, which provide a novel way to efficiently deal with synchronization errors, i.e., insertions and deletions. Synchronization errors are strictly more general and much harder to cope with than more commonly considered Hamming-type errors, i.e., symbol substitutions and erasures. For every ε > 0, synchronization strings allow us to index a sequence with an ε-O(1)-size alphabet, such that one can efficiently transform k synchronization errors into (1 + ε)k Hamming-type errors. This powerful new technique has many applications. In this article, we focus on designing insdel codes, i.e., error correcting block codes (ECCs) for insertion-deletion channels. While ECCs for both Hamming-type errors and synchronization errors have been intensely studied, the latter has largely resisted progress. As Mitzenmacher puts it in his 2009 survey [30]: “Channels with synchronization errors...are simply not adequately understood by current theory. Given the near-complete knowledge, we have for channels with erasures and errors...our lack of understanding about channels with synchronization errors is truly remarkable.” Indeed, it took until 1999 for the first insdel codes with constant rate, constant distance, and constant alphabet size to be constructed and only since 2016 are there constructions of constant rate insdel codes for asymptotically large noise rates. Even in the asymptotically large or small noise regimes, these codes are polynomially far from the optimal rate-distance tradeoff. This makes the understanding of insdel codes up to this work equivalent to what was known for regular ECCs after Forney introduced concatenated codes in his doctoral thesis 50 years ago. A straightforward application of our synchronization strings-based indexing method gives a simple black-box construction that transforms any ECC into an equally efficient insdel code with only a small increase in the alphabet size. This instantly transfers much of the highly developed understanding for regular ECCs into the realm of insdel codes. Most notably, for the complete noise spectrum, we obtain efficient “near-MDS” insdel codes, which get arbitrarily close to the optimal rate-distance tradeoff given by the Singleton bound. In particular, for any δ ∈ (0,1) and ε > 0, we give a family of insdel codes achieving a rate of 1 - δ - ε over a constant-size alphabet that efficiently corrects a δ fraction of insertions or deletions.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75462891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Michael Blondin, Matthias Englert, A. Finkel, Stefan Göller, C. Haase, R. Lazic, P. McKenzie, Patrick Totzke
We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.
{"title":"The Reachability Problem for Two-Dimensional Vector Addition Systems with States","authors":"Michael Blondin, Matthias Englert, A. Finkel, Stefan Göller, C. Haase, R. Lazic, P. McKenzie, Patrick Totzke","doi":"10.1145/3464794","DOIUrl":"https://doi.org/10.1145/3464794","url":null,"abstract":"We prove that the reachability problem for two-dimensional vector addition systems with states is NL-complete or PSPACE-complete, depending on whether the numbers in the input are encoded in unary or binary. As a key underlying technical result, we show that, if a configuration is reachable, then there exists a witnessing path whose sequence of transitions is contained in a bounded language defined by a regular expression of pseudo-polynomially bounded length. This, in turn, enables us to prove that the lengths of minimal reachability witnesses are pseudo-polynomially bounded.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74602097","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Invited Articles section of this issue consists of two articles. The article “Uniform, Integral, and Feasible Proofs for the Determinant Identities,” by Iddo Tzameret and Stephen Cook, was invited from the 32nd ACM/ IEEE Symposium on Logic in Computer Science (LICS’17). We want to thank the LICS’17 Program Committee for their help in selecting this invited article, and editor Andrew Pitts for handling the article. The second article, “Exploiting Spontaneous Transmissions for Broadcasting and Leader Election in Radio Networks,” by Artur Czumaj and Peter Davies, won the best student paper award at the 36th ACM Symposium on Principles of Distributed Computing (POCD’17). We want to thank the PODC’17 Program Committee for their help in selecting this invited article, and editor Christian Scheideler for handling the article.
{"title":"Invited Articles Foreword","authors":"É. Tardos","doi":"10.1145/3456290","DOIUrl":"https://doi.org/10.1145/3456290","url":null,"abstract":"The Invited Articles section of this issue consists of two articles. The article “Uniform, Integral, and Feasible Proofs for the Determinant Identities,” by Iddo Tzameret and Stephen Cook, was invited from the 32nd ACM/ IEEE Symposium on Logic in Computer Science (LICS’17). We want to thank the LICS’17 Program Committee for their help in selecting this invited article, and editor Andrew Pitts for handling the article. The second article, “Exploiting Spontaneous Transmissions for Broadcasting and Leader Election in Radio Networks,” by Artur Czumaj and Peter Davies, won the best student paper award at the 36th ACM Symposium on Principles of Distributed Computing (POCD’17). We want to thank the PODC’17 Program Committee for their help in selecting this invited article, and editor Christian Scheideler for handling the article.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91079741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chi Jin, Praneeth Netrapalli, Rong Ge, S. Kakade, Michael I. Jordan
Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient—their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.
{"title":"On Nonconvex Optimization for Machine Learning","authors":"Chi Jin, Praneeth Netrapalli, Rong Ge, S. Kakade, Michael I. Jordan","doi":"10.1145/3418526","DOIUrl":"https://doi.org/10.1145/3418526","url":null,"abstract":"Gradient descent (GD) and stochastic gradient descent (SGD) are the workhorses of large-scale machine learning. While classical theory focused on analyzing the performance of these methods in convex optimization problems, the most notable successes in machine learning have involved nonconvex optimization, and a gap has arisen between theory and practice. Indeed, traditional analyses of GD and SGD show that both algorithms converge to stationary points efficiently. But these analyses do not take into account the possibility of converging to saddle points. More recent theory has shown that GD and SGD can avoid saddle points, but the dependence on dimension in these analyses is polynomial. For modern machine learning, where the dimension can be in the millions, such dependence would be catastrophic. We analyze perturbed versions of GD and SGD and show that they are truly efficient—their dimension dependence is only polylogarithmic. Indeed, these algorithms converge to second-order stationary points in essentially the same time as they take to converge to classical first-order stationary points.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90049589","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Ω(D log n/D + log2 n) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O(D log n log log n/log D + logO(1) n)-time broadcasting algorithm. In this article, we give a new and faster algorithm that completes broadcasting in O(D log n/log D + logO(1) n) time, succeeding with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever n is polynomial in D. Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have used broadcasting as a subroutine and their complexity has been asymptotically strictly larger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O(D log n/D min {log log n, log n/D} + logO(1) n)-time, succeeding with high probability. Our new algorithm again requires O(D log n/log D + logO(1) n) time, also succeeding with high probability.
研究了拓扑未知、无碰撞检测机制的经典多跳无线网络模型中的广播和leader选举两个基本通信原语。近20年来，人们已经知道，在具有n个节点和直径为D的无向网络中，随机广播需要Ω(D log n/D + log2 n)轮，假设不知情的节点不允许通信(直到它们被告知)。直到最近，Haeupler和Wajc (PODC'2016)才表明，对于具有自发传输的模型，可以改进该边界，提供O(D log n log n/log D + logO(1) n)时间广播算法。在本文中，我们给出了一种新的更快的算法，该算法在O(D log n/log D + logO(1) n)时间内完成广播，成功率高。当n是D中的多项式时，这产生了第一个最优的O(D)时间广播算法。此外，我们的方法可以应用于设计一个新的领导人选举算法，该算法与我们的广播算法的性能相匹配。在此之前，所有快速随机领导人选举算法都将广播作为子程序，其复杂度渐近严格大于广播复杂度。特别是，目前已知最快的Ghaffari和Haeupler (SODA’2013)随机领导人选举算法需要O(D log n/D min {log log n, log n/D} + logO(1) n)时间，成功率高。我们的新算法同样需要O(D log n/log D + logO(1) n)时间，同样有高概率成功。
{"title":"Exploiting Spontaneous Transmissions for Broadcasting and Leader Election in Radio Networks","authors":"A. Czumaj, Peter Davies","doi":"10.1145/3446383","DOIUrl":"https://doi.org/10.1145/3446383","url":null,"abstract":"We study two fundamental communication primitives: broadcasting and leader election in the classical model of multi-hop radio networks with unknown topology and without collision detection mechanisms. It has been known for almost 20 years that in undirected networks with n nodes and diameter D, randomized broadcasting requires Ω(D log n/D + log2 n) rounds, assuming that uninformed nodes are not allowed to communicate (until they are informed). Only very recently, Haeupler and Wajc (PODC'2016) showed that this bound can be improved for the model with spontaneous transmissions, providing an O(D log n log log n/log D + logO(1) n)-time broadcasting algorithm. In this article, we give a new and faster algorithm that completes broadcasting in O(D log n/log D + logO(1) n) time, succeeding with high probability. This yields the first optimal O(D)-time broadcasting algorithm whenever n is polynomial in D. Furthermore, our approach can be applied to design a new leader election algorithm that matches the performance of our broadcasting algorithm. Previously, all fast randomized leader election algorithms have used broadcasting as a subroutine and their complexity has been asymptotically strictly larger than the complexity of broadcasting. In particular, the fastest previously known randomized leader election algorithm of Ghaffari and Haeupler (SODA'2013) requires O(D log n/D min {log log n, log n/D} + logO(1) n)-time, succeeding with high probability. Our new algorithm again requires O(D log n/log D + logO(1) n) time, also succeeding with high probability.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75362515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Invited Article section of this issue includes “The Reachability Problem for Petri Nets Is Not Elementary,” by Wojciech Czerwiński, Sławomir Lasota, Ranko Lazić, Jérôme Leroux, and Flip Mazowiecki, which won the best paper award at the 51st Annual ACM Symposium on the Theory of Computing (STOC’19). We thank the STOC’19 Program Committee for their help in selecting this invited article, and we thank editor Martin Grohe for handling this article.
{"title":"Invited Article Foreword","authors":"É. Tardos","doi":"10.1145/3442687","DOIUrl":"https://doi.org/10.1145/3442687","url":null,"abstract":"The Invited Article section of this issue includes “The Reachability Problem for Petri Nets Is Not Elementary,” by Wojciech Czerwiński, Sławomir Lasota, Ranko Lazić, Jérôme Leroux, and Flip Mazowiecki, which won the best paper award at the 51st Annual ACM Symposium on the Theory of Computing (STOC’19). We thank the STOC’19 Program Committee for their help in selecting this invited article, and we thank editor Martin Grohe for handling this article.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79222530","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the geometry of units and ideals of cyclotomic rings and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, given an ideal lattice of the cyclotomic ring of conductor m, the algorithm finds an approximation of the shortest vector by a factor exp (Õ(√ m)). This result exposes an unexpected hardness gap between these structured lattices and general lattices: The best known polynomial time generic lattice algorithms can only reach an approximation factor exp (Õ(m)). Following a recent series of attacks, these results call into question the hardness of various problems over structured lattices, such as Ideal-SVP and Ring-LWE, upon which relies the security of a number of cryptographic schemes. NOTE. This article is an extended version of a conference paper [11]. The results are generalized to arbitrary cyclotomic fields. In particular, we also extend some results of Reference [10] to arbitrary cyclotomic fields. In addition, we prove the numerical stability of the method of Reference [10]. These extended results appeared in the Ph.D. dissertation of the third author [46].
{"title":"Mildly Short Vectors in Cyclotomic Ideal Lattices in Quantum Polynomial Time","authors":"R. Cramer, L. Ducas, B. Wesolowski","doi":"10.1145/3431725","DOIUrl":"https://doi.org/10.1145/3431725","url":null,"abstract":"In this article, we study the geometry of units and ideals of cyclotomic rings and derive an algorithm to find a mildly short vector in any given cyclotomic ideal lattice in quantum polynomial time, under some plausible number-theoretic assumptions. More precisely, given an ideal lattice of the cyclotomic ring of conductor m, the algorithm finds an approximation of the shortest vector by a factor exp (Õ(√ m)). This result exposes an unexpected hardness gap between these structured lattices and general lattices: The best known polynomial time generic lattice algorithms can only reach an approximation factor exp (Õ(m)). Following a recent series of attacks, these results call into question the hardness of various problems over structured lattices, such as Ideal-SVP and Ring-LWE, upon which relies the security of a number of cryptographic schemes. NOTE. This article is an extended version of a conference paper [11]. The results are generalized to arbitrary cyclotomic fields. In particular, we also extend some results of Reference [10] to arbitrary cyclotomic fields. In addition, we prove the numerical stability of the method of Reference [10]. These extended results appeared in the Ph.D. dissertation of the third author [46].","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82129226","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Dughmi, Jason D. Hartline, Robert D. Kleinberg, Rad Niazadeh
We provide a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories. In a Bernoulli factory problem, one is given a function mapping the bias of an “input coin” to that of an “output coin,” and the challenge is to efficiently simulate the output coin given only sample access to the input coin. This is the key ingredient in designing an incentive compatible mechanism for bipartite matching, which can be used to make the approximately incentive compatible reduction of Hartline et al. [18] exactly incentive compatible.
{"title":"Bernoulli Factories and Black-box Reductions in Mechanism Design","authors":"S. Dughmi, Jason D. Hartline, Robert D. Kleinberg, Rad Niazadeh","doi":"10.1145/3440988","DOIUrl":"https://doi.org/10.1145/3440988","url":null,"abstract":"We provide a polynomial time reduction from Bayesian incentive compatible mechanism design to Bayesian algorithm design for welfare maximization problems. Unlike prior results, our reduction achieves exact incentive compatibility for problems with multi-dimensional and continuous type spaces. The key technical barrier preventing exact incentive compatibility in prior black-box reductions is that repairing violations of incentive constraints requires understanding the distribution of the mechanism’s output, which is typically #P-hard to compute. Reductions that instead estimate the output distribution by sampling inevitably suffer from sampling error, which typically precludes exact incentive compatibility. We overcome this barrier by employing and generalizing the computational model in the literature on Bernoulli Factories. In a Bernoulli factory problem, one is given a function mapping the bias of an “input coin” to that of an “output coin,” and the challenge is to efficiently simulate the output coin given only sample access to the input coin. This is the key ingredient in designing an incentive compatible mechanism for bipartite matching, which can be used to make the approximately incentive compatible reduction of Hartline et al. [18] exactly incentive compatible.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74170892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marthe Bonamy, Édouard Bonnet, N. Bousquet, Pierre Charbit, P. Giannopoulos, Eun Jung Kim, Paweł Rzaͅżewski, F. Sikora, Stéphan Thomassé
A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for MAXIMUM CLIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2Õ(n2/3) for MAXIMUM CLIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for MAX CLIQUE on disk and unit ball graphs. MAX CLIQUE on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, MAXIMUM CLIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2n1−ɛ, unless the Exponential Time Hypothesis fails.
{"title":"EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs","authors":"Marthe Bonamy, Édouard Bonnet, N. Bousquet, Pierre Charbit, P. Giannopoulos, Eun Jung Kim, Paweł Rzaͅżewski, F. Sikora, Stéphan Thomassé","doi":"10.1145/3433160","DOIUrl":"https://doi.org/10.1145/3433160","url":null,"abstract":"A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for MAXIMUM CLIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2Õ(n2/3) for MAXIMUM CLIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for MAX CLIQUE on disk and unit ball graphs. MAX CLIQUE on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, MAXIMUM CLIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2n1−ɛ, unless the Exponential Time Hypothesis fails.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80199353","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article shows how to solve linear programs of the form minAx=b,x≥ 0 c⊤ x with n variables in time O*((nω+n2.5−α/2+n2+1/6) log (n/δ)), where ω is the exponent of matrix multiplication, α is the dual exponent of matrix multiplication, and δ is the relative accuracy. For the current value of ω δ 2.37 and α δ 0.31, our algorithm takes O*(nω log (n/δ)) time. When ω = 2, our algorithm takes O*(n2+1/6 log (n/δ)) time. Our algorithm utilizes several new concepts that we believe may be of independent interest: • We define a stochastic central path method. • We show how to maintain a projection matrix √ WA⊤ (AWA⊤)−1A√ W in sub-quadratic time under ell2 multiplicative changes in the diagonal matrix W.
{"title":"Solving Linear Programs in the Current Matrix Multiplication Time","authors":"Michael B. Cohen, Y. Lee, Zhao Song","doi":"10.1145/3424305","DOIUrl":"https://doi.org/10.1145/3424305","url":null,"abstract":"This article shows how to solve linear programs of the form minAx=b,x≥ 0 c⊤ x with n variables in time O*((nω+n2.5−α/2+n2+1/6) log (n/δ)), where ω is the exponent of matrix multiplication, α is the dual exponent of matrix multiplication, and δ is the relative accuracy. For the current value of ω δ 2.37 and α δ 0.31, our algorithm takes O*(nω log (n/δ)) time. When ω = 2, our algorithm takes O*(n2+1/6 log (n/δ)) time. Our algorithm utilizes several new concepts that we believe may be of independent interest: • We define a stochastic central path method. • We show how to maintain a projection matrix √ WA⊤ (AWA⊤)−1A√ W in sub-quadratic time under ell2 multiplicative changes in the diagonal matrix W.","PeriodicalId":17199,"journal":{"name":"Journal of the ACM (JACM)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82147837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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