Given a (multi) set S of n positive integers and a target integer u, the subset sum problem is to decide if there is a subset of S that sums up to u. We present a series of new algorithms that compute and return all the realizable subset sums up to the integer u in Õ(min { √nu,u5/4,σ }), where σ is the sum of all elements of S and Õ hides polylogarithmic factors. We also present a modified algorithm for integers modulo m, which computes all the realizable subset sums modulo m in Õ(min { √nm,m5/4}) time. Our contributions improve upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithms are the fastest deterministic algorithms for this problem. The new results can be employed in various algorithmic problems, from graph bipartition to computational social choice. Finally, we also improve a result on covering Zm, which might be of independent interest.
{"title":"Faster Pseudopolynomial Time Algorithms for Subset Sum","authors":"Konstantinos Koiliaris, Chao Xu","doi":"10.1145/3329863","DOIUrl":"https://doi.org/10.1145/3329863","url":null,"abstract":"Given a (multi) set S of n positive integers and a target integer u, the subset sum problem is to decide if there is a subset of S that sums up to u. We present a series of new algorithms that compute and return all the realizable subset sums up to the integer u in Õ(min { √nu,u5/4,σ }), where σ is the sum of all elements of S and Õ hides polylogarithmic factors. We also present a modified algorithm for integers modulo m, which computes all the realizable subset sums modulo m in Õ(min { √nm,m5/4}) time. Our contributions improve upon the standard dynamic programming algorithm that runs in O(nu) time. To the best of our knowledge, the new algorithms are the fastest deterministic algorithms for this problem. The new results can be employed in various algorithmic problems, from graph bipartition to computational social choice. Finally, we also improve a result on covering Zm, which might be of independent interest.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127771041","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 design a simple ascending-price algorithm to compute a (1 + ε)-approximate equilibrium in Arrow-Debreu markets with weak gross substitute property. It applies to an unknown market setting without exact knowledge about the number of agents, their individual utilities, and endowments. Instead, our algorithm only uses price queries to a global demand oracle. This is the first polynomial-time algorithm for most of the known tractable classes of Arrow-Debreu markets, which computes such an equilibrium with a number of calls to the demand oracle that is polynomial in log 1/ε and avoids heavy machinery such as the ellipsoid method. Demands can be real-valued functions of prices, but the oracles only return demand values of bounded precision. Due to this more realistic assumption, precision and representation of prices and demands become a major technical challenge, and we develop new tools and insights that may be of independent interest. Furthermore, we give the first polynomial-time algorithm to compute an exact equilibrium for markets with spending constraint utilities. This resolves an open problem posed by Duan and Mehlhorn.
{"title":"Ascending-Price Algorithms for Unknown Markets","authors":"Xiaohui Bei, J. Garg, M. Hoefer","doi":"10.1145/3319394","DOIUrl":"https://doi.org/10.1145/3319394","url":null,"abstract":"We design a simple ascending-price algorithm to compute a (1 + ε)-approximate equilibrium in Arrow-Debreu markets with weak gross substitute property. It applies to an unknown market setting without exact knowledge about the number of agents, their individual utilities, and endowments. Instead, our algorithm only uses price queries to a global demand oracle. This is the first polynomial-time algorithm for most of the known tractable classes of Arrow-Debreu markets, which computes such an equilibrium with a number of calls to the demand oracle that is polynomial in log 1/ε and avoids heavy machinery such as the ellipsoid method. Demands can be real-valued functions of prices, but the oracles only return demand values of bounded precision. Due to this more realistic assumption, precision and representation of prices and demands become a major technical challenge, and we develop new tools and insights that may be of independent interest. Furthermore, we give the first polynomial-time algorithm to compute an exact equilibrium for markets with spending constraint utilities. This resolves an open problem posed by Duan and Mehlhorn.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129871994","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}
Given a drawing of a read-once formula (called the blueprint), and a blackbox implementation with the same topology as the blueprint that purports to compute the formula, can we tell if it does? Under a fault model, where the only faults in the implementation are gates that complement their outputs, we show that there is an efficient algorithm that makes a linear number of probes to the blackbox implementation and determines if the blueprint and implementation are identical. We also show a matching lower bound. We further ask whether we can diagnose where the faults are, using blackbox testing. We prove that if the implementation has a property called polynomial balance, then it is possible to do this efficiently. To complement this result, we show that even if the blueprint is polynomially balanced and there are only logarithmically many errors in the implementation, the implementation could be unbalanced and the diagnosis problem provably requires super-polynomially many tests. We point out that this problem is one instance of a general class of problems of learning deviations from a blueprint, which we call conformance learning. Conformance learning seems worthy of further investigation in a broader context.
{"title":"Locating Errors in Faulty Formulas","authors":"Sampath Kannan, Kevin Tian","doi":"10.1145/3313776","DOIUrl":"https://doi.org/10.1145/3313776","url":null,"abstract":"Given a drawing of a read-once formula (called the blueprint), and a blackbox implementation with the same topology as the blueprint that purports to compute the formula, can we tell if it does? Under a fault model, where the only faults in the implementation are gates that complement their outputs, we show that there is an efficient algorithm that makes a linear number of probes to the blackbox implementation and determines if the blueprint and implementation are identical. We also show a matching lower bound. We further ask whether we can diagnose where the faults are, using blackbox testing. We prove that if the implementation has a property called polynomial balance, then it is possible to do this efficiently. To complement this result, we show that even if the blueprint is polynomially balanced and there are only logarithmically many errors in the implementation, the implementation could be unbalanced and the diagnosis problem provably requires super-polynomially many tests. We point out that this problem is one instance of a general class of problems of learning deviations from a blueprint, which we call conformance learning. Conformance learning seems worthy of further investigation in a broader context.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114233797","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}
Xiaoming Sun, David P. Woodruff, Guang Yang, Jialin Zhang
We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products, namely, the algorithm chooses vectors v1, ⃛ , vq, and observes Mv1, ⃛ , Mvq. Here the vi can be randomized as well as chosen adaptively as a function of Mv1, ⃛ , Mvi-1. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.
我们考虑通过矩阵-向量积访问未知矩阵M ε F n×d的算法，即算法选择向量v1，， vq，并观察Mv1，， Mvq。在这里，vi可以随机化，也可以自适应地选择，作为Mv1, ick1, Mvi-1的函数。由于素描在分布式计算、线性代数和流模型中的应用，以及与通信复杂性和属性测试等领域的联系，我们开始研究解决各种基本问题所需的查询数量q。我们研究三大类问题，包括线性代数问题、统计问题和图问题。例如，我们考虑近似矩阵M的秩、迹、最大特征值和范数所需的查询数量;计算M的每一列或每一行的AND/OR/奇偶性，判断M中是否有相同的列或行，M是对称的、对角的还是酉的;或者计算由M定义的图是连通的还是无三角形的。我们还展示了只允许通过查询右边的向量来获得矩阵向量积的算法与可以同时查询左边和右边的向量的算法的分离。我们还显示了依赖于矩阵-向量乘积发生的底层场的分离。对于图问题，我们根据矩阵的形式(二部邻接矩阵与带符号边顶点关联矩阵)显示分离来表示图。令人惊讶的是，很少有作品讨论这个基本模型，我们相信对这个模型中的问题进行彻底的调查将有利于许多不同的应用领域。
{"title":"Querying a Matrix through Matrix-Vector Products","authors":"Xiaoming Sun, David P. Woodruff, Guang Yang, Jialin Zhang","doi":"10.1145/3470566","DOIUrl":"https://doi.org/10.1145/3470566","url":null,"abstract":"We consider algorithms with access to an unknown matrix M ε F n×d via matrix-vector products, namely, the algorithm chooses vectors v1, ⃛ , vq, and observes Mv1, ⃛ , Mvq. Here the vi can be randomized as well as chosen adaptively as a function of Mv1, ⃛ , Mvi-1. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, very few works discuss this fundamental model, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133515970","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 are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of seven papers that were presented at the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017) in Barcelona, Spain, on January 16–19, 2017. These papers, selected on the basis of their high rating by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “A (2 + ɛ)-Approximation for Maximum Weight Matching in the Semi-Streaming Model,” Ami Paz and Gregory Schwartzman study the maximum weight matching problem in graphs where the edges appear one by one, and after seeing an edge, the algorithm needs to decide, using small space and time, how to update the current approximate matching. The main result is that for every constant ɛ > 0 using only O(n log n) space, one can maintain a (2 + ɛ)-approximation to the maximum weight matching in the current graph by spending only O(log n) time per edge, improving vastly over the previous known 3.5 + ɛ-approximation in this setting. In “Beating Approximation Factor Two for Weighted Tree Augmentation with Bounded Costs,” David Adjiashvili gives the first polynomial-time algorithm with approximation ratio better than two for a classic graph-augmentation problem: given a spanning tree in an edge-weighted graph, augment the tree with a minimum-weight subset of edges to achieve 2-edge connectivity. In “Firefighting on Trees Beyond Integrality Gaps,” David Adjiashvili, Andrea Baggio, and Rico Zenklusen give the first polynomial-time approximation scheme (PTAS) for the Firefighter problem (introduced in 1995 by Hartnell) on trees, and the first polynomial-time constantapproximation algorithm for the related problem of Resource Minimization for Fire Containment (RMFC) on trees, matching the known hardness results. In “Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs,” Sergio Cabello presents the first algorithm for the diameter problem in planar graphs that runs in time that is truly subquadratic in the number of vertices, resolving a longstanding open problem. The diameter problem asks for the largest shortest paths distance in a graph. A truly subquadratic time algorithm for diameter in general graphs is known to have strong implications in complexity. In “Even Delta-Matroids and the Complexity of Planar Boolean CSPs,” Alexandr Kazda, Vladimir Kolmogorov, and Michal Rolínek study Boolean Constraint Satisfaction Problems (CSPs) under the restriction that every variable appears in at most two constraints. The main result is new tractable class of such CSPs: if all constraints are even Δ-matroids, then the problem is solvable in polynomial time. As a consequence, this result completes the dichotomy for planar Boolean CSPs. In “Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications,” Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and Ryan Williams relate the comp
我们很高兴地发表《ACM算法汇刊》特刊，其中包含2017年1月16日至19日在西班牙巴塞罗那举行的第28届ACM- siam离散算法研讨会(SODA 2017)上发表的七篇论文的完整版本。这些论文是由会议计划委员会根据其高评级选出的，并根据该杂志的最高标准进行了彻底的审查。Ami Paz和Gregory Schwartzman在“A (2 + i)- approximate for Maximum Weight Matching In half - streaming Model”中研究了一条边一条边出现的图的最大权值匹配问题，在看到一条边后，算法需要在很小的空间和时间内决定如何更新当前的近似匹配。主要结果是，对于每一个常数，只使用O(n log n)空间，就可以维持当前图中最大权值匹配的(2 + log)-近似值，每条边只花费O(log n)时间，大大改善了之前已知的3.5 + log -近似值。David Adjiashvili在“为有界成本的加权树增强击败近似因子2”中给出了第一个近似比优于2的多项式时间算法，用于经典的图增强问题:给定边加权图中的生成树，用最小权重的边子集增强树以实现2边连接。在“超越完整性差距的树木上的消防”中，David Adjiashvili, Andrea Baggio和Rico Zenklusen给出了树木上消防员问题(由Hartnell于1995年引入)的第一个多项式时间近似方案(PTAS)，以及树木上防火资源最小化(RMFC)相关问题的第一个多项式时间常数近似算法，与已知的硬度结果相匹配。在“平面图中直径和两两距离和的次二次算法”中，Sergio Cabello提出了平面图中直径问题的第一个算法，该算法在时间上运行，在顶点数量上是真正的次二次，解决了一个长期存在的开放问题。直径问题要求图中最大的最短路径距离。已知一般图中直径的真正次二次时间算法在复杂性方面具有很强的含义。Alexandr Kazda, Vladimir Kolmogorov和Michal Rolínek在“偶三角矩阵和平面布尔csp的复杂性”一文中，研究了每个变量最多出现在两个约束条件下的布尔约束满足问题(csp)。主要结果是一类新的可处理的csp:如果所有约束都是偶数Δ-matroids，那么问题在多项式时间内可解。因此，该结果完成了平面布尔csp的二分法。在“稀疏结构一阶性质的完备性及其算法应用”一文中，Gao Jiawei, Russell Impagliazzo, Antonina Kolokolova和Ryan Williams将一阶模型检验的计算复杂性与正交向量问题的各种变体的复杂性联系起来。他们表明，一个多项式速度更快的正交向量算法不仅会打破强指数时间假设，而且还会产生更快的算法来解决大量的图问题和更多的问题。
{"title":"Introduction to the Special Issue on SODA 2017","authors":"Dániel Marx, V. V. Williams, N. Young","doi":"10.1145/3319426","DOIUrl":"https://doi.org/10.1145/3319426","url":null,"abstract":"We are delighted to present a Special Issue of ACM Transactions on Algorithms, containing full versions of seven papers that were presented at the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017) in Barcelona, Spain, on January 16–19, 2017. These papers, selected on the basis of their high rating by the conference program committee, have been thoroughly reviewed according to the journal’s highest standards. In “A (2 + ɛ)-Approximation for Maximum Weight Matching in the Semi-Streaming Model,” Ami Paz and Gregory Schwartzman study the maximum weight matching problem in graphs where the edges appear one by one, and after seeing an edge, the algorithm needs to decide, using small space and time, how to update the current approximate matching. The main result is that for every constant ɛ > 0 using only O(n log n) space, one can maintain a (2 + ɛ)-approximation to the maximum weight matching in the current graph by spending only O(log n) time per edge, improving vastly over the previous known 3.5 + ɛ-approximation in this setting. In “Beating Approximation Factor Two for Weighted Tree Augmentation with Bounded Costs,” David Adjiashvili gives the first polynomial-time algorithm with approximation ratio better than two for a classic graph-augmentation problem: given a spanning tree in an edge-weighted graph, augment the tree with a minimum-weight subset of edges to achieve 2-edge connectivity. In “Firefighting on Trees Beyond Integrality Gaps,” David Adjiashvili, Andrea Baggio, and Rico Zenklusen give the first polynomial-time approximation scheme (PTAS) for the Firefighter problem (introduced in 1995 by Hartnell) on trees, and the first polynomial-time constantapproximation algorithm for the related problem of Resource Minimization for Fire Containment (RMFC) on trees, matching the known hardness results. In “Subquadratic Algorithms for the Diameter and the Sum of Pairwise Distances in Planar Graphs,” Sergio Cabello presents the first algorithm for the diameter problem in planar graphs that runs in time that is truly subquadratic in the number of vertices, resolving a longstanding open problem. The diameter problem asks for the largest shortest paths distance in a graph. A truly subquadratic time algorithm for diameter in general graphs is known to have strong implications in complexity. In “Even Delta-Matroids and the Complexity of Planar Boolean CSPs,” Alexandr Kazda, Vladimir Kolmogorov, and Michal Rolínek study Boolean Constraint Satisfaction Problems (CSPs) under the restriction that every variable appears in at most two constraints. The main result is new tractable class of such CSPs: if all constraints are even Δ-matroids, then the problem is solvable in polynomial time. As a consequence, this result completes the dichotomy for planar Boolean CSPs. In “Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications,” Jiawei Gao, Russell Impagliazzo, Antonina Kolokolova, and Ryan Williams relate the comp","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116955440","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}
P. Chan, L. Lau, Aaron Schild, Sam Chiu-wai Wong, Hong Zhou
We consider a new problem of designing a network with small s-t effective resistance. In this problem, we are given an undirected graph G = (V,E), two designated vertices s,t ∈ V, and a budget k. The goal is to choose a subgraph of G with at most k edges to minimize the s-t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.
{"title":"Network Design for s-t Effective Resistance","authors":"P. Chan, L. Lau, Aaron Schild, Sam Chiu-wai Wong, Hong Zhou","doi":"10.1145/3522588","DOIUrl":"https://doi.org/10.1145/3522588","url":null,"abstract":"We consider a new problem of designing a network with small s-t effective resistance. In this problem, we are given an undirected graph G = (V,E), two designated vertices s,t ∈ V, and a budget k. The goal is to choose a subgraph of G with at most k edges to minimize the s-t effective resistance. This problem is an interpolation between the shortest path problem and the minimum cost flow problem and has applications in electrical network design. We present several algorithmic and hardness results for this problem and its variants. On the hardness side, we show that the problem is NP-hard, and the weighted version is hard to approximate within a factor smaller than two assuming the small-set expansion conjecture. On the algorithmic side, we analyze a convex programming relaxation of the problem and design a constant factor approximation algorithm. The key of the rounding algorithm is a randomized path-rounding procedure based on the optimality conditions and a flow decomposition of the fractional solution. We also use dynamic programming to obtain a fully polynomial time approximation scheme when the input graph is a series-parallel graph, with better approximation ratio than the integrality gap of the convex program for these graphs.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"43 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124985538","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}
P. Agarwal, Ravid Cohen, D. Halperin, Wolfgang Mulzer
We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O((k+1)log2 n) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x-monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O(log2n) time; for a query point x0∈ ℝ, it can report, in O(log n) time, the point on the lower envelope with x-coordinate x0; and for a query point q∈ ℝ2, it can return all k pseudo-lines lying below q in time O(log n+klog2 n). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D, all input arcs intersecting D can be reported in O(n1/2+ɛ + k) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O(log2 n) time.
{"title":"Maintaining the Union of Unit Discs under Insertions with Near-Optimal Overhead","authors":"P. Agarwal, Ravid Cohen, D. Halperin, Wolfgang Mulzer","doi":"10.1145/3527614","DOIUrl":"https://doi.org/10.1145/3527614","url":null,"abstract":"We present efficient dynamic data structures for maintaining the union of unit discs and the lower envelope of pseudo-lines in the plane. More precisely, we present three main results in this paper: (i) We present a linear-size data structure to maintain the union of a set of unit discs under insertions. It can insert a disc and update the union in O((k+1)log2 n) time, where n is the current number of unit discs and k is the combinatorial complexity of the structural change in the union due to the insertion of the new disc. It can also compute, within the same time bound, the area of the union after the insertion of each disc. (ii) We propose a linear-size data structure for maintaining the lower envelope of a set of x-monotone pseudo-lines. It can handle insertion/deletion of a pseudo-line in O(log2n) time; for a query point x0∈ ℝ, it can report, in O(log n) time, the point on the lower envelope with x-coordinate x0; and for a query point q∈ ℝ2, it can return all k pseudo-lines lying below q in time O(log n+klog2 n). (iii) We present a linear-size data structure for storing a set of circular arcs of unit radius (not necessarily on the boundary of the union of the corresponding discs), so that for a query unit disc D, all input arcs intersecting D can be reported in O(n1/2+ɛ + k) time, where k is the output size and ɛ > 0 is an arbitrarily small constant. A unit-circle arc can be inserted or deleted in O(log2 n) time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"735 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133256655","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}
Convergence rate and stability of a solution concept are classically measured in terms of “eventually” and “forever,” respectively. In the wake of recent computational criticisms to this approach, we study whether these timeframes can be updated to have states computed “quickly” and stable for “long enough”. Logit dynamics allows irrationality in players’ behavior and may take time exponential in the number of players n to converge to a stable state (i.e., a certain distribution over pure strategy profiles). We prove that every potential game, for which the behavior of the logit dynamics is not chaotic as n increases, admits distributions stable for a super-polynomial number of steps in n no matter the players’ irrationality and the starting profile of the dynamics. The convergence rate to these metastable distributions is polynomial in n when the players are not too rational. Our proofs build upon the new concept of partitioned Markov chains, which might be of independent interest, and a number of involved technical contributions.
{"title":"Metastability of the Logit Dynamics for Asymptotically Well-Behaved Potential Games","authors":"Diodato Ferraioli, Carmine Ventre","doi":"10.1145/3301315","DOIUrl":"https://doi.org/10.1145/3301315","url":null,"abstract":"Convergence rate and stability of a solution concept are classically measured in terms of “eventually” and “forever,” respectively. In the wake of recent computational criticisms to this approach, we study whether these timeframes can be updated to have states computed “quickly” and stable for “long enough”. Logit dynamics allows irrationality in players’ behavior and may take time exponential in the number of players n to converge to a stable state (i.e., a certain distribution over pure strategy profiles). We prove that every potential game, for which the behavior of the logit dynamics is not chaotic as n increases, admits distributions stable for a super-polynomial number of steps in n no matter the players’ irrationality and the starting profile of the dynamics. The convergence rate to these metastable distributions is polynomial in n when the players are not too rational. Our proofs build upon the new concept of partitioned Markov chains, which might be of independent interest, and a number of involved technical contributions.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116809223","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}
V. Mäkinen, Alexandru I. Tomescu, A. Kuosmanen, Topi Paavilainen, T. Gagie, R. Chikhi
The minimum path cover problem asks us to find a minimum-cardinality set of paths that cover all the nodes of a directed acyclic graph (DAG). We study the case when the size k of a minimum path cover is small, that is, when the DAG has a small width. This case is motivated by applications in pan-genomics, where the genomic variation of a population is expressed as a DAG. We observe that classical alignment algorithms exploiting sparse dynamic programming can be extended to the sequence-against-DAG case by mimicking the algorithm for sequences on each path of a minimum path cover and handling an evaluation order anomaly with reachability queries. Namely, we introduce a general framework for DAG-extensions of sparse dynamic programming. This framework produces algorithms that are slower than their counterparts on sequences only by a factor k. We illustrate this on two classical problems extended to DAGs: longest increasing subsequence and longest common subsequence. For the former, we obtain an algorithm with running time O(k|E|log |V|). This matches the optimal solution to the classical problem variant when the input sequence is modeled as a path. We obtain an analogous result for the longest common subsequence problem. We then apply this technique to the co-linear chaining problem, which is a generalization of the above two problems. The algorithm for this problem turns out to be more involved, needing further ingredients, such as an FM-index tailored for large alphabets and a two-dimensional range search tree modified to support range maximum queries. We also study a general sequence-to-DAG alignment formulation that allows affine gap costs in the sequence. The main ingredient of the proposed framework is a new algorithm for finding a minimum path cover of a DAG (V,E) in O(k|E|log |V|) time, improving all known time-bounds when k is small and the DAG is not too dense. In addition to boosting the sparse dynamic programming framework, an immediate consequence of this new minimum path cover algorithm is an improved space/time tradeoff for reachability queries in arbitrary directed graphs.
{"title":"Sparse Dynamic Programming on DAGs with Small Width","authors":"V. Mäkinen, Alexandru I. Tomescu, A. Kuosmanen, Topi Paavilainen, T. Gagie, R. Chikhi","doi":"10.1145/3301312","DOIUrl":"https://doi.org/10.1145/3301312","url":null,"abstract":"The minimum path cover problem asks us to find a minimum-cardinality set of paths that cover all the nodes of a directed acyclic graph (DAG). We study the case when the size k of a minimum path cover is small, that is, when the DAG has a small width. This case is motivated by applications in pan-genomics, where the genomic variation of a population is expressed as a DAG. We observe that classical alignment algorithms exploiting sparse dynamic programming can be extended to the sequence-against-DAG case by mimicking the algorithm for sequences on each path of a minimum path cover and handling an evaluation order anomaly with reachability queries. Namely, we introduce a general framework for DAG-extensions of sparse dynamic programming. This framework produces algorithms that are slower than their counterparts on sequences only by a factor k. We illustrate this on two classical problems extended to DAGs: longest increasing subsequence and longest common subsequence. For the former, we obtain an algorithm with running time O(k|E|log |V|). This matches the optimal solution to the classical problem variant when the input sequence is modeled as a path. We obtain an analogous result for the longest common subsequence problem. We then apply this technique to the co-linear chaining problem, which is a generalization of the above two problems. The algorithm for this problem turns out to be more involved, needing further ingredients, such as an FM-index tailored for large alphabets and a two-dimensional range search tree modified to support range maximum queries. We also study a general sequence-to-DAG alignment formulation that allows affine gap costs in the sequence. The main ingredient of the proposed framework is a new algorithm for finding a minimum path cover of a DAG (V,E) in O(k|E|log |V|) time, improving all known time-bounds when k is small and the DAG is not too dense. In addition to boosting the sparse dynamic programming framework, an immediate consequence of this new minimum path cover algorithm is an improved space/time tradeoff for reachability queries in arbitrary directed graphs.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129525345","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}
Tien-Nam Le, D. Lokshtanov, Saket Saurabh, Stéphan Thomassé, M. Zehavi
We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3-Packing. For these four problems, kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this article, we give the first kernels for FVST, CVD, TPT, and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with O(k3/2) vertices. • CVD admits a kernel with O(k5/3) vertices. • TPT admits a kernel with O(k3/2) vertices. • Induced P3-Packing admits a kernel with O(k5/3) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k2−ε) vertices for FVST and CVD. All of our results are based on novel uses of old and new “expansion lemmas” and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).
我们考虑了图上四个研究得很好的np完全填充/覆盖问题:反馈顶点集竞赛(FVST)、聚类顶点删除(CVD)、三角形填充竞赛(TPT)和诱导p3填充。对于这四个问题，有O(k2)个顶点的核已经知道很长时间了。事实上，这样的核可以通过将这些问题解释为寻找k个大小为3的成对不相交集合的包装(3- set packing)或对于大小不超过3的集合族(3- hit set)寻找大小不超过k的命中集来获得。在本文中，我们给出了具有次二次顶点数的FVST、CVD、TPT和诱导P3-Packing的第一核。具体而言，我们得到以下结果:•FVST允许有O(k3/2)个顶点的核。•CVD允许有O(k5/3)个顶点的核。•TPT允许有O(k3/2)个顶点的核。•诱导P3-Packing允许有O(k5/3)个顶点的核。我们的结果解决了WorKer 2010中关于FVST和CVD的O(k2−ε)顶点核的存在性的开放问题。我们所有的结果都是基于新旧“展开引理”的新使用和弱形式的冠分解，其中(i)几乎所有的冠都被溶液使用(而不是全部)，(ii)几乎没有冠被溶液使用(而不是没有)，以及(iii)如果H从G中移除，那么头部和其余部分之间几乎没有相互作用(而不是根本没有相互作用)。
{"title":"Subquadratic Kernels for Implicit 3-Hitting Set and 3-Set Packing Problems","authors":"Tien-Nam Le, D. Lokshtanov, Saket Saurabh, Stéphan Thomassé, M. Zehavi","doi":"10.1145/3293466","DOIUrl":"https://doi.org/10.1145/3293466","url":null,"abstract":"We consider four well-studied NP-complete packing/covering problems on graphs: Feedback Vertex Set in Tournaments (FVST), Cluster Vertex Deletion (CVD), Triangle Packing in Tournaments (TPT) and Induced P3-Packing. For these four problems, kernels with O(k2) vertices have been known for a long time. In fact, such kernels can be obtained by interpreting these problems as finding either a packing of k pairwise disjoint sets of size 3 (3-Set Packing) or a hitting set of size at most k for a family of sets of size at most 3 (3-Hitting Set). In this article, we give the first kernels for FVST, CVD, TPT, and Induced P3-Packing with a subquadratic number of vertices. Specifically, we obtain the following results. • FVST admits a kernel with O(k3/2) vertices. • CVD admits a kernel with O(k5/3) vertices. • TPT admits a kernel with O(k3/2) vertices. • Induced P3-Packing admits a kernel with O(k5/3) vertices. Our results resolve an open problem from WorKer 2010 on the existence of kernels with O(k2−ε) vertices for FVST and CVD. All of our results are based on novel uses of old and new “expansion lemmas” and a weak form of crown decomposition where (i) almost all of the head is used by the solution (as opposed to all), (ii) almost none of the crown is used by the solution (as opposed to none), and (iii) if H is removed from G, then there is almost no interaction between the head and the rest (as opposed to no interaction at all).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130486838","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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