ACM Transactions on Algorithms (TALG)最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}