ACM Transactions on Algorithms (TALG)最新文献

{"title":"r-Simple k-Path and Related Problems Parameterized by k/r","authors":"G. Gutin, Magnus Wahlström, M. Zehavi","doi":"10.1145/3439721","DOIUrl":"https://doi.org/10.1145/3439721","url":null,"abstract":"Abasi et al. (2014) introduced the following two problems. In the r-Simple k-Path problem, given a digraph G on n vertices and positive integers r, k, decide whether G has an r-simple k-path, which is a walk where every vertex occurs at most r times and the total number of vertex occurrences is k. In the (r, k)-Monomial Detection problem, given an arithmetic circuit that succinctly encodes some polynomial P on n variables and positive integers k, r, decide whether P has a monomial of total degree k where the degree of each variable is at most r. Abasi et al. obtained randomized algorithms of running time 4(k/r)log r⋅ nO(1) for both problems. Gabizon et al. (2015) designed deterministic 2O((k/r)log r)⋅ nO(1)-time algorithms for both problems (however, for the (r, k)-Monomial Detection problem the input circuit is restricted to be non-canceling). Gabizon et al. also studied the following problem. In the P-Set (r, q)-Packing Problem, given a universe V, positive integers (p, q, r), and a collection H of sets of size P whose elements belong to V, decide whether there exists a subcollection H′ of H of size q where each element occurs in at most r sets of H′. Gabizon et al. obtained a deterministic 2O((pq/r)log r) ⋅ nO(1)-time algorithm for P-Set (r, q)-Packing. The above results prove that the three problems are single-exponentially fixed-parameter tractable (FPT) parameterized by the product of two parameters, that is, k/r and log r, where k=pq for P-Set (r, q)-Packing. Abasi et al. and Gabizon et al. asked whether the log r factor in the exponent can be avoided. Bonamy et al. (2017) answered the question for (r, k)-Monomial Detection by proving that unless the Exponential Time Hypothesis (ETH) fails there is no 2o((k/r) log r) ⋅ (n + log k)O(1)-time algorithm for (r, k)-Monomial Detection, i.e., (r, k)-Monomial Detection is unlikely to be single-exponentially FPT when parameterized by k/r alone. The question remains open for r-Simple k-Path and P-Set (r, q)-Packing. We consider the question from a wider perspective: are the above problems FPT when parameterized by k/r only, i.e., whether there exists a computable function f such that the problems admit a f(k/r)(n+log k)O(1)-time algorithm? Since r can be substantially larger than the input size, the algorithms of Abasi et al. and Gabizon et al. do not even show that any of these three problems is in XP parameterized by k/r alone. We resolve the wider question by (a) obtaining a 2O((k/r)2 log(k/r)) ⋅ (n + log k)O(1)-time algorithm for r-Simple k-Path on digraphs and a 2O(k/r) &sdot (n + log k)O(1)-time algorithm for r-Simple k-Path on undirected graphs (i.e., for undirected graphs, we answer the original question in affirmative), (b) showing that P-Set (r, q)-Packing is FPT (in contrast, we prove that P-Multiset (r, q)-Packing is W[1]-hard), and (c) proving that (r, k)-Monomial Detection is para-NP-hard even if only two distinct variables are in polynomial P and the circuit is non-canceling. For the ","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117009123","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":"4 vs 7 Sparse Undirected Unweighted Diameter Is SETH-hard at Time n4/3","authors":"Édouard Bonnet","doi":"10.1145/3494540","DOIUrl":"https://doi.org/10.1145/3494540","url":null,"abstract":"We show, assuming the Strong Exponential Time Hypothesis, that for every ε > 0, approximating undirected unweighted Diameter on n-vertex m-edge graphs within ratio 7/4 - ε requires m4/3 - o(1) time, even when m = Õ(n). This is the first result that conditionally rules out a near-linear time 5/3-approximation for undirected Diameter.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121103397","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":"Deterministic APSP, Orthogonal Vectors, and More","authors":"Timothy M. Chan, R. R. Williams","doi":"10.1145/3402926","DOIUrl":"https://doi.org/10.1145/3402926","url":null,"abstract":"We show how to solve all-pairs shortest paths on n nodes in deterministic n3>/2>Ω ( √ log n) time, and how to count the pairs of orthogonal vectors among n 0−1 vectors in d = clog n dimensions in deterministic n2−1/O(log c) time. These running times essentially match the best known randomized algorithms of Williams [46] and Abboud, Williams, and Yu [8], respectively, and the ability to count was open even for randomized algorithms. By reductions, these two results yield faster deterministic algorithms for many other problems. Our techniques can also be used to deterministically count k-satisfiability (k-SAT) assignments on n variable formulas in 2n-n/O(k) time, roughly matching the best known running times for detecting satisfiability and resolving an open problem of Santhanam [24]. A key to our constructions is an efficient way to deterministically simulate certain probabilistic polynomials critical to the algorithms of prior work, carefully applying small-biased sets and modulus-amplifying polynomials.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129483493","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":"A Faster Algorithm for Finding Tarski Fixed Points","authors":"John Fearnley, Rahul Savani","doi":"10.1145/3524044","DOIUrl":"https://doi.org/10.1145/3524044","url":null,"abstract":"Dang et al. have given an algorithm that can find a Tarski fixed point in a k-dimensional lattice of width n using O(log k n) queries [2]. Multiple authors have conjectured that this algorithm is optimal [2, 7], and indeed this has been proven for two-dimensional instances [7]. We show that these conjectures are false in dimension three or higher by giving an O(log2 n) query algorithm for the three-dimensional Tarski problem. We also give a new decomposition theorem for k-dimensional Tarski problems which, in combination with our new algorithm for three dimensions, gives an O(log2 ⌈k/3⌉ n) query algorithm for the k-dimensional problem.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121705111","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":"A Complexity Theoretical Study of Fuzzy K-Means","authors":"Johannes Blömer, Sascha Brauer, Kathrin Bujna","doi":"10.1145/3409385","DOIUrl":"https://doi.org/10.1145/3409385","url":null,"abstract":"The fuzzy K-means problem is a popular generalization of the well-known K-means problem to soft clusterings. In this article, we present the first algorithmic study of the problem going beyond heuristics. Our main result is that, assuming a constant number of clusters, there is a polynomial time approximation scheme for the fuzzy K-means problem. As a part of our analysis, we also prove the existence of small coresets for fuzzy K-means. At the heart of our proofs are two novel techniques developed to analyze the otherwise notoriously difficult fuzzy K-means objective function.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"125 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131702179","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":"Polynomial-time trace reconstruction in the smoothed complexity model","authors":"Xi Chen, Anindya De, Chin Ho Lee, R. Servedio, S. Sinha","doi":"10.1145/3560819","DOIUrl":"https://doi.org/10.1145/3560819","url":null,"abstract":"In the trace reconstruction problem, an unknown source string x ∈ {0, 1}n is sent through a probabilistic deletion channel which independently deletes each bit with probability δ and concatenates the surviving bits, yielding a trace of x. The problem is to reconstruct x given independent traces. This problem has received much attention in recent years both in the worst-case setting where x may be an arbitrary string in {0, 1}n [DOS19, NP17, HHP18, HL20, Cha21a, Cha21b] and in the average-case setting where x is drawn uniformly at random from {0, 1}n [PZ17, HPP18, HL20, Cha21a, Cha21b]. This paper studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” string xworst is chosen arbitrarily from {0, 1}n, and then a perturbed version x of xworst is formed by independently replacing each coordinate by a uniform random bit with probability σ. The problem is to reconstruct x given independent traces from it. Our main result is an algorithm which, for any constant perturbation rate 0 < σ < 1 and any constant deletion rate 0 < δ < 1, uses poly(n) running time and traces and succeeds with high probability in reconstructing the string x. This stands in contrast with the worst-case version of the problem, for which (text{exp}(tilde{O}(n^{1/5})) ) is the best known time and sample complexity [Cha21b]. Our approach is based on reconstructing x from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly(n)-time procedure for reconstructing the multiset of all O(log n)-length subwords of any source string x ∈ {0, 1}n given access to traces of x.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129955961","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":"Improving the Dilation of a Metric Graph by Adding Edges","authors":"Joachim Gudmundsson, Sampson Wong","doi":"10.1145/3517807","DOIUrl":"https://doi.org/10.1145/3517807","url":null,"abstract":"Most of the literature on spanners focuses on building the graph from scratch. This article instead focuses on adding edges to improve an existing graph. A major open problem in this field is: Given a graph embedded in a metric space, and a budget of k edges, which k edges do we add to produce a minimum-dilation graph? The special case where k=1 has been studied in the past, but no major breakthroughs have been made for k > 1. We provide the first positive result, an O(k)-approximation algorithm that runs in O(n3 log n) time.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116945454","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":"Rapid Mixing from Spectral Independence beyond the Boolean Domain","authors":"Weiming Feng, Heng Guo, Yitong Yin, Chihao Zhang","doi":"10.1145/3531008","DOIUrl":"https://doi.org/10.1145/3531008","url":null,"abstract":"We extend the notion of spectral independence (introduced by Anari, Liu, and Oveis Gharan [4]) from the Boolean domain to general discrete domains. This property characterises distributions with limited correlations and implies that the corresponding Glauber dynamics is rapidly mixing. As a concrete application, we show that Glauber dynamics for sampling proper q-colourings mixes in polynomial-time for the family of triangle-free graphs with maximum degree Δ provided q≥ (α* + δ)Δ where α*≈ 1.763 is the unique solution to α* = exp (1/α*) and δ Þ 0 is any constant. This is the first efficient algorithm for sampling proper q-colourings in this regime with possibly unbounded Δ. Our main tool of establishing spectral independence is the recursive coupling by Goldberg, Martin, and Paterson [25].","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128518224","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":"The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains","authors":"C. Viola, Stanislav Živný","doi":"10.1145/3458041","DOIUrl":"https://doi.org/10.1145/3458041","url":null,"abstract":"Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129084460","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":"A Colored Path Problem and Its Applications","authors":"E. Eiben, Iyad A. Kanj","doi":"10.1145/3396573","DOIUrl":"https://doi.org/10.1145/3396573","url":null,"abstract":"Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? Equivalently, can we remove k obstacles so that there is an obstacle-free path between the two designated points? This is a fundamental NP-hard problem that has undergone a tremendous amount of research work. The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t ∈ V(G), and k ∈ N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, then the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph. We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, including a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms. We also show that our hardness results translate to the geometric instances of the problem. We then focus on graphs satisfying the color-connectivity property. We design an FPT algorithm for this problem parameterized by both k and the treewidth of the graph and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result implies and explains previous FPT results for various obstacle shapes.","PeriodicalId":154047,"journal":{"name":"ACM Transactions on Algorithms (TALG)","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134117689","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}