14,199 research outputs found

    Faster Detours in Undirected Graphs

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    The kk-Detour problem is a basic path-finding problem: given a graph GG on nn vertices, with specified nodes ss and tt, and a positive integer kk, the goal is to determine if GG has an stst-path of length exactly dist(s,t)+k\text{dist}(s, t) + k, where dist(s,t)\text{dist}(s, t) is the length of a shortest path from ss to tt. The kk-Detour problem is NP-hard when kk is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in f(k)poly(n)f(k)\text{poly}(n) time, for ff as slow-growing as possible. We present faster algorithms for kk-Detour in undirected graphs, running in 1.853kpoly(n)1.853^k \text{poly}(n) randomized and 4.082kpoly(n)4.082^k \text{poly}(n) deterministic time. The previous fastest algorithms for this problem took 2.746kpoly(n)2.746^k \text{poly}(n) randomized and 6.523kpoly(n)6.523^k \text{poly}(n) deterministic time [Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a "bipartitioned" subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length kk in undirected graphs [Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017]. Our work has direct implications for the kk-Longest Detour problem: in this problem, we are given the same input as in kk-Detour, but are now tasked with determining if GG has an stst-path of length at least dist(s,t)+k.\text{dist}(s, t) + k. Our results for k-Detour imply that we can solve kk-Longest Detour in 3.432kpoly(n)3.432^k \text{poly}(n) randomized and 16.661kpoly(n)16.661^k \text{poly}(n) deterministic time. The previous fastest algorithms for this problem took 7.539kpoly(n)7.539^k \text{poly}(n) randomized and 42.549kpoly(n)42.549^k \text{poly}(n) deterministic time [Fomin et al., STACS 2022]

    Patching Colors with Tensors

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    Parameterized Distributed Algorithms

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    In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds. Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2

    On the Equivalence among Problems of Bounded Width

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    In this paper, we introduce a methodology, called decomposition-based reductions, for showing the equivalence among various problems of bounded-width. First, we show that the following are equivalent for any α>0\alpha > 0: * SAT can be solved in O(2αtw)O^*(2^{\alpha \mathrm{tw}}) time, * 3-SAT can be solved in O(2αtw)O^*(2^{\alpha \mathrm{tw}}) time, * Max 2-SAT can be solved in O(2αtw)O^*(2^{\alpha \mathrm{tw}}) time, * Independent Set can be solved in O(2αtw)O^*(2^{\alpha \mathrm{tw}}) time, and * Independent Set can be solved in O(2αcw)O^*(2^{\alpha \mathrm{cw}}) time, where tw and cw are the tree-width and clique-width of the instance, respectively. Then, we introduce a new parameterized complexity class EPNL, which includes Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and Independent Set parameterized by path-width are EPNL-complete. This implies that if one of these EPNL-complete problems can be solved in O(ck)O^*(c^k) time, then any problem in EPNL can be solved in O(ck)O^*(c^k) time.Comment: accepted to ESA 201

    Faster Deterministic Algorithms for Packing, Matching and tt-Dominating Set Problems

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    In this paper, we devise three deterministic algorithms for solving the mm-set kk-packing, mm-dimensional kk-matching, and tt-dominating set problems in time O(5.44mk)O^*(5.44^{mk}), O(5.44(m1)k)O^*(5.44^{(m-1)k}) and O(5.44t)O^*(5.44^{t}), respectively. Although recently there has been remarkable progress on randomized solutions to those problems, our bounds make good improvements on the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with arXiv:1303.047

    Fast Witness Extraction Using a Decision Oracle

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    The gist of many (NP-)hard combinatorial problems is to decide whether a universe of nn elements contains a witness consisting of kk elements that match some prescribed pattern. For some of these problems there are known advanced algebra-based FPT algorithms which solve the decision problem but do not return the witness. We investigate techniques for turning such a YES/NO-decision oracle into an algorithm for extracting a single witness, with an objective to obtain practical scalability for large values of nn. By relying on techniques from combinatorial group testing, we demonstrate that a witness may be extracted with O(klogn)O(k\log n) queries to either a deterministic or a randomized set inclusion oracle with one-sided probability of error. Furthermore, we demonstrate through implementation and experiments that the algebra-based FPT algorithms are practical, in particular in the setting of the kk-path problem. Also discussed are engineering issues such as optimizing finite field arithmetic.Comment: Journal version, 16 pages. Extended abstract presented at ESA'1
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