A Deterministic Algorithm for Finding All Minimum <i>k</i>‐Way Cuts

Kamidoi, Yoko; Yoshida, Noriyoshi; Nagamochi, Hiroshi

doi:10.1137/050631616

Cited by 53 publications

(36 citation statements)

References 24 publications

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“…Karger and Stein improved this to an O(n (2−o(1))k )-time randomized MonteCarlo algorithm using the idea of random edgecontractions [13]. After Kamidoi et al [10] gave an O(n 4k+o(1) )-time deterministic algorithm based on divide-and-conquer, Thorup gave anÕ(n 2k )-time deterministic algorithm based on tree packings [25]. Small values of k ∈ [2, 6] also have been separately studied [19,9,2,12,20,21,16].…”

Section: Other Related Workmentioning

confidence: 99%

An FPT Algorithm Beating 2-Approximation for k-Cut

Gupta¹,

Lee²,

Li³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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In the k-Cut problem, we are given an edge-weighted graph G and an integer k, and have to remove a set of edges with minimum total weight so that G has at least k connected components. Prior work on this problem gives, for all h ∈ [2, k], a (2 − h/k)-approximation algorithm for k-cut that runs in time n O(h) . Hence to get a (2−ε)-approximation algorithm for some absolute constant ε, the best runtime using prior techniques is n O(kε) . Moreover, it was recently shown that getting a (2−ε)-approximation for general k is NP-hard, assuming the Small Set Expansion Hypothesis.If we use the size of the cut as the parameter, an FPT algorithm to find the exact k-Cut is known, but solving the k-Cut problem exactly is W [1]-hard if we parameterize only by the natural parameter of k. An immediate question is: can we approximate k-Cut better in FPT-time, using k as the parameter?We answer this question positively. We show that for some absolute constant ε > 0, there exists a (2 − ε)-approximation algorithm that runs in time 2. This is the first FPT algorithm that is parameterized only by k and strictly improves the 2-approximation.

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Section: Other Related Workmentioning

confidence: 99%

An FPT Algorithm Beating 2-Approximation for k-Cut

Gupta¹,

Lee²,

Li³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…This problem has applications in numerous areas of computer science, such as finding cutting planes for the traveling salesman problem, clusteringrelated settings (e.g., VLSI design), or network reliability [9]. In general, k-Way Cut is NPcomplete [31] but solvable in polynomial time for fixed k: a long line of research [31,42,44,65] led to a deterministic algorithm running in time O(mn 2k−2 ). The dependency on k in the exponent is probably unavoidable: from the parameterized perspective, the k-Way Cut problem parameterized by k is W [1]-hard [24].…”

Section: Introductionmentioning

confidence: 99%

Clique Cover and Graph Separation

Cygan

Kratsch

Pilipczuk

et al. 2014

ACM Trans. Comput. Theory

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The field of kernelization studies polynomial-time preprocessing routines for hard problems in the framework of parameterized complexity. In this paper we show that, unless the polynomial hierarchy collapses to its third level, the following parameterized problems do not admit a polynomial-time preprocessing algorithm that reduces the size of an instance to polynomial in the parameter:• Edge Clique Cover, parameterized by the number of cliques,• Directed Edge/Vertex Multiway Cut, parameterized by the size of the cutset, even in the case of two terminals,• Edge/Vertex Multicut, parameterized by the size of the cutset, and• k-Way Cut, parameterized by the size of the cutset.

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“…For general graphs, there is a simple 2-approximation algorithm disconnecting each terminal from the others by a min-cut, but for any fixed k ≥ 3, the problem is APX-hard. Result 1 spawned many papers giving improved running times for the case of planar graphs and fixed k; see, e.g., [22,10,24,4]. Polyhedral work, e.g.…”

Section: Introductionmentioning

confidence: 99%

A Polynomial-time Approximation Scheme for Planar Multiway Cut

Bateni¹,

Hajiaghayi²,

Klein³

et al. 2012

Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms

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Given an undirected graph with edge lengths and a subset of nodes (called the terminals), a multiway cut (also called a multi-terminal cut) problem asks for a subset of edges with minimum total length and whose removal disconnects each terminal from the others. It generalizes the min st-cut problem but is NP-hard for planar graphs and APX-hard for general graphs. We give a polynomial-time approximation scheme for this problem on planar graphs. We prove the result by building a novel "spanner" for multiway cut on planar graphs which is of independent interest.

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A Deterministic Algorithm for Finding All Minimum k‐Way Cuts

Cited by 53 publications

References 24 publications

An FPT Algorithm Beating 2-Approximation for k-Cut

An FPT Algorithm Beating 2-Approximation for k-Cut

Clique Cover and Graph Separation

A Polynomial-time Approximation Scheme for Planar Multiway Cut

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