Fixed-Parameter Algorithms for Cluster Vertex Deletion

Hüffner, Falk; Komusiewicz, Christian; Moser, Hannes; Niedermeier, Rolf

doi:10.1007/978-3-540-78773-0_61

Cited by 32 publications

(35 citation statements)

References 27 publications

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“…The first step of our algorithm is to use the technique of iterative compression introduced by Reed et al [43]. It has been used to obtain faster FPT algorithms for various problems [6,8,16,22,23,24,35,42]. We transform the Subset-DFVS problem into the following problem:…”

Section: Iterative Compressionmentioning

confidence: 99%

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Chitnis

Cygan

Hajiaghayi

et al. 2012

Automata, Languages, and Programming

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The classical FEEDBACK VERTEX SET problem asks, for a given undirected graph G and an integer k, to find a set of at most k vertices that hits all the cycles in the graph G. FEEDBACK VERTEX SET has attracted a large amount of research in the parameterized setting, and subsequent kernelization and fixed-parameter algorithms have been a rich source of ideas in the field.In this paper we consider a more general and difficult version of the problem, named SUBSET FEEDBACK VER-TEX SET (SUBSET-FVS in short) where an instance comes additionally with a set S ⊆ V of vertices, and we ask for a set of at most k vertices that hits all simple cycles passing through S. Because of its applications in circuit testing and genetic linkage analysis SUBSET-FVS was studied from the approximation algorithms perspective by Even et al. [SICOMP'00, SIDMA'00].The question whether the SUBSET-FVS problem is fixed-parameter tractable was posed independently by Kawarabayashi and Saurabh in 2009. We answer this question affirmatively. We begin by showing that this problem is fixed-parameter tractable when parametrized by |S|. Next we present an algorithm which reduces the given instance to 2 k n O(1) instances with the size of S bounded by O(k 3 ), using kernelization techniques such as the 2-Expansion Lemma, Menger's theorem and Gallai's theorem. These two facts allow us to give a 2 O(k log k) n O(1) time algorithm solving the SUBSET FEEDBACK VERTEX SET problem, proving that it is indeed fixed-parameter tractable.

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Section: Iterative Compressionmentioning

confidence: 99%

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Chitnis

Cygan

Hajiaghayi

et al. 2012

Automata, Languages, and Programming

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“…Two immediate challenges arising from our work are to determine whether there is an O(k)-vertex problem kernel for Transitivity Editing or Transitivity Deletion in the case of general digraphs (see [8,14,6] for corresponding results in the case of undirected graphs, that is, Cluster Editing) or to improve the running time of the kernelization, which so far takes cubic time in the number of vertices (for Cluster Editing, even linear-time kernelization is possible [24]). Finally, note that we focused on arc modifications to make a given digraph transitive-it might be of similar interest to start an investigation of the Transitivity Vertex Deletion problem, where the graph shall be made transitive by as few vertex deletions as possible (see [17] for corresponding results in the case of undirected graphs, that is, Cluster Vertex Deletion). Finally, from a more general point of view, there seems to be a rich field of studying further modification problems on digraphs.…”

Section: Resultsmentioning

confidence: 99%

On making directed graphs transitive

Weller

Komusiewicz

Niedermeier

et al. 2012

Journal of Computer and System Sciences

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We present the first thorough theoretical analysis of the Transitivity Editing problem on digraphs. Herein, the task is to make a given digraph transitive by a minimum number of arc insertions or deletions. Transitivity Editing has recently been identified as important for the detection of hierarchical structure in molecular characteristics of disease. We present a first proof of NP-hardness, which also extends to the restricted cases where the input digraph is acyclic or has maximum degree three. Moreover, we improve previous fixed-parameter algorithms, now achieving a running time of O(2.57 k + n 3) for an n-vertex digraph if k arc modifications are sufficient to make it transitive. By providing an O(k 2)-vertex problem kernel, we also answer an open question from the literature. In case of digraphs with maximum degree d, an O(k • d)-vertex problem kernel can be shown. We also demonstrate that if the input digraph does not contain "diamonds", then there is an optimal solution that performs only arc deletions. Our hardness as well as algorithmic results transfer to Transitivity Deletion, where only arc deletions are allowed.

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“…Several methods for vertex deletion problems have been proposed. A typical method that has been employed [13,23,24] can be described as follows [25] , where Triangle Vertex Deletion (TVD) is taken as an example.…”

Section: The Main Technical Pointsmentioning

confidence: 99%

An overview of kernelization algorithms for graph modification problems

Liu

Wang

Guo

2014

Tinshhua Sci. Technol.

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Kernelization algorithms for graph modification problems are important ingredients in parameterized computation theory. In this paper, we survey the kernelization algorithms for four types of graph modification problems, which include vertex deletion problems, edge editing problems, edge deletion problems, and edge completion problems. For each type of problem, we outline typical examples together with recent results, analyze the main techniques, and provide some suggestions for future research in this field.

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Fixed-Parameter Algorithms for Cluster Vertex Deletion

Cited by 32 publications

References 27 publications

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

Directed Subset Feedback Vertex Set Is Fixed-Parameter Tractable

On making directed graphs transitive

An overview of kernelization algorithms for graph modification problems

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