A survey of parameterized algorithms and the complexity of edge modification

Crespelle, Christophe; Drange, Pål Grønås; Fomin, Fedor V.; Golovach, Petr A.

doi:10.48550/arxiv.2001.06867

Cited by 7 publications

(8 citation statements)

References 206 publications

(392 reference statements)

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“…For a fixed graph class F and a fixed set M of allowed graph modification operations, such as vertex deletion, edge deletion, edge addition, edge editing or edge contraction, the F-M-Modification problem takes as input a graph G and a positive integer k, and the goal is to decide whether at most k operations from M can be applied to G so that the resulting graph belongs to the class F. For most natural graph classes F and modification operations M, the F-M-Modification problem is NP-hard [33,42,43]. To cope up with this hardness, these problems have been examined via the lens of parameterized complexity [8,13]. With an appropriate choice of F and the allowed modification operations M, F-M-Modification can encapsulate well-studied problems like Vertex Cover, Feedback Vertex Set (FVS), Odd Cycle Transversal (OCT), Chordal Completion, or Cluster Editing, to name just a few.…”

Section: Introductionmentioning

confidence: 99%

Reducing the Vertex Cover Number via Edge Contractions

Lima¹,

dos²,

Sau³

et al. 2022

Preprint

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The Contraction(vc) problem takes as input a graph G on n vertices and two integers k and d, and asks whether one can contract at most k edges to reduce the size of a minimum vertex cover of G by at least d. Recently, Lima et al. [JCSS 2021] proved, among other results, that unlike most of the so-called blocker problems, Contraction(vc) admits an XP algorithm running in time f (d) • n O(d) . They left open the question of whether this problem is FPT under this parameterization. In this article, we continue this line of research and prove the following results:Moreover, unless the ETH fails, the problem does not admit an algorithm running in time f (k + d) • n o(k+d) for any function f . In particular, this answers the open question stated in Lima et al. [JCSS 2021] in the negative. It is NP-hard to decide whether an instance (G, k, d) of Contraction(vc) is a Yes-instance even when k = d, hence enhancing our understanding of the classical complexity of the problem.Contraction(vc) can be solved in time 2 O(d) • n k−d+O(1) . This XP algorithm improves the one of Lima et al. [JCSS 2021], which uses Courcelle's theorem as a subroutine and hence, the f (d)-factor in the running time is non-explicit and probably very large. On the other hard, it shows that when k = d, the problem is FPT parameterized by d (or by k).

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Section: Introductionmentioning

confidence: 99%

Reducing the Vertex Cover Number via Edge Contractions

Lima¹,

dos²,

Sau³

et al. 2022

Preprint

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“…This makes a sharp contrast with the vertex deletion problems (deleting vertices instead of edges), for which a polynomial kernel is guaranteed when the number of forbidden induced subgraphs is finite [7]. We refer the reader to the recent survey of Crespelle et al [4], particularly its Section 2.1 and Table 1, for the most relevant results.…”

Section: Introductionmentioning

confidence: 99%

Improved Kernels for Edge Modification Problems

Cao,

2021

Preprint

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In an edge modification problem, we are asked to modify at most k edges to a given graph to make the graph satisfy a certain property. Depending on the operations allowed, we have the completion problems and the edge deletion problems. A great amount of efforts have been devoted to understanding the kernelization complexity of these problems. We revisit several well-studied edge modification problems, and develop improved kernels for them:• a 2k-vertex kernel for the cluster edge deletion problem,• a 3k 2 -vertex kernel for the trivially perfect completion problem,• a 5k 1.5 -vertex kernel for the split completion problem and the split edge deletion problem, and• a 5k 1.5 -vertex kernel for the pseudo-split completion problem and the pseudo-split edge deletion problem.Moreover, our kernels for split completion and pseudo-split completion have only O(k 2.5 ) edges. Our results also include a 2k-vertex kernel for the strong triadic closure problem, which is related to cluster edge deletion.

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“…In the last decades, edge modification problems received considerable attention from the point of view of parameterized complexity [3,4,6,12,15,19,27] (see [10] for a recent survey). All along the paper, we will consider problems parameterized by the size k of the solution.…”

Section: Introductionmentioning

confidence: 99%

“…While Clique + IS Addition is trivial, both Clique + IS Deletion and Clique + IS Edition are NP-complete (reduction from the Clique problem), and both can be solved in subexponential time (O * (1.64 [12]. Both problems also admit a simple 2k-kernel, based on twin reduction rules [10].…”

Section: Introductionmentioning

confidence: 99%

(Sub)linear kernels for edge modification problems towards structured graph classes

Bathie,

Bousquet,

Pierron

2021

Preprint

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In a (parameterized) graph edge modification problem, we are given a graph G, an integer k and a (usually well-structured) class of graphs G, and ask whether it is possible to transform G into a graph G ′ ∈ G by adding and/or removing at most k edges. Parameterized graph edge modification problems received considerable attention in the last decades.In this paper we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if a 2k kernel exists for Cluster Editing, this kernel does not reduce the size of the instance in most cases. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size O(k/ log k).We also obtain small kernels for several other edge modification problems. We prove that Split Addition (and the equivalent Split Deletion) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. [19]. We also prove that Trivially Perfect Addition admits a quadratic kernel (improving the cubic kernel of Guo [21]), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under ETH.

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A survey of parameterized algorithms and the complexity of edge modification

Cited by 7 publications

References 206 publications

Reducing the Vertex Cover Number via Edge Contractions

Reducing the Vertex Cover Number via Edge Contractions

Improved Kernels for Edge Modification Problems

(Sub)linear kernels for edge modification problems towards structured graph classes

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