Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Jansen, Bmp Bart; Bodlaender, Hans L.

doi:10.4230/lipics.stacs.2011.177

Cited by 5 publications

(6 citation statements)

References 26 publications

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“…We remark that using a more involved argument based on a decomposition theorem describing independent sets in forests by Zito [45], it is possible to show that Lemma 3 holds even if F is a forest that does not admit a perfect matching. This argument can be found in an earlier version of this work [30,Lemma 4].…”

Section: Reduction Rules For Clean Instancessupporting

confidence: 52%

“…We first show that a single application of the Nemhauser-Trotter decomposition theorem [35], used for kernelization of the vertex cover problem by Chen et al [9], allows us to restrict our attention to instances of fvs-Vertex Cover where the forest G−X has a perfect matching. This will greatly simplify the analysis of the kernel size as compared to the extended abstract of this work [30] where we worked with arbitrary forests G − X. In Section 3.1 we will then introduce a set of reduction rules and prove they are correct.…”

Section: Cubic Kernel For Fvs-vertex Covermentioning

confidence: 99%

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Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Jansen,

Bodlaender

2010

Preprint

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An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G, k) of Vertex Cover outputs an equivalent instance (G , k ) in polynomial time with the guarantee that G has at most 2k vertices (and thus O((k ) 2 ) edges) with k ≤ k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k 2 ) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number vc(G) since fvs(G) ≤ vc(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G, X, k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G , X , k ) such that |V (G )| ≤ 2k and |V (G )| ∈ O(|X | 3 ). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ⊆ coNP/poly and the polynomial hierarchy collapses to the third level.

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Section: Reduction Rules For Clean Instancessupporting

confidence: 52%

Section: Cubic Kernel For Fvs-vertex Covermentioning

confidence: 99%

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Jansen,

Bodlaender

2010

Preprint

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“…Input: A graph G, an independent set Y in G such that each component of G − Y is isomorphic to P 2 , and an integer k. Question: Does G have an independent set of size at least k? Jansen et al [35,Lemma 10] proved that Independent Set on P 2 -Split Graphs is NP-complete, and used it to prove a kernel lower bound for a weighted version of Vertex Cover. By adapting their construction, we prove a lower bound for Induced Ψ s,t -Subgraph Test (vc).…”

Section: Independent Set On P 2 -Split Graphsmentioning

confidence: 99%

“…For example, it has been shown that several graph problems such as Treewidth [9], η-Transversal [19], and 3-Coloring [36], admit polynomial kernels parameterized by the size of a given vertex cover. On the other hand, under certain complexity-theoretic assumptions it is possible to show that a number of problems including Dominating Set [22], Clique [8], Chromatic Number [8], Cutwidth [18], and Weighted Vertex Cover [35], do not admit polynomial kernels for this parameter. As the vertex cover number is one of the largest structural graph parameters, being at least as large as treewidth and the feedback vertex number, a superpolynomial kernel lower bound for a parameterization by vertex cover immediately rules out the possibility of obtaining polynomial kernels for these smaller parameters (cf.…”

Section: Introductionmentioning

confidence: 99%

Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

Fomin¹,

Jansen²,

Pilipczuk³

2012

Preprint

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We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like q-Coloring, Odd Cycle Transversal, Chordal Deletion, η-Transversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like F-Minor-Free Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F.While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP ⊆ coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.

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“…does not admit a polynomial kernel under this complexity assumption. This work has been followed by a flurry of results refining this technology [12,19,20,23,34] and using it to prove negative results for concrete parameterized problems (e.g., [6,10,13,17,21,26,37,39,38,43,44], see also the recent survey of Lokshtanov et al [47]). We continue this line of research by trying to characterize which F-Subgraph Test and F-Packing problems admit polynomial kernels.…”

Section: Introductionmentioning

confidence: 99%

Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels

Jansen

Marx

2014

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

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Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Cited by 5 publications

References 26 publications

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?

Characterizing the easy-to-find subgraphs from the viewpoint of polynomial-time algorithms, kernels, and Turing kernels

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