Vertex Cover: Further Observations and Further Improvements

Chen, Jianer; Kanj, Iyad A.; Jia, Weijia

doi:10.1006/jagm.2001.1186

Cited by 385 publications

(125 citation statements)

References 18 publications

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“…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.…”

Section: Cubic Kernel For Fvs-vertex Covermentioning

confidence: 99%

“…There has been an impressive series of ever-faster parameterized algorithms to solve k-VERTEX COVER, 1 which led to the current-best algorithm by Chen et al that can decide whether a graph G has a vertex cover of size k in O(1.2738 k + kn) time and polynomial space [9,10,20,38]. Mishra et al [34] studied the role of König deletion sets (vertex sets whose removal ensure that the size of a maximum matching in the remaining graph equals the vertex cover number of that graph) for the complexity of the VERTEX COVER problem, and showed that VERTEX COVER parameterized above the size of a maximum matching is fixed-parameter tractable by exhibiting a connection to ALMOST 2-SAT [40].…”

Section: Introductionmentioning

confidence: 99%

“…The k-VERTEX COVER problem admits a kernel with 2k vertices and O(k 2 ) edges, which can be obtained through crown reduction [2,11,12] or by applying a linear-programming theorem due to Nemhauser and Trotter [9,35]. These kernelization algorithms have been a subject of repeated study and experimentation [1,7,16].…”

Section: Introductionmentioning

confidence: 99%

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Vertex Cover Kernelization Revisited

Jansen

Bodlaender

2012

Theory Comput Syst

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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 (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 instanceA 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: Cubic Kernel For Fvs-vertex Covermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vertex Cover Kernelization Revisited

Jansen

Bodlaender

2012

Theory Comput Syst

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“…As a result, we obtain the transformation illustrated in Figure 2: This transformation is known as vertex folding. It was used to improve the worst case time complexity for the vertex cover and stable set problems [11,17].…”

Section: Simplicial Vertex Reductionmentioning

confidence: 99%

Stability preserving transformations of graphs

Lozin

2008

Ann Oper Res

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Graph transformations proved useful for many algorithmic problems. In the present paper, we study this tool with respect to the maximum stable set problem. We first review available results on this topic and then propose an approach to uniformly describe and systematically develop graph transformations that do not change the size of a maximum stable set in the graph. The approach is illustrated by a number of new transformations.

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“…Vertex Cover There is now an algorithm running in time O (1.286 k + k|G|) ( [12]) for determining if a graph G has a vertex cover of size k. This has been implemented and is practical for |G| of unlimited size and k up to around 400 [41].…”

Section: Introductionmentioning

confidence: 99%