Vertex Cover Kernelization Revisited

Abstract

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.

A Turing Kernelization Dichotomy for Structural Parameterizations of $$\mathcal {F}$$ -Minor-Free Deletion

“…Earlier work [8,23,25,37] has shown that several F-Minor-Free Deletion problems admit polynomial kernelizations when parameterized by the feedback vertex number. In this paper we showed that when F contains a forest and each graph in F has a connected component of at least three vertices, the F-Minor-Free Deletion problem does not admit such a polynomial kernel unless NP ⊆ coNP/poly.…”

“…In terms of the decision problem, this means G has a solution of size at most with U and R as witness partite sets, if and only if G has a vertex cover of size at most − |X |. Since fvs(G ) ≤ fvs(G), we can apply the known [25] kernel for Vertex Cover parameterized by the feedback vertex number to reduce (G , − |X |) to an equivalent instance with O(fvs(G) 3 ) vertices, which is queried to the oracle. If the oracle answers positively to any query, then (G, ) has answer yes; otherwise the answer is no.…”

“…However, this method does not give any nontrivial guarantees when the solution size is known to be proportional to the total size of the input. For that reason, there is an alternative line of research [9,10,13,21,25,26,27,38] that focuses on parameterizations based on a measure of nontriviality of the instance (cf. [33]).…”

“…Previous research has shown that for the Vertex Cover problem, there is a polynomial kernel parameterized by the feedback vertex number [25]. This preprocessing algorithm guarantees that inputs which are large with respect to their feedback vertex number, can be efficiently reduced.…”

“…Apply the kernelization by Jansen and Bodlaender [25] for Vertex Cover parameterized by feedback vertex set to the instance (G 0 , 0 ) and the approximate feedback vertex set S 0 . This takes polynomial time, and results in an instance (…”