Polynomial Kernelizations for MIN F+Π1 and MAX NP

Abstract

It has been observed in many places that constant-factor approximable problems often admit polynomial or even linear problem kernels for their decision versions, e.g., VERTEX COVER, FEEDBACK VERTEX SET, and TRIANGLE PACK-ING. While there exist examples like BIN PACKING, which does not admit any kernel unless P = NP, there apparently is a strong relation between these two polynomialtime techniques. We add to this picture by showing that the natural decision versions of all problems in two prominent classes of constant-factor approximable problems, namely MIN F + 1 and MAX NP, admit polynomial problem kernels. Problems in MAX SNP, a subclass of MAX NP, are shown to admit kernels with a linear base set, e.g., the set of vertices of a graph. This extends results of Cai and Chen (J. Comput. Syst. Sci. 54(3): 465-474, 1997), stating that the standard parameterizations of problems in MAX SNP and MIN F + 1 are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al. in J. Comput. Syst. Sci. 75(8): 423-434, 2009).

Structure of Polynomial-Time Approximation

“…Then the running time of the given algorithm improves to O(k · 2 O(k) ) plus a polynomial in the input size. Kratsch [31] recently showed that problems in MAX-NP admit a polynomial kernel, strengthening the above result.…”

Approximation and Tidying—A Problem Kernel for s-Plex Cluster Vertex Deletion

“…This is a rich class of graphs, among others containing various cluster graphs. When applicable, our method may allow for significantly smaller problem kernel sizes than the more general method by Kratsch [19].…”

“…While the technique of Kratsch [19] is more general, the approach presented here seems to be useful to obtain smaller problem kernels. For example, the forbidden induced subgraphs for s-PLEX CLUSTER VERTEX DELETION consist of at most s+ s+1 vertices [15]; therefore, Kratsch's technique [19] yields an O(k s+ s+1 )-vertex problem kernel. In contrast, we obtain an O(k 2 s 3 )-vertex problem kernel using the approximation and tidying method described below.…”

“…Our main conceptual contribution is the "approximation and tidying" kernelization method outlined in Section 3. A decisive advantage over the O(k s+ s+1 )-vertex problem kernel for s-PLEX CLUSTER VERTEX DELETION that can be obtained from a general result from Kratsch [19] is that, in our O(k 2 s 3 )-vertex bound, the value of s does not influence the degree of the polynomial in k.…”

“…Moreover, one may observe that its minimization version is in MIN F + Π 1 , a class of minimization problems whose optimum solution can be expressed using a certain type of universally quantified firstorder logic formulae [18]. Therefore, using a technique due to Kratsch [19], F -FREE VERTEX DELETION admits a problem kernel containing O(k h ) vertices, where h is the maximum number of vertices of a graph in F . We present an alternative technique to kernelize F -FREE VERTEX DELETION problems for finite sets F .…”