Exploring the Kernelization Borders for Hitting Cycles

Abstract

A generalization of classical cycle hitting problems, called conflict version of the problem, is defined as follows. An input is undirected graphs G and H on the same vertex set, and a positive integer k, and the objective is to decide whether there exists a vertex subset X subseteq V(G) such that it intersects all desired "cycles" (all cycles or all odd cycles or all even cycles) and X is an independent set in H. In this paper we study the conflict version of classical Feedback Vertex Set, and Odd Cycle Transversal problems, from the view point of kernelization complexity. In particular, we obtain the following results, when the conflict graph H belongs to the family of d-degenerate graphs. 
1) CF-FVS admits a O(k^{O(d)}) kernel.
2) CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP subseteq coNP/poly.
 For our kernelization algorithm we exploit ideas developed for designing polynomial kernels for the classical Feedback Vertex Set problem, as well as, devise new reduction rules that exploit degeneracy crucially. Our main conceptual contribution here is the notion of "k-independence preserver". Informally, it is a set of "important" vertices for a given subset X subseteq V(H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size k^{O(d)} exists, and can be used in designing polynomial kernel.