Exploring the Kernelization Borders for Hitting Cycles
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.
Parameterized Complexity
Kernelization
Conflict-free problems
Feedback Vertex Set
Even Cycle Transversal
Odd Cycle Transversal
Theory of computation~Parameterized complexity and exact algorithms
14:1-14:14
Regular Paper
This research has received funding from the European Research Council under ERC grant no. 306992 PARAPPROX, ERC grant no. 715744 PaPaALG and ERC grant no. 725978 SYSTEMATIC-GRAPH, and DST, India for SERB-NPDF fellowship [PDF/2016/003508].
Akanksha
Agrawal
Akanksha Agrawal
Institute of Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary
Pallavi
Jain
Pallavi Jain
Institute of Mathematical Sciences, HBNI, Chennai, India
Lawqueen
Kanesh
Lawqueen Kanesh
Institute of Mathematical Sciences, HBNI, Chennai, India
Pranabendu
Misra
Pranabendu Misra
University of Bergen, Bergen, Norway
Saket
Saurabh
Saket Saurabh
Institute of Mathematical Sciences, HBNI, Chennai, India
10.4230/LIPIcs.IPEC.2018.14
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