TY - GEN

T1 - Exploring the kernelization borders for hitting cycles

AU - Agrawal, Akanksha

AU - Jain, Pallavi

AU - Kanesh, Lawqueen

AU - Misra, Pranabendu

AU - Saurabh, Saket

N1 - Publisher Copyright:
© A. Agrawal, and P. Jain, and L. Kanesh, and P. Misra, and S. Saurabh; licensed under Creative Commons License CC-BY.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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 ? 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(kO(d)) kernel. 2. CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP ? coNPpoly . 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 ? V (H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size kO(d) exists, and can be used in designing polynomial kernel.

AB - 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 ? 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(kO(d)) kernel. 2. CF-OCT does not admit polynomial kernel (even when H is 1-degenerate), unless NP ? coNPpoly . 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 ? V (H), that is enough to capture the independent set property in H. We show that for d-degenerate graph independence preserver of size kO(d) exists, and can be used in designing polynomial kernel.

KW - Conflict-free problems

KW - Even Cycle Transversal

KW - Feedback Vertex Set

KW - Kernelization

KW - Odd Cycle Transversal

KW - Parameterized Complexity

UR - http://www.scopus.com/inward/record.url?scp=85092566688&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2018.14

DO - 10.4230/LIPIcs.IPEC.2018.14

M3 - Conference contribution

AN - SCOPUS:85092566688

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 13th International Symposium on Parameterized and Exact Computation, IPEC 2018

A2 - Paul, Christophe

A2 - Pilipczuk, Michal

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 13th International Symposium on Parameterized and Exact Computation, IPEC 2018

Y2 - 22 August 2018 through 24 August 2018

ER -