On the hardness of losing width

  • Authors:

  • Marek Cygan;Daniel Lokshtanov;Marcin Pilipczuk;Michał Pilipczuk;Saket Saurabh

  • Affiliations:

  • Institute of Informatics, University of Warsaw, Poland;University of California, San Diego, La Jolla, CA;Institute of Informatics, University of Warsaw, Poland;Institute of Informatics, University of Warsaw, Poland;The Institute of Mathematical Sciences, Chennai, India

  • Venue:
  • IPEC'11 Proceedings of the 6th international conference on Parameterized and Exact Computation
  • Year:
  • 2011

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Abstract

Let η≥0 be an integer and G be a graph. A set X⊆V(G) is called a η-transversal inG if G∖X has treewidth at most η. Note that a 0-transversal is a vertex cover, while a 1-transversal is a feedback vertex set of G. In the η/ρ-transversal problem we are given an undirected graph G, a ρ-transversal X⊆V(G) in G, and an integer ℓ and the objective is to determine whether there exists an η-transversal Z⊆V(G) in G of size at most ℓ. In this paper we study the kernelization complexity of η/ρ-transversalparameterized by the size of X. We show that for every fixed η and ρ that either satisfy 1≤ηρ, or η=0 and 2≤ρ, the η/ρ-transversal problem does not admit a polynomial kernel unless NP ⊆ coNP/poly. This resolves an open problem raised by Bodlaender and Jansen in [STACS 2011]. Finally, we complement our kernelization lower bounds by showing that ρ/0-transversal admits a polynomial kernel for any fixed ρ.