On the Hardness of Losing Width

Authors:
Marek Cygan;Daniel Lokshtanov;Marcin Pilipczuk;Michał Pilipczuk;Saket Saurabh
Affiliations:
Institute of Informatics, University of Warsaw, Warsaw, Poland;University of California, San Diego, La Jolla, USA 92093-0404;Institute of Informatics, University of Warsaw, Warsaw, Poland;Institute of Informatics, University of Warsaw, Warsaw, Poland;Institute of Mathematical Sciences, Chennai, India 600113
Venue:
Theory of Computing Systems
Year:
2014

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Abstract

Let 驴驴0 be an integer and G be a graph. A set X⊆V(G) is called a 驴-treewidth modulator in G if G驴X has treewidth at most 驴. Note that a 0-treewidth modulator is a vertex cover, while a 1-treewidth modulator is a feedback vertex set of G. In the 驴/驴-Treewidth Modulator problem we are given an undirected graph G, a 驴-treewidth modulator X⊆V(G) in G, and an integer ℓ and the objective is to determine whether there exists an 驴-treewidth modulator Z⊆V(G) in G of size at most ℓ. In this paper we study the kernelization complexity of 驴/驴-Treewidth Modulator parameterized by the size of X. We show that for every fixed 驴 and 驴 that either satisfy 1≤驴驴, or 驴=0 and 2≤驴, the 驴/驴-Treewidth Modulator problem does not admit a polynomial kernel unless NP⊆coNP/poly. This resolves an open problem raised by Bodlaender and Jansen (STACS, pp. 177---188, 2011). Finally, we complement our kernelization lower bounds by showing that 驴/0-Treewidth Modulator admits a polynomial kernel for any fixed 驴.