Linear kernels and single-exponential algorithms via Protrusion Decompositions

Kim, E.J. and Langer, A. and Paul, C. and Reidl, Felix and Rossmanith, P. and Sau, I. and Sikdar, S. (2015) Linear kernels and single-exponential algorithms via Protrusion Decompositions. ACM Transactions on Algorithms 12 (2), 21:1-21:41. ISSN 1549-6325.

Abstract

A \emph{t-treewidth-modulator} of a graph G is a set X⊆V(G) such that the treewidth of G−X is at most some constant t−1. In this paper, we present a novel algorithm to compute a decomposition scheme for graphs G that come equipped with a t-treewidth-modulator. This decomposition, called a \emph{protrusion decomposition}, is the cornerstone in obtaining the following two main results. We first show that any parameterized graph problem (with parameter k) that has \emph{finite integer index} and is \emph{treewidth-bounding} admits a linear kernel on H-topological-minor-free graphs, where H is some arbitrary but fixed graph. A parameterized graph problem is called treewidth-bounding if all positive instances have a t-treewidth-modulator of size O(k), for some constant t. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus [Bodlaender et al., FOCS 2009] and H-minor-free graphs [Fomin et al., SODA 2010]. Our second application concerns the Planar-F-Deletion problem. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar-F-Deletion asks whether G has a set X⊆V(G) such that |X|≤k and G−X is H-minor-free for every H∈F. Very recently, an algorithm for Planar-F-Deletion with running time 2O(k)nlog2n (such an algorithm is called \emph{single-exponential}) has been presented in [Fomin et al., FOCS 2012] under the condition that every graph in F is connected. Using our algorithm to construct protrusion decompositions as a building block, we get rid of this connectivity constraint and present an algorithm for the general Planar-F-Deletion problem running in time 2O(k)n2.

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