Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes

Polynomial Kernelizations For {MIN F+Pi1} And {MAX NP}

The relation of constant-factor approximability to fixed-parameter tractability and kernelization is a long-standing open question. We prove that two large classes of constant-factor approximable problems, namely~\textsc{MIN F}^+\Pi_1 and~\textsc{MAX NP}, including the well-known subclass~\textsc{MAX SNP}, admit polynomial kernelizations for their natural decision versions. This extends results of Cai and Chen (JCSS 1997), stating that the standard parameterizations of problems in~\textsc{MAX SNP} and~\textsc{MIN F}^+\Pi_1 are fixed-parameter tractable, and complements recent research on problems that do not admit polynomial kernelizations (Bodlaender et al.\ ICALP 2008)