Hitting forbidden minors: Approximation and Kernelization
Résumé
We study a general class of problems called \fd{} problems. In an \fd{} 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 ${\cal F}$ of forbidden minors. We obtain a number of algorithmic results on the \fd{}~problem when $\mathcal{F}$ contains a planar graph. We give \begin{itemize} \setlength{\itemsep}{-2pt} \item 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. \item 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. \end{itemize} Finally, we obtain polynomial kernels for the case when $\cal F$ only contains graph \jc{} as a minor for a fixed integer $c$. The graph \jc{} 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 \dhs{}. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes.
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