Hitting forbidden minors: Approximation and Kernelization

TL;DR

This paper investigates F-deletion problems involving forbidden minors, providing new kernelization and approximation algorithms for specific classes of graphs, including planar and claw-free graphs, with implications for classical problems like Vertex Cover.

Contribution

It introduces linear kernels for graphs excluding t-claw as an induced subgraph, an approximation algorithm with ratio O(log^{3/2} OPT), and polynomial kernels for problems involving theta_c minors.

Findings

Linear vertex kernel for graphs excluding t-claw as an induced subgraph

Approximation algorithm with ratio O(log^{3/2} OPT)

Polynomial kernels for problems with theta_c minors

Abstract

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.