A unified local ratio approximation of node-deletion problems

Abstract

In this paper we consider a unified approximation method for node-deletion problems with nontrivial and hereditary graph properties. It was proved 16 years ago that every node-deletion problems for a nontrivial hereditary property is NP-complete via a few generic approximation preserving reductions from the Vertex Cover problem. An open problem posed at that time is concerned with the other direction of approximability: can other node-deletion problems be approximated as good as the Vertex Cover ? The goal of the current paper is to take a first step along the direction of research suggested above. More specifically, one generic approximation algorithm is presented, which is applicable to every node-deletion problem for a hereditary property. It will be seen then that under simple and natural weighting schemes, serving as a parameter of the algorithm, various node-deletion problems can be approximated with ratio 2, or otherwise with some nontrivial performance ratios. Two types of graph properties are considered in this paper: one with a finite number of minimal forbidden graphs, and the other in which all the edge sets of satisfying (sub)graphs form a family of independent sets for some matroid.