A Uni ed Local Ratio Approximation of Node { Deletion Problems ( Extended

In this paper we consider a uni ed 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 rst step along the direction of research suggested above. More speci cally, 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 nite 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.