# A primal-dual approach to approximation of node-deletion problems for matroidal properties

## Abstract

This paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. We will focus on such graph properties that are derived from matroids definable on the edge set of any graph. It will be shown first that all the node-deletion problem for such properties can be uniformly formulated by a simple but non-standard form of the integer program. A primaldual approximation algorithm based on this and the dual of its linear relaxation is then presented.

When a property has infinitely many minimal forbidden graphs no constant factor approximation for the corresponding node-deletion problem has been known except for the case of the Feedback Vertex Set (FVS) problem in undirected graphs. It will be shown next that FVS is not the sole exceptional case and that there exist infinitely many graph (hereditary) properties with an infinite number of minimal forbidden graphs, for which the node-deletion problems are efficiently approximable to within a factor of 2. Such properties are derived from the notion of matroidal families of graphs and relaxing the definitions for them.

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