Approximating Node-Deletion Problems for Matroidal Properties

The paper is concerned with the polynomial time approximability of node-deletion problems for hereditary properties. It is observed that, when such a property derives a matroid on any graph, the problem can be formulated as a matroid set cover optimization problem, and this leads us naturally to consider two well-known approaches, greedy and primal-dual. The heuristics based on these principles are analyzed and general approximation bounds are obtained. Next, more specific types of hereditary properties, calleduniformly sparse, are introduced and, for any of them, the primal-dual heuristic is shown to approximate the corresponding node-deletion problem within a factor of 2. We conclude that there exist infinitely many (at least countably many) node-deletion problems, each of which approximable to a factor of 2, for hereditary properties with an infinite number of minimal forbidden graphs.