Density Functions subject to a Co-Matroid Constraint

Abstract

In this paper we consider the problem of finding the densest subset subject to co-matroid constraints. We are given a monotone supermodular set function f defined over a universe U, and the density of a subset S is defined to be f(S)/|S|. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid M a set S is feasible, iff the complement of S is independent in the matroid. Under such constraints, the problem becomes NP-hard. The specific case of graph density has been considered in literature under specific co-matroid constraints, for example, the cardinality matroid and the partition matroid. We show a 2-approximation for finding the densest subset subject to co-matroid constraints. Thereby we improve the approximation guarantees for the result for partition matroids in the literature