Density Functions subject to a Co-Matroid Constraint
  In this paper we consider the problem of finding the {\em densest} subset subject to {\em co-matroid constraints}. We are given a {\em monotone supermodular} set function $f$ defined over a universe $U$, and the density of a subset $S$ is defined to be $f(S)/\crd{S}$. This generalizes the concept of graph density. Co-matroid constraints are the following: given matroid $\calM$ a set $S$ is feasible, iff the complement of $S$ is {\em 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. Thus, for instance, we improve the approximation guarantees for the result for partition matroids in the literature. 
Venkatesan T. Chakaravarthy and Natwar Modani and Sivaramakrishnan R.   Natarajan and Sambuddha Roy and Yogish Sabharwal
cs.DS cs.DM