Density Functions subject to a Co-Matroid Constraint

TL;DR

This paper introduces a 2-approximation algorithm for finding the densest subset under co-matroid constraints, generalizing graph density and improving previous approximation guarantees for specific matroid cases.

Contribution

It presents the first general 2-approximation algorithm for the densest subset problem with co-matroid constraints, extending prior work on specific matroids.

Findings

Achieved a 2-approximation ratio for the problem.

Generalized the densest subset problem to co-matroid constraints.

Improved approximation guarantees over previous results for partition matroids.

Abstract

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.