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.

💡 Research Summary

The paper studies the problem of selecting a subset of a ground set U that maximizes the density defined as ρ(S)=f(S)/|S|, where f is a monotone super‑modular set function. The feasible subsets are those whose complement is independent in a given matroid 𝔐; this is called a co‑matroid constraint. The authors first show that the problem is NP‑hard by a reduction from the classic densest subgraph problem, which is a special case when f counts edges in a graph and the matroid forces the complement to be a single vertex. Consequently, exact polynomial‑time algorithms are unlikely, motivating approximation approaches.

The core of the algorithm is a parametric search on a candidate density value λ. For a fixed λ, the authors consider the auxiliary function g_λ(S)=f(S)−λ·|S|. Because f is super‑modular and the linear term is modular, g_λ remains super‑modular. The feasibility test asks whether there exists a feasible set S with g_λ(S)≥0. This is equivalent to solving a super‑modular minimization problem under the co‑matroid constraint, which can be done in polynomial time using the Iwata‑Fleischer‑Fujishige (IFF) algorithm or its later improvements.

By binary searching over λ in the interval