Fair Submodular Cover

TL;DR

This paper introduces the Fair Submodular Cover problem, proposing new algorithms that balance coverage and fairness in subset selection, with theoretical guarantees and empirical validation on maximum coverage tasks.

Contribution

The paper formulates the Fair Submodular Cover problem and develops discrete and continuous algorithms with strong approximation guarantees, advancing fair optimization methods.

Findings

Discrete algorithms achieve a bicriteria approximation of (1/ε, 1-O(ε)).

Continuous algorithm matches the best approximation guarantee of unconstrained submodular cover.

Empirical results demonstrate the effectiveness of the proposed algorithms on maximum coverage instances.

Abstract

Submodular optimization is a fundamental problem with many applications in machine learning, often involving decision-making over datasets with sensitive attributes such as gender or age. In such settings, it is often desirable to produce a diverse solution set that is fairly distributed with respect to these attributes. Motivated by this, we initiate the study of Fair Submodular Cover (FSC), where given a ground set $U$, a monotone submodular function $f:2^U\to\mathbb{R}_{\ge 0}$, a threshold $\tau$, the goal is to find a balanced subset of $S$ with minimum cardinality such that $f(S)\ge\tau$. We first introduce discrete algorithms for FSC that achieve a bicriteria approximation ratio of $(\frac{1}{\epsilon}, 1-O(\epsilon))$. We then present a continuous algorithm that achieves a $(\ln\frac{1}{\epsilon}, 1-O(\epsilon))$-bicriteria approximation ratio, which matches the best approximation guarantee of submodular cover without a fairness constraint. Finally, we complement our theoretical results with a number of empirical evaluations that demonstrate the effectiveness of our algorithms on instances of maximum coverage.