Fair Submodular Cover

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 $τ$, the goal is to find a balanced subset of $S$ with minimum cardinality such that $f(S)\geτ$. We first introduce discrete algorithms for FSC that achieve a bicriteria approximation ratio of $(\frac{1}ε, 1-O(ε))$. We then present a continuous algorithm that achieves a $(\ln\frac{1}ε, 1-O(ε))$-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.