The minimum entropy submodular set cover problem

Abstract

We study Minimum Entropy Submodular Set Cover, a variant of the Submodular Set Cover problem (Wolsey [21], Fujito [8], etc.) that generalizes the Minimum Entropy Set Cover problem (Halperin and Karp [11], Cardinal et al. [4]) We give a general bound on the approximation performance of the greedy algorithm using an approach that can be interpreted in terms of a particular type of biased network flows. As an application we rederive known results for the Minimum Entropy Set Cover and Minimum Entropy Orientation problems, and obtain a nontrivial bound for a new problem called the Minimum Entropy Spanning Tree problem. The problem can be applied to (and is partly motivated by) a worst-case approach to fairness in concave cooperative games.