The Minimum Entropy Submodular Set Cover Problem

We study minimum entropy submodular set cover, a variant of the submodular set cover problem (Wolsey [22], Fujito [11], etc) that generalizes the minimum entropy set cover problem (Halperin and Karp [12], Cardinal et al. [5]) We give a general bound of 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.