Minimum Entropy Submodular Optimization (and Fairness in Cooperative Games)

We study minimum entropy submodular optimization, a common generalization of the minimum entropy set cover problem, studied earlier by Cardinal et al., and the submodular set cover problem (Wolsey [Wol82], Fujishige [BIKP01], etc). 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) the definition of worst-case approaches to fairness in concave cooperative games, similar to the notion of price of anarchy in noncooperative settings.