Minimum Entropy Submodular Optimization (and Fairness in Cooperative Games)

TL;DR

This paper introduces bounds for minimum entropy submodular optimization, generalizing previous problems, and explores applications to fairness in cooperative games using a biased network flow approach.

Contribution

It provides a unified approximation bound for greedy algorithms in minimum entropy submodular problems and extends results to new applications like the Minimum Entropy Spanning Tree.

Findings

Derived bounds for greedy algorithm performance in submodular optimization

Reestablished results for set cover and orientation problems

Established a new bound for the Minimum Entropy Spanning Tree problem

Abstract

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. 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.