Encouraging Cooperation in Sharing Supermodular Costs

Abstract

The least core value of a cooperative game is the minimum penalty we need to charge a coalition for defecting that ensures the existence of a fair and efficient cost allocation. The set of all such cost allocations is called the least core. In this paper, we study the computational complexity and algorithmic aspects of computing the least core value of supermodular cost cooperative games, and uncover some structural properties of the least core of these games. We motivate the study of these games by showing that a particular class of optimization problems has supermodular optimal costs. This class includes a variety of problems in combinatorial optimization, especially in machine scheduling. We show that computing the least core value of supermodular cost cooperative games is NP-hard, and build a framework to approximate the least core value of these games using oracles that approximately determine maximally violated constraints. With recent work on maximizing submodular functions, our framework yields a 3-approximation algorithm for computing the least core value of general supermodular cost games. We also apply our approximation framework to two particular classes of cooperative games: schedule planning games and matroid profit games. Schedule planning games are cooperative games in which the cost to a coalition is derived from the minimum sum of weighted completion times on a single machine. By specializing some of the results for general supermodular cost cooperative games, we are able to show that the Shapley value is an element of the least core of schedule planning games, and design a fully polynomial time approximation scheme for computing the least core value of these games. Matroid profit games are cooperative games with submodular profits: the profit to a coalition arises from the maximum weight of a matroid basis. We show that an element of the least core and the least core value of matroid profit games can be computed in polynomial time.