Encouraging Cooperation in Sharing Supermodular Costs

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Abstract

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 provide motivation for studying 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 design approximation algorithms based on oracles that approximately determine maximally violated constraints. We apply our results to schedule planning games, or cooperative games where the costs arise from the minimum sum of weighted completion times on a single machine. By improving upon some of the results for general supermodular cost cooperative games, we are able to give an explicit formula for 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.