Optimal Partitions in Additively Separable Hedonic Games

We conduct a computational analysis of partitions in additively separable hedonic games that satisfy standard criteria of fairness and optimality. We show that computing a partition with maximum egalitarian or utilitarian social welfare is NP-hard in the strong sense whereas a Pareto optimal partition can be computed in polynomial time when preferences are strict. Perhaps surprisingly, checking whether a given partition is Pareto optimal is coNP-complete in the strong sense, even when preferences are symmetric and strict. We also show that checking whether there exists a partition which is both Pareto optimal and envy-free is Σp2-complete. Furthermore, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete when preferences are symmetric.