Optimal Partitions in Additively Separable Hedonic Games

TL;DR

This paper analyzes the computational complexity of finding fair and optimal partitions in additively separable hedonic games, revealing polynomial-time solvability for some cases and NP-hardness for others.

Contribution

It provides the first comprehensive complexity classification of various fairness and optimality concepts in additively separable hedonic games.

Findings

Pareto optimal partition can be found in polynomial time for strict preferences.

Verifying Pareto optimality is coNP-complete.

Computing maximum social welfare or both Pareto optimality and individual rationality is NP-hard.

Abstract

We conduct a computational analysis of fair and optimal partitions in additively separable hedonic games. We show that, for strict preferences, a Pareto optimal partition can be found in polynomial time while verifying whether a given partition is Pareto optimal is coNP-complete, even when preferences are symmetric and strict. Moreover, computing a partition with maximum egalitarian or utilitarian social welfare or one which is both Pareto optimal and individually rational is NP-hard. We also prove that checking whether there exists a partition which is both Pareto optimal and envy-free is $\Sigma_{2}^{p}$-complete. Even though an envy-free partition and a Nash stable partition are both guaranteed to exist for symmetric preferences, checking whether there exists a partition which is both envy-free and Nash stable is NP-complete.