  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.
