On the complexity of efficiency and envy-freeness in fair division of indivisible goods with additive preferences

Abstract

We study the problem of allocating a set of indivisible goods to a set of agents having additive preferences. We introduce two new important complexity results concerning efficiency and fairness in resource allocation problems: we prove that the problem of deciding whether a given allocation is Pareto-optimal is coNP-complete, and that the problem of deciding whether there is a Pareto-efficient and envy-free allocation is ∑ 2p-complete. © 2009 Springer-Verlag Berlin Heidelberg.