On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences
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 Σp2 -complete.
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De Keijzer, B, Bouveret, S., Klos, T.B, & Zhang, Y. (2009). On the Complexity of Efficiency and Envy-Freeness in Fair Division of Indivisible Goods with Additive Preferences. doi:10.1007/978-3-642-04428-1_9