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 <font face="Sans Serif">coNP<!-- /fontSF --></font>-complete, and that the problem of deciding whether there is a Pareto-efficient and envy-free allocation is <math>&Sigma;<sub>2</sub><sup class="exponent">p</sup></math>-complete.