  We investigate the query complexity of the fair allocation of indivisible
goods. For two agents with arbitrary monotonic utilities, we design an
algorithm that computes an allocation satisfying envy-freeness up to one good
(EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We
show that the logarithmic query complexity bound also holds for three agents
with additive utilities, and that a polylogarithmic bound holds for three
agents with monotonic utilities. These results suggest that it is possible to
fairly allocate goods in practice even when the number of goods is extremely
large. By contrast, we prove that computing an allocation satisfying
envy-freeness and another of its relaxations, envy-freeness up to any good
(EFX), requires a linear number of queries even when there are only two agents
with identical additive utilities.
