We investigate the query complexity of the fair allocation of indivisible
goods. For two agents with arbitrary monotonic valuations, 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 valuations. 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 valuations