  We initiate the study of the communication complexity of fair division with
indivisible goods. We focus on some of the most well-studied fairness notions
(envy-freeness, proportionality, and approximations thereof) and valuation
classes (submodular, subadditive and unrestricted). Within these parameters,
our results completely resolve whether the communication complexity of
computing a fair allocation (or determining that none exist) is polynomial or
exponential (in the number of goods), for every combination of fairness notion,
valuation class, and number of players, for both deterministic and randomized
protocols.
