On the Complexity of Envy-Free Cake Cutting

We study the envy-free cake-cutting problem for d+1 players with d cuts, for both the oracle function model and the polynomial time function model. For the former, we derive a θ(( ǫ )) time matching bound for the query complexity of d + 1 player cake cutting with Lipschitz utilities for any d > 1. When the utility functions are given by a polynomial time algorithm, we prove the problem to be PPADcomplete. For measurable utility functions, we find a fully polynomial-time algorithm for finding an approximate envy-free allocation of a cake among three people using two cuts.