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 $theta(({1overepsilon})^{d-1})$ time matching bound for the query comp
lexity 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 PPAD-complete. 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.
