Cutting a Cake for Five People

Given a heterogeneous cake and 5 players, we give the first bounded algorithm for computing an envy-free division of the cake, such that each person thinks he gets the largest piece. The case with 4 players was solved in a famous paper by Brams et al. in 1997. Our algorithm can be discretized to obtain an *** envy-free division in $O({\rm polylog} \left( 1 / \epsilon \right))$ time. The algorithm is based on augmenting the irrevocable advantage graph in a new way. We also look at the open problem of finding discrete procedures for computing envy-free division among 4 players. We present a simple algorithm that finds an envy-free division of a portion of the cake, such that each player gets at least 1/4 of the whole cake (in his valuation).