Computing socially-efficient cake divisions

We consider a setting in which a single divisible good (“cake”) needs to be divided between n players, each with a possibly different valuation function over pieces of the cake. For this setting, we address the problem of finding divisions that maximize the utilitarian social welfare, focusing on divisions where each player needs to get one contiguous piece of the cake. We provide a constant factor approximation algorithm for the problem, and show that it is NP-hard to find the optimal division and prove that no FPTAS is possible unless P=NP. We show that these results hold both when the full valuations of all the players are given to the algorithm, and when the algorithm has only oracle access to the valuations. Unlike in the case of “connected pieces”, when the contiguity requirement is dropped, we show that the results vary greatly depending on the input model. Specifically, we show that when the algorithm gets the full valuation functions of the players, maximizing welfare is very easy. In contrast, when the algorithm needs to query the players for information on their valuations, it is impossible to guarantee a division in which the welfare is more than 1/n of the optimum.