On Cutting Cakes and Crossing Curves
Resolves the open problem of computational complexity for three-agent envy-free cake-cutting, showing it is hard for a subclass of PPAD.
The paper proves that computing an envy-free cake-cutting allocation with contiguous pieces for three agents is intractable, by establishing a novel connection to the Jordan curve theorem problem and providing query lower bounds and UEOPL-hardness for that problem.
We consider the classic envy-free cake-cutting problem where the goal is to cut and allocate a divisible resource among a set of agents in a way that avoids any envy between them. When the agents' valuation functions are continuous and nonnegative, an envy-free solution is guaranteed to exist where each agent is allocated a contiguous piece of the resource. Such a solution can be efficiently computed using the standard cut-and-choose algorithm for two agents, but the problem is known to be hard when there are at least four agents. The setting with three agents has remained open. We show that the problem remains intractable for three agents. We obtain this result by uncovering a novel connection between cake-cutting and a computational problem corresponding to the Jordan curve theorem, introduced by Adler, Daskalakis, and Demaine (2016). As our main technical contribution, we provide the first lower bounds for the Jordan curve problem in the form of a query lower bound as well as hardness for the class UEOPL, a subclass of PPAD containing notoriously challenging problems such as Simple Stochastic Games and the P-matrix Linear Complementarity Problem.