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.

翻译：我们研究经典的无嫉妒公平分配问题，其目标是将可分割资源切割并分配给一组智能体，使任何智能体之间不产生嫉妒。当智能体的估值函数连续且非负时，总存在一种无嫉妒的解决方案，使得每个智能体获得资源的一段连续部分。对于两个智能体，这种方案可通过标准切-选算法高效计算，但已知当智能体数量不少于四个时问题具有难度。三个智能体的情形此前尚未解决。我们证明对于三个智能体该问题仍属于难解问题。该结论源于我们揭示了蛋糕切割问题与Adler、Daskalakis和Demaine（2016）提出的基于约当曲线定理的计算问题之间存在新的关联。作为主要技术贡献，我们首次给出约当曲线问题的下界：包括查询复杂度下界，以及属于UEOPL类（PPAD的子类，包含简单随机博弈和P矩阵线性互补问题等公认难题）的难度结论。