On Cutting Cakes and Crossing Curves

2026-06-11

AI Summary

This work investigates the computational complexity of envy-free cake cutting among three agents. By establishing a novel reduction between this problem and a computational formulation of the Jordan Curve Theorem, it uncovers a deep connection between fair division and topological computation. Leveraging query-based models and complexity classes UEOPL and PPAD, the paper demonstrates the computational intractability of three-agent envy-free cake cutting. Furthermore, it establishes the first query complexity lower bound for the Jordan Curve problem and proves its completeness for the class UEOPL. These results simultaneously advance the theoretical foundations of both fair division and topological computational complexity.

Abstract

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.

Problem

• envy-free cake-cutting
• computational hardness
• Jordan curve theorem
• UEOPL
• fair division

Innovation

• envy-free cake-cutting
• Jordan curve theorem
• UEOPL
• query complexity
• computational hardness