Cutting Down the Tower: Single-Exponential Envy-Free Cake Cutting

Abstract

Envy-free cake cutting is a central problem in fair division with a striking divide between existence and computation. Classical topology guarantees that envy-free allocations exist, yet finding one efficiently turned out to be much harder, and this problem has resisted decades of work. A well-known result by Aziz and Mackenzie established the existence of a bounded protocol for every $n$, but its query bound is $n^{n^{n^{n^{n^n}}}}$. A tighter analysis by Sokolov subsequently reduced this upper bound to $n^{8n^2(1+o(1))}$, the best known prior to this work. In contrast, the general lower bound, due to Procaccia, is merely $Ω(n^2)$. We close much of this massive gap with a protocol using at most $n^{O(1)}2^n$ queries. At a high level, our protocol repeatedly allocates some cake without creating envy until the remaining problem involves fewer agents. The main difficulty is to ensure that, when we later put these allocations together, we neither assign any cake twice nor create envy. To overcome this difficulty, we develop a new construction using only polynomially many partial allocations, replacing the $n^{n^{n^n}}$ partial allocations used in previous work. Overall, our protocol gives the first single-exponential query bound for finding a complete envy-free allocation with arbitrary nonatomic, additive valuations.