AI 中文总结
该研究针对无嫉妒蛋糕分割问题，提出查询复杂度为n^{O(1)}2^n的协议，首次实现单指数查询上界，大幅缩小了此前巨大的上下界差距。
AI 中文摘要
无嫉妒蛋糕分割是公平分配领域的核心问题，其存在性与计算复杂度间存在显著鸿沟。经典拓扑学保证无嫉妒分配存在，但高效找到该分配的难度大得多，该问题已困扰研究者数十年。Aziz和Mackenzie的著名结果为每个n确定了有界协议的存在性，但其查询复杂度为n^{n^{n^{n^{n^n}}}}。Sokolov的更紧分析随后将该上界降至n^{8n^2(1+o(1))}，这是本研究之前的最佳结果。相比之下，Procaccia给出的一般下界仅为Ω(n^2)。我们使用至多n^{O(1)}2^n次查询的协议大幅缩小了这一巨大差距。从高层看，我们的协议在不产生嫉妒的情况下反复分配部分蛋糕，直至剩余问题涉及更少的参与者。主要难点在于确保后续合并这些分配时，既不会重复分配任何蛋糕，也不会产生嫉妒。为克服这一难点，我们开发了一种仅使用多项式多个部分分配的新构造，替代了以往工作中使用的n^{n^{n^n}}个部分分配。总体而言，我们的协议首次为寻找具有任意非原子加法估值的完整无嫉妒分配提供了单指数查询复杂度。
英文摘要
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.
