AI 中文总结
研究n个代理对m个分三类的不可分割物品的无嫉妒分配问题，提出首个运行时间为(n·m)^O(√n)的次指数算法，利用几何表示和定理递归分解代理集，还扩展算法处理预固定分配代理以保持无嫉妒性。
AI 中文摘要
我们研究了在物品分为三种不同类型时，为n个代理计算m个不可分割物品的无嫉妒（EF）分配问题。每个代理对物品类型持有可加估值，可能为正（商品）、负（琐事）或混合。我们提出了首个运行时间为(n·m)^O(√n)的次指数时间算法，若存在EF分配则找到它，否则正确报告不存在。我们的方法利用EF分配在R^3中作为凸多面体的几何表示，并应用米勒平面循环分离器定理将代理集递归分解为平衡子组。我们还扩展算法以处理预先固定分配的代理，保持所有代理的无嫉妒性。
英文摘要
We study the problem of computing an envy-free (EF) allocation of $m$ indivisible items among $n$ agents when items come in three distinct types. Each agent holds additive valuations over item types that may be positive (goods), negative (chores), or mixed. We present the first subexponential-time algorithm with running time time $(n \cdot m)^{O(\sqrt{n})}$ that finds an EF allocation whenever one exists, or correctly reports that none exists. Our approach exploits a geometric representation of EF allocations as convex polyhedra in $\mathbb{R}^3$ and applies Miller's planar cycle-separator theorem to recursively decompose the agent set into balanced subgroups. We further extend the algorithm to handle agents whose allocations are fixed in advance, preserving envy-freeness across all agents.
