比例分组公平分配与单侧偏差的近紧边界
Nearly Tight Bounds for Proportional Group Fair Divisions and One-Sided Discrepancy
AI总结：
本文针对k组智能体的不可分割物品公平分配问题，改进了已有边界，证明比例分组公平分配的最坏向下偏差为Θ̃(√(n/k))，并开发了处理单侧偏差约束的新工具。
AI中文摘要：
本文研究将不可分割物品公平分配给k个分别有n₁,…,nₖ个智能体的分组的问题，考察分组内智能体与其1/k份额的最坏向下偏差PROP(n₁,…,nₖ)。我们改进了Manurangsi和Meka（2026）的边界，证明PROP(n₁,…,nₖ)=Θ̃(√(n/k))，其中n=n₁+…+nₖ为智能体总数。对于上界证明，我们开发了新颖的偏差型工具，尤其是有效处理单侧偏差约束的方法。
英文摘要：
This paper studies the problem of fair division of indivisible goods among $k$ groups of $n_1,\ldots, n_k$ agents. We look at the worst downward deviation $\textit{PROP}(n_1,\ldots, n_k)$ of an agent in a group from its $1/k$-share. We improve the bounds of (Manurangsi and Meka, 2026) and show that $\textit{PROP}(n_1,\ldots, n_k) = \tildeΘ(\sqrt{n/k})$, where $n = n_1 + \ldots + n_k$ is the total number of agents. For the proof of the upper bound, we develop novel discrepancy-type tools and, in particular, a way to efficiently work with one-sided discrepancy constraints.
