加权混合甘露按比例分配的紧致补贴界
Tight Subsidy Bounds for Weighted Proportional Allocation of Mixed Manna
- University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
- University of Macau（澳门大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本研究针对混合甘露环境下加权比例分配问题，证明总补贴上界为 n/4，填补了先前差距，并提出了满足 WPROP1 的紧致补贴界及多项式时间算法。
AI中文摘要：
我们研究在混合甘露环境中，将 m 个不可分割物品公平分配给 n 个可能具有不平等权利的代理人的问题，其中每个物品可能被不同代理人视为商品或苦差事。我们关注比例性这一基本公平概念。由于在此环境中比例分配不一定存在，我们允许货币补贴以恢复比例性，同时最小化总补贴。当每个物品的（负）效用以 1 为界时，总补贴至少需要 \tau(n) \approx n/4。对于仅商品或仅苦差事的实例，先前已知的最佳上界是 Wu 和 Zhou（2024）给出的 n/3-1/6。我们通过证明总补贴至多 \tau(n) 总是足够的，从而填补了这一差距，确立了紧致补贴界。我们的结果甚至在更一般的加权混合甘露环境中也成立，解决了 Wu 等人（2023）和 Garg 等人（2026）提出的开放问题。该分配还满足加权比例性直至一个物品（WPROP1）。我们的证明开发了 Knaster-Kuratowski-Mazurkiewicz（KKM）不动点定理的新应用，将 KKM 框架扩展到基于份额的公平概念。最后，我们为任意固定数量的代理人设计了计算此类分配的多项式时间算法。
英文摘要：
We study the problem of fairly allocating m indivisible items among n agents with possibly unequal entitlements in the mixed manna setting, where each item may be perceived as a good or a chore by different agents. We focus on the fundamental fairness notion of proportionality. Since proportional allocations need not exist in this setting, we allow monetary subsidies to restore proportionality while minimizing the total subsidy. When each item's (dis)utility is bounded by 1, a total subsidy of at least τ(n) \approx n/4 may be necessary. For goods-only or chores-only instances, the best previously known upper bound was n/3-1/6 due to Wu and Zhou~(2024). We close this gap by proving that a total subsidy of at most τ(n) always suffices, thereby establishing the tight subsidy bound. Our results hold even in the more general setting of weighted mixed manna, resolving an open question posed by~Wu et al. (2023) and Garg et al. (2026). The allocation also satisfies weighted proportionality up to one item (WPROP1). Our proof develops a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) fixed-point theorem, extending the KKM framework to share-based fairness notions. Finally, we design a polynomial-time algorithm to compute such allocations for any fixed number of agents.
