AI 中文总结
本文研究受群体公平约束的设施分配问题的度量失真，提出全信息与有序信息算法，推导最大和、和最大社会目标的失真上下界，且下界与上界匹配，覆盖单向匹配与聚类问题。
AI 中文摘要
我们研究设施分配问题的群体公平度量失真，其中一组被划分为未知群体的智能体，必须被分配到一组设施中，可能受容量或其他可行性约束。给定分配方案，每个智能体产生的成本取决于其与所分配设施的距离，以及通过亲和因子计算的其群体中其他成员与各自所分配设施的平均距离。我们考虑两类算法：全信息算法，其完全知晓度量空间；有序信息算法，其知晓设施间的距离，仅知晓智能体对设施的排名（按距离递增排序）。我们针对最大和（Max-of-Sum）与和最大（Sum-of-Max）社会目标，建立了最坏情况下的失真上界，这两个目标结合了经典的功利主义和平等主义社会成本度量。我们还针对单向匹配和聚类这两个被我们的模型涵盖的基础且被广泛研究的问题，推导了信息论下界，该下界与最大和的上界完全匹配，与和最大的上界渐近匹配。
英文摘要
We study the group-fair distortion of metric facility assignment problems, where a set of agents, partitioned into unknown groups, must be assigned to a collection of facilities, possibly subject to capacity or other feasibility constraints. Given an assignment, each agent incurs a cost that depends on both its distance to its assigned facility and, via an affinity factor, the average distance of the other members in its group to their assigned facilities. We consider full-information algorithms, which have complete knowledge of the metric space, and ordinal-information algorithms, which know the distances between facilities and only the rankings of the agents over facilities (sorted by increasing distance). We establish worst-case distortion upper bounds in terms of the Max-of-Sum and Sum-of-Max social objectives, which combine the classic utilitarian and egalitarian social cost measures. We also derive informational lower bounds for one-sided matching and clustering, two fundamental and well-studied problems captured by our model, that match our upper bounds exactly for Max-of-Sum and asymptotically for Sum-of-Max.
