We study the facility location problems where agents are located on a real line and divided into groups based on criteria such as ethnicity or age. Our aim is to design mechanisms to locate a facility to approximately minimize the costs of groups of agents to the facility fairly while eliciting the agents' locations truthfully. We first explore various well-motivated group fairness cost objectives for the problems and show that many natural objectives have an unbounded approximation ratio. We then consider minimizing the maximum total group cost and minimizing the average group cost objectives. For these objectives, we show that existing classical mechanisms (e.g., median) and new group-based mechanisms provide bounded approximation ratios, where the group-based mechanisms can achieve better ratios. We also provide lower bounds for both objectives. To measure fairness between groups and within each group, we study a new notion of intergroup and intragroup fairness (IIF) . We consider two IIF objectives and provide mechanisms with tight approximation ratios.

翻译：我们研究设施选址问题，其中代理位于实线上，并根据种族或年龄等标准划分为不同群体。我们的目标是设计机制，在真实获取代理位置信息的同时，将设施选址以近似公平地最小化各群体代理成本。首先，我们探讨了多种具有充分动机的群体公平成本目标，并发现许多自然目标具有无界的近似比。随后，我们考虑最小化最大群体总成本与最小化平均群体成本这两个目标。针对这些目标，我们证明现有经典机制（如中位数机制）与新型群体基机制能提供有界近似比，其中群体基机制可实现更优的近似比，同时给出了两个目标的下界。为衡量群体间与群体内的公平性，我们提出了一种新的群体间与群体内公平（IIF）概念，并针对两种IIF目标设计了具有紧近似比的机制。