We study a distributed facility location problem in which a set of agents, each with a private position on the real line, is partitioned into a collection of fixed, disjoint groups. The goal is to open $k$ facilities at locations chosen from the set of positions reported by the agents. This decision is made by mechanisms that operate in two phases. In Phase 1, each group selects the position of one of its agents to serve as the group's representative location. In Phase 2, $k$ representatives are chosen as facility locations. Once the facility locations are determined, each agent incurs an individual cost, defined either as the sum of its distances to all facilities (sum-variant) or as the distance to its farthest facility (max-variant). We focus on the class of strategyproof mechanisms, which preclude the agents from benefiting through strategic misreporting, and establish tight bounds on the approximation ratio with respect to the social cost (the total individual agent cost) in both variants.

翻译：本文研究一个分布式设施选址问题，其中一组位于实轴私有位置的代理被划分为若干固定且互不相交的群体。目标是从代理报告的位置集合中选择$k$个位置开设设施。该决策通过分两阶段运行的机制实现：第一阶段，每个群体从其代理中选择一个位置作为该群体的代表位置；第二阶段，从所有代表位置中选取$k$个作为设施选址点。设施位置确定后，每个代理将产生个体成本，其定义分为两种变体：到所有设施距离之和（求和变体）或到最远设施的距离（最大变体）。我们聚焦于策略证明机制类，这类机制能防止代理通过策略性虚报位置获益，并针对两种变体在社会成本（即所有代理个体成本之和）上的近似比建立了紧确界。