We study a truthful two-facility location problem in which a set of agents have private positions on the line of real numbers and known approval preferences over two facilities. Given the locations of the two facilities, the cost of an agent is the total distance from the facilities she approves. The goal is to decide where to place the facilities from a given finite set of candidate locations so as to (a) approximately optimize desired social objectives, and (b) incentivize the agents to truthfully report their private positions. We focus on the class of deterministic strategyproof mechanisms and pinpoint the ones with the best possible approximation ratio in terms of the social cost (i.e., the total cost of the agents) and the max cost. In particular, for the social cost, we show a tight bound of $1+\sqrt{2}$ when the preferences of the agents are homogeneous (i.e., all agents approve both facilities), and a tight bound of $3$ when the preferences might be heterogeneous. For the max cost, we show tight bounds of $2$ and $3$ for homogeneous and heterogeneous preferences, respectively.

翻译：我们研究了一个真实双设施选址问题，其中一组代理在实数线上拥有私有位置，并且对两个设施有已知的批准偏好。给定两个设施的位置，代理的成本是其批准的设施的总距离。目标是从给定的有限候选地点集合中决定设施的位置，以（a）近似优化期望的社会目标，以及（b）激励代理如实报告其私有位置。我们聚焦于确定性策略证明机制，并找出在社会成本（即代理的总成本）和最大成本方面具有最佳近似比的机制。特别地，对于社会成本，当代理的偏好是同质的（即所有代理都批准两个设施）时，我们展示了紧界为$1+\sqrt{2}$，而当偏好可能异质时，紧界为$3$。对于最大成本，我们在同质和异质偏好下分别展示了紧界为$2$和$3$。