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：研究了一个诚实双设施选址问题，其中一组代理拥有实数直线上的私人位置，并对两个设施具有已知的认可偏好。给定两个设施的位置，代理的成本是其认可的设施总距离。目标是从给定的有限候选位置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人位置。我们聚焦于确定性防策略机制类，并找出在总成本（即所有代理的成本之和）和最大成本方面具有最佳可能近似比的机制。具体来说，对于总成本，当代理偏好同质（即所有代理都认可两个设施）时，我们证明了紧界为$1+\sqrt{2}$；当偏好可能异质时，紧界为$3$。对于最大成本，我们分别证明了同质和异质偏好下的紧界为$2$和$3$。