Truthful Two-Facility Location with Candidate Locations in Metric Spaces

We study a truthful facility location problem in which agents have private positions in a metric space and known approval preferences over two distinct facilities. Each agent incurs a cost equal to her total distance from the facilities she approves. The goal is to locate the facilities at a given set of candidate locations so as to approximately minimize either the social cost or the max cost, while incentivizing the agents to report their positions truthfully. This problem was previously studied for line metrics and deterministic mechanisms. We first settle the approximation ratio of deterministic mechanisms for the social cost on the line by establishing a lower bound of $7$, matching the upper bound from previous work, and show that randomization leads to a strict improvement. For the max cost, we show that the previously known optimal deterministic guarantee of $3$ remains tight even for universally strategyproof mechanisms. We then turn to general metric spaces. For deterministic strategyproof mechanisms, we obtain tight approximation ratios of $2n-1$ for the social cost and $5$ for the max cost. We also show that randomization yields substantial improvements for both objectives: for the social cost, we obtain a tight characterization, while for the max cost we establish nearly matching upper and lower