Product Gap Mechanisms for Multi-Facility Location

We study randomized strategyproof mechanisms for locating multiple facilities on the real line. We introduce the \emph{Product-Gap mechanism}, which selects $k$ reported locations with probability proportional to the product of the consecutive gaps between them and opens facilities at the selected locations. We prove that, for every $k\ge 2$, the mechanism achieves a tight approximation ratio of $2k$ for social cost. We then study its incentive properties and show that it is strategyproof in expectation for $k=2$ and $k=3$, but is not strategyproof for $k\ge 4$. In particular, the mechanism gives strategyproof $4$- and $6$-approximations for two and three facilities, respectively. Finally, for two facilities, we combine Product-Gap with the Proportional mechanism of Lu et al. We show that an optimized report-independent mixture is strategyproof and has a tight approximation ratio of $(74+4\sqrt{3})\mathrm{/}23\approx 3.519$ on the line, improving upon the previous factor of $4$.