Breaking the 4-Approximation Barrier in Strategyproof Two-Facility Location

Abstract

We study strategyproof mechanism design without transfers for the two-facility location problem in metric spaces. A mechanism selects two facility locations based on agents' reported locations; each agent incurs her distance to the nearer facility, and the objective is to minimize the expected social cost. A mechanism is strategyproof if no agent ever benefits from misreporting her location. The best approximation ratio achieved by a randomized strategyproof mechanism has been $4$, attained by the Proportional mechanism of Lu, Sun, Wang, and Zhu (EC 2010), and the best lower bound has been $1.045$, due to Lu, Wang, and Zhou (WINE 2009). Neither bound has moved since then, even on the line $\mathbb{R}$. We improve both bounds. Our main result is a randomized strategyproof mechanism with approximation ratio $11/3 \approx 3.667$ on every Ptolemaic metric space, a rich class containing all Euclidean spaces. The mechanism randomizes between the Proportional mechanism and a new mechanism that we call Global Pair. Global Pair draws an unordered pair of agents with probability proportional to their distance and opens facilities at their reported locations. Although Global Pair and Proportional each have approximation ratio $4$, the two mechanisms attain their worst-case approximation ratios on complementary instances. Randomizing between them balances these complementary weaknesses and breaks the $4$-approximation barrier. On the lower-bound side, we construct a new two-profile instance that yields a lower bound of $(1+\sqrt{2})/2 \approx 1.207$, improving upon the previous lower bound of $1.045$.