Strategyproof Facility Location in Perturbation Stable Instances

We study the approximability of k-Facility Location games on the real line by strategyproof mechanisms without payments. To circumvent impossibility results for k >= 3, we focus on gamma-(perturbation) stable instances, where the optimal agent clustering is not affected by moving any subset of consecutive agent locations closer to each other by a factor at most gamma >= 1. We show that the optimal solution is strategyproof in (2+root 3)-stable instances, if it does not include any singleton clusters, and that allocating the facility to the agent next to the rightmost one in each optimal cluster is strategyproof and (n - 2)/2-approximate for 5-stable instances (even if singleton clusters are present), where n is the number of agents. On the negative side, we show that for any k >= 3 and any delta > 0, deterministic anonymous strategyproof mechanisms suffer an unbounded approximation ratio in (root 2 - delta)-stable instances. Moreover, we prove that allocating the facility to a random agent of each optimal cluster is strategyproof and 2-approximate in 5-stable instances.