Strategyproof Facility Location in Perturbation Stable Instances

Dimitris Fotakis, Panagiotis Patsilinakos

Abstract: We study the approximability of k-Facility Location games on the real line by strategyproof mechanisms without payments. To circumvent impossibility results for $$k \ge 3$$ , 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 $$\gamma \ge 1$$ . We show that the optimal solution is strategyproof in $$(2+\sqrt{3})$$ -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 $$k \ge 3$$ and any $$\delta > 0$$ , deterministic anonymous strategyproof mechanisms suffer an unbounded approximation ratio in $$(\sqrt{2}-\delta )$$ -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.