We consider $k$-Facility Location games, where $n$ strategic agents report their locations on the real line, and a mechanism maps them to $k\ge 2$ facilities. Each agent seeks to minimize her distance to the nearest facility. We are interested in (deterministic or randomized) strategyproof mechanisms without payments that achieve a reasonable approximation ratio to the optimal social cost of the agents. To circumvent the inapproximability of $k$-Facility Location by deterministic strategyproof mechanisms, we restrict our attention to perturbation stable instances. An instance of $k$-Facility Location on the line is $\gamma$-perturbation stable (or simply, $\gamma$-stable), for some $\gamma\ge 1$, if the optimal agent clustering is not affected by moving any subset of consecutive agent locations closer to each other by a factor at most $\gamma$. We show that the optimal solution is strategyproof in $(2+\sqrt{3})$-stable instances whose optimal solution does not include any singleton clusters, and that allocating the facility to the agent next to the rightmost one in each optimal cluster (or to the unique agent, for singleton clusters) is strategyproof and $(n-2)/2$-approximate for $5$-stable instances (even if their optimal solution includes singleton clusters). On the negative side, we show that for any $k\ge 3$ and any $\delta > 0$, there is no deterministic anonymous mechanism that achieves a bounded approximation ratio and is strategyproof in $(\sqrt{2}-\delta)$-stable instances. We also prove that allocating the facility to a random agent of each optimal cluster is strategyproof and $2$-approximate in $5$-stable instances. To the best of our knowledge, this is the first time that the existence of deterministic (resp. randomized) strategyproof mechanisms with a bounded (resp. constant) approximation ratio is shown for a large and natural class of $k$-Facility Location instances.

翻译：我们考虑$k$-设施选址博弈问题，其中$n$个策略性代理在实数线上报告其位置，机制将其映射到$k\ge 2$个设施。每个代理旨在最小化其到最近设施的距离。我们关注无需支付的（确定性或随机化）策略性机制，这些机制能够实现代理最优社会成本的合理近似比。为规避确定性策略性机制在$k$-设施选址问题中的不可近似性，我们将注意力限制在扰动稳定实例上。对于某个$\gamma\ge 1$，若将任意连续代理子集的位置以不超过$\gamma$的因子相互靠近后，最优代理聚类结果不变，则称该线型$k$-设施选址实例为$\gamma$-扰动稳定的（简称为$\gamma$-稳定）。我们证明：在最优解不含单点聚类的$(2+\sqrt{3})$稳定实例中，最优解本身具有策略性；对于$5$-稳定实例（即使其最优解包含单点聚类），将每个最优聚类中次右侧代理作为设施选址（单点聚类则选取唯一代理）的机制是策略性的，且近似比为$(n-2)/2$。在否定方面，我们证明：对于任意$k\ge 3$及$\delta>0$，不存在能在$(\sqrt{2}-\delta)$-稳定实例中实现有界近似比且满足策略性的确定性匿名机制。我们同时证明：在$5$-稳定实例中，从每个最优聚类中随机选取一个代理作为设施选址的机制是策略性的，且近似比为$2$。据我们所知，这是首次在$k$-设施选址问题的自然大类实例上，证明存在具有有界（或常数）近似比的确定性（或随机化）策略性机制。