We study the mechanism design problem of facility location on a metric space in the learning-augmented framework, where mechanisms have access to imperfect predictions of the optimal facility locations. Our objective is to design strategyproof (SP) mechanisms that truthfully elicit agents' preferences over facility locations and, using the given prediction, select a facility location that approximately minimizes the maximum cost among all agents. In particular, we seek SP mechanisms whose approximation guarantees depend on the prediction error: they should achieve improved performance when the prediction is accurate (the property of \emph{consistency}) while still ensuring strong worst-case guarantees when the prediction is arbitrarily inaccurate (the property of \emph{robustness}). On the real line, we characterize all deterministic SP mechanisms with consistency strictly better than 2 and bounded robustness for the maximum cost. We show that any such mechanism must coincide with the MinMaxP mechanism, which returns the prediction if it lies between the two extreme agent locations and otherwise returns the agent location closest to the prediction. For any prediction error $η\ge 0$, we prove that MinMaxP achieves a $(1+\min(1, η))$-approximation and that no deterministic SP mechanism can obtain a better approximation ratio. In addition, for two-dimensional spaces with the $\ell_p$ distance, we analyze the approximation guarantees of a deterministic mechanism that applies MinMaxP independently on each coordinate, as well as a randomized mechanism that selects between two deterministic mechanisms with carefully chosen probabilities. We further extend these results to the $L_p$-norm social cost objective on the line metric and the maximum cost objective on the tree metric. Finally, we examine the group strategyproofness of the mechanisms.

翻译：我们研究了在增强学习框架下的度量空间设施选址的机制设计问题，其中机制能够获取关于最优设施位置的不完美预测。我们的目标是设计防策略性（SP）机制，该机制能真实地诱导出代理人关于设施选址的偏好，并利用给定的预测选择一个近似最小化所有代理人中最大成本的设施位置。具体而言，我们寻求近似保证依赖于预测误差的防策略性机制：当预测准确时，这些机制能实现改进的性能（即一致性属性），同时当预测任意不准确时，仍能确保强的最坏情况保证（即鲁棒性属性）。在实数轴上，我们刻画了所有具有严格优于2的一致性和有界鲁棒性的确定性防策略性机制，其针对最大成本目标。我们证明了任何此类机制必须等同于MinMaxP机制，该机制在预测位于两个极端代理人位置之间时返回预测值，否则返回最接近预测值的代理人位置。对于任意预测误差$η\ge 0$，我们证明了MinMaxP达到$(1+\min(1, η))$近似比，且没有任何确定性防策略性机制能获得更好的近似比。此外，对于具有$\ell_p$距离的二维空间，我们分析了一种在每个坐标上独立应用MinMaxP的确定性机制以及一种以精心选择的概率在两个确定性机制之间进行选择的随机机制的近似保证。我们进一步将这些结果扩展到直线度量上的$L_p$范数社会成本目标和树度量上的最大成本目标。最后，我们检验了这些机制的抗群体操纵性。