Learning-Augmented Mechanism Design for Facility Location under $L_p$-Norm Social Costs

We consider learning-augmented mechanism design for facility location in the standard setting of locating a single facility on the line. Leveraging the given (imperfect) prediction on the optimal facility location, our goal is to design strategyproof (SP) mechanisms that truthfully elicit agent location preferences and determine facility locations that approximately minimize the $L_p$-norm social cost for $p \in (1, +\infty)$, achieving high \emph{consistency} (i.e., approximation ratios when the prediction is correct) and \emph{robustness} (i.e., approximation ratios when the prediction is incorrect). For deterministic SP mechanisms, we propose a family of generalized median mechanisms parameterized by a fraction \(t \in (0,1)\) of phantom points placed at the predicted location \(\pi\). We show that this mechanism achieves $\big(1+\big(\frac{1-t}{1+t}\big)^{\frac{1}{p-1}}\big)^{\frac{p-1}{p}}$-consistency and $\big(1+\big(\frac{1+t}{1-t}\big)^{\frac{1}{p-1}}\big)^{\frac{p-1}{p}}$-robustness and that no deterministic SP mechanism with the same consistency can obtain a better robustness. For randomized SP mechanisms, we analyze the upper bounds for several special cases, including (1) two-agent and (2) the squared cost settings, and provide a lower bound on consistency-robustness tradeoffs.