In the strategic facility location problem, a set of agents report their locations in a metric space and the goal is to use these reports to open a new facility, minimizing an aggregate distance measure from the agents to the facility. However, agents are strategic and may misreport their locations to influence the facility's placement in their favor. The aim is to design truthful mechanisms, ensuring agents cannot gain by misreporting. This problem was recently revisited through the learning-augmented framework, aiming to move beyond worst-case analysis and design truthful mechanisms that are augmented with (machine-learned) predictions. The focus of this prior work was on mechanisms that are deterministic and augmented with a prediction regarding the optimal facility location. In this paper, we provide a deeper understanding of this problem by exploring the power of randomization as well as the impact of different types of predictions on the performance of truthful learning-augmented mechanisms. We study both the single-dimensional and the Euclidean case and provide upper and lower bounds regarding the achievable approximation of the optimal egalitarian social cost.

翻译：在策略性设施选址问题中，一组代理报告其在度量空间中的位置，目标是根据这些报告开设一个新设施，以最小化代理到该设施的聚合距离度量。然而，代理具有策略性，可能虚报其位置以使设施选址对自身有利。目标在于设计真实机制，确保代理无法通过虚报获益。近期，该问题通过学习增强框架被重新审视，旨在超越最坏情况分析，设计结合（机器学习）预测的真实机制。先前工作的重点在于确定性机制，并辅以关于最优设施位置的预测。本文通过探索随机化的能力以及不同类型预测对真实学习增强机制性能的影响，对该问题提供更深入的理解。我们研究单维情形与欧几里得情形，并就可达的最优平等主义社会成本近似比给出上界与下界。