Algorithms with predictions have attracted much attention in the last years across various domains, including variants of facility location, as a way to surpass traditional worst-case analyses. We study the $k$-facility location mechanism design problem, where the $n$ agents are strategic and might misreport their location. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. However, these predictions are only "mostly" and "approximately" correct (or MAC for short) -- i.e., some $\delta$-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are allowed to be correct up to an $\varepsilon$-error. We make no assumption on the independence of the errors. Can such predictions allow us to beat the current best bounds for strategyproof facility location? We show that the $1$-median (geometric median) of a set of points is naturally robust under corruptions, which leads to an algorithm for single-facility location with MAC predictions. We extend the robustness result to a "balanced" variant of the $k$ facilities case. Without balancedness, we show that robustness completely breaks down, even for the setting of $k=2$ facilities on a line. For this "unbalanced" setting, we devise a truthful random mechanism that outperforms the best known result of Lu et al. [2010], which does not use predictions. En route, we introduce the problem of "second" facility location (when the first facility's location is already fixed). Our findings on the robustness of the $1$-median and more generally $k$-medians may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.

翻译：近年来，带有预测的算法作为超越传统最坏情况分析的方法，在包括设施定位变体在内的多个领域引起了广泛关注。本文研究k-设施定位机制设计问题，其中n个智能体具有策略性，可能虚报自身位置。与现有模型中预测k个最优设施位置不同，我们接收针对每个智能体位置的n个预测。但这些预测仅具有"大致"与"近似"正确性（简称MAC）——即允许预测位置中有δ比例完全错误，其余预测可容忍ε误差。我们不对误差独立性做任何假设。此类预测能否帮助我们突破策略性设施定位的现有最优界限？研究表明，点集的1-中位数（几何中位数）天然具有抗干扰鲁棒性，由此可提出适用于MAC预测的单设施定位算法。我们将该鲁棒性结果推广至k设施情形的"均衡"变体。若不满足均衡性，即使在线上的2设施情形中，鲁棒性也会完全失效。针对此非均衡情形，我们设计了一种诚实随机机制，其性能优于Lu等人[2010]未使用预测的最优已知结果。在此过程中，我们引入了"第二"设施定位问题（当第一设施位置已固定时）。本文关于1-中位数及更一般k-中位数鲁棒性的发现，可作为鲁棒统计中经典崩溃点定理的量化版本，具有独立研究价值。