MAC Advice for Facility Location Mechanism Design

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 δ-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.