MAC Advice for Facility Location Mechanism Design

Algorithms with predictions are gaining traction across various domains, as a way to surpass traditional worst-case bounds through (machine-learned) advice. We study the canonical problem of k-facility location mechanism design, where the n agents are strategic and might misreport their locations. We receive a prediction for each agent's location, and these predictions are crucially allowed to be only “mostly” and “approximately” correct (MAC for short): a δ-fraction of the predicted locations are allowed to be arbitrarily incorrect, and the remainder of the predictions are required to be correct up to an ε-error. Moreover, we make no assumption on the independence of the errors. Can such “flawed” predictions allow us to beat the current best bounds for strategyproof facility location? We show how natural robustness of the 1-median (also known as the geometric median) of a set of points leads to an algorithm for single-facility location with MAC predictions. We extend our results to a natural “balanced” variant of the k-facility case, and show that without balancedness, robustness completely breaks down even for k = 2 facilities on a line. As our main result, for this “unbalanced” setting we devise a truthful random mechanism, which outperforms the best known mechanism (with no predictions) by Lu et al. [2010]. En route, we introduce the problem of “second” facility location, in which the first facility location is already fixed. Our robustness findings may be of independent interest, as quantitative versions of classic breakdown-point results in robust statistics.