MAC Advice for Facility Location Mechanism Design
arxiv(2024)
摘要
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 ε-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.
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