MAC Advice for Facility Location Mechanism Design

• URL: http://arxiv.org/abs/2403.12181v1
• Date: Mon, 18 Mar 2024 18:52:04 GMT
• Title: MAC Advice for Facility Location Mechanism Design
• Authors: Zohar Barak, Anupam Gupta, Inbal Talgam-Cohen,
• Abstract summary: We study the $k$-facility location mechanism design problem. Unlike previous models, where predictions are for the $k$ optimal facility locations, we receive $n$ predictions for the locations of each of the agents. 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.
• Score: 12.136427429093395
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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