Learning-Augmented Facility Location Mechanisms for Envy Ratio

TL;DR

This paper introduces learning-augmented mechanisms for facility location on a line, optimizing fairness via envy ratio, and demonstrates improved theoretical guarantees and new randomized approaches leveraging predictions.

Contribution

It proposes the $ extalpha$-Bounding Interval Mechanism with optimal bounds, resolves open questions with a new randomized mechanism, and introduces the Bias-Aware Mechanism leveraging predictions for better guarantees.

Findings

The $ extalpha$-BIM achieves optimal consistency and robustness bounds.

A randomized mechanism improves approximation ratio from 2 to ~1.8944.

The Bias-Aware Mechanism enhances fairness guarantees using predictions.

Abstract

The augmentation of algorithms with predictions of the optimal solution, such as from a machine-learning algorithm, has garnered significant attention in recent years, particularly in facility location problems. Moving beyond the traditional focus on utilitarian and egalitarian objectives, we design learning-augmented facility location mechanisms on a line for the envy ratio objective, a fairness metric defined as the maximum ratio between the utilities of any two agents. For the deterministic setting, we propose the $\alpha$-Bounding Interval Mechanism ($\alpha$-BIM), which utilizes predictions to achieve $\alpha$-consistency and $\frac{\alpha}{\alpha - 1}$-robustness for a selected parameter $\alpha \in [1,2]$, and prove its optimality. We also resolve open questions raised by Ding et al. [10], devising a randomized mechanism without predictions to improve upon the best-known approximation ratio from $2$ to approximately $1.8944$. Building upon these advancements, we construct a novel randomized mechanism, the Bias-Aware Mechanism (BAM), which incorporates predictions to achieve improved consistency and robustness guarantees.