Learning-Augmented Facility Location Mechanisms for Envy Ratio

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 $α$-Bounding Interval Mechanism ($α$-BIM), which utilizes predictions to achieve $α$-consistency and $\fracα{α- 1}$-robustness for a selected parameter $α\in [1,2]$, 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.

💡 Research Summary

This paper studies the one‑dimensional facility location problem under the envy‑ratio objective, a fairness measure defined as the maximum ratio between the utilities of any two agents. Unlike the classic worst‑off (maximum distance) objective, the envy ratio captures pairwise relative welfare, making it a more sensitive indicator of inequality. The authors adopt the learning‑augmented algorithmic paradigm: mechanisms receive, in addition to the agents’ reported locations, a prediction of the optimal facility location (e.g., produced by a machine‑learning model trained on historical data). The goal is to design strategy‑proof, anonymous mechanisms that achieve good “best‑of‑both‑worlds” guarantees—high consistency when the prediction is accurate and strong robustness when it is not.

Deterministic Mechanism (α‑Bounding Interval Mechanism, α‑BIM).
The authors introduce a tunable parameter α∈