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.

翻译：近年来，利用最优解的预测（例如来自机器学习算法）来增强算法性能的方法受到广泛关注，尤其在设施选址问题中。本文超越了传统对功利主义与平等主义目标的关注，针对嫉妒比目标——一种定义为任意两个智能体效用比最大值的公平性度量——设计了直线上的学习增强型设施选址机制。在确定性场景下，我们提出了$α$-边界区间机制（$α$-BIM），该机制利用预测实现选定参数$α\in [1,2]$对应的$α$-一致性与$\fracα{α- 1}$-鲁棒性，并证明了其最优性。同时，我们解决了Ding等人[10]提出的开放性问题，设计了一种无需预测的随机机制，将最佳已知近似比从$2$提升至约$1.8944$。基于这些进展，我们构建了一种新颖的随机机制——偏置感知机制（BAM），该机制通过整合预测实现了更优的一致性与鲁棒性保证。