Optimal Metric Distortion for Learning-Augmented Matching on the Line

Abstract

We revisit the problem of matching on the line with ordinal preferences. In the classic setting, there are $n$ agents and $n$ items in a shared unknown line metric, and the goal is to find a low-cost perfect matching using only the agents' rankings of the items by distance. A mechanism has distortion $\alpha$ if it always outputs a matching whose cost is within a factor of $\alpha$ of the optimum, in every consistent line metric. In the learning-augmented setting, the mechanism is also supplied with a prediction that conveys additional information about the instance. The quality of this prediction is unknown, and the goal is to optimize the mechanism's distortion when the prediction is accurate (consistency), while preserving worst-case guarantees when the prediction is arbitrarily inaccurate (robustness). We propose a mechanism that takes a matching as its prediction and guarantees $1$-consistency and $3$-robustness. By recovering an optimal matching when the prediction is perfectly accurate while retaining the optimal prediction-free distortion guarantee when it is arbitrarily inaccurate, we resolve an open question of Filos-Ratsikas et al. (IJCAI, 2025).