Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion

TL;DR

This paper presents an $O(1)$-competitive algorithm for online metric matching in Euclidean spaces with adversarial servers and requests drawn from smooth distributions, using only a single sample per distribution.

Contribution

It introduces the first algorithm achieving an $o(\log n)$ competitive ratio for non-trivial metrics beyond the i.i.d. setting, bypassing traditional probabilistic embedding barriers.

Findings

Achieves $O(1)$-competitiveness for $d eq 2$ in Euclidean spaces.

Requires only a single sample from each request distribution.

Breaks the $\Omega(\log n)$ barrier in metric embedding analysis.

Abstract

In the online metric matching problem, $n$ servers and $n$ requests lie in a metric space. Servers are available upfront, and requests arrive sequentially. An arriving request must be matched immediately and irrevocably to an available server, incurring a cost equal to their distance. The goal is to minimize the total matching cost. We study this problem in the Euclidean metric $[0, 1]^d$, when servers are adversarial and requests are independently drawn from distinct distributions that satisfy a mild smoothness condition. Our main result is an $O(1)$-competitive algorithm for $d \neq 2$ that requires no distributional knowledge, relying only on a single sample from each request distribution. To our knowledge, this is the first algorithm to achieve an $o(\log n)$ competitive ratio for non-trivial metrics beyond the i.i.d. setting. Our approach bypasses the $\Omega(\log n)$ barrier introduced by probabilistic metric embeddings: instead of analyzing the embedding distortion and the algorithm separately, we directly bound the cost of the algorithm on the target metric of a simple deterministic embedding. We then combine this analysis with lower bounds on the offline optimum for Euclidean metrics, derived via majorization arguments, to obtain our guarantees.