Smoothed Analysis of Online Metric Matching with a Single Sample: Beyond Metric Distortion
Yingxi Li, Ellen Vitercik, Mingwei Yang
TL;DR
This work studies online metric matching with adversarial servers and smoothed, independent request distributions in the Euclidean unit cube, using only one sample per distribution. By embedding the space into a simple 2-HST via a canonical dyadic partition and analyzing the RS and BBGN algorithms directly on the tree, it bypasses the conventional embedding distortion barrier and achieves $O(1)$-competitive performance for $d \neq 2$. The analysis combines Poisson–Binomial fluctuation bounds with majorization-based lower bounds on the offline optimum and leverages a sample-to-semi reduction to integrate the single-sample access model. The results reveal dimension-dependent behavior (notably the critical role of $d=2$) and open avenues for extending smoothed-analysis techniques to correlated arrivals and related online matching problems.
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 $Ω(\log n)$ 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.