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.

翻译：在线度量匹配问题中，$n$个服务器与$n$个请求位于度量空间内。服务器位置预先确定，请求按序到达。每个到达的请求必须被即时且不可撤销地匹配至可用服务器，产生的成本等于两者间的距离。目标是最小化总匹配成本。本研究在欧几里得度量空间 $[0, 1]^d$ 中探讨该问题，其中服务器位置由对抗性设定，而请求则从满足温和平滑条件的独立分布中抽取。我们的核心成果是：对于 $d \neq 2$ 的情形，提出一种仅需从每个请求分布中获取单一样本、无需分布先验知识的 $O(1)$ 竞争比算法。据我们所知，这是在非独立同分布设定下，首个对非平凡度量空间实现 $o(\log n)$ 竞争比的算法。我们的方法绕过了概率度量嵌入引入的 $Ω(\log n)$ 障碍：不同于分别分析嵌入失真与算法性能，我们直接在简单确定性嵌入的目标度量上界定算法成本。随后，通过主序化论证推导欧几里得度量下离线最优解的下界，将二者结合以证明算法保证。