We study the stochastic online metric matching problem. In this problem, $m$ servers and $n$ requests are located in a metric space, where all servers are available upfront and requests arrive one at a time. In particular, servers are adversarially chosen, and requests are independently drawn from a known distribution. Upon the arrival of a new request, it needs to be immediately and irrevocably matched to a free server, resulting in a cost of their distance. The objective is to minimize the total matching cost. In this paper, we show that the problem can be reduced to a more accessible setting where both servers and requests are drawn from the same distribution by incurring a moderate cost. Combining our reduction with previous techniques, for $[0, 1]^d$ with various choices of distributions, we achieve improved competitive ratios and nearly optimal regrets in both balanced and unbalanced markets. In particular, we give $O(1)$-competitive algorithms for $d \geq 3$ in both balanced and unbalanced markets with smooth distributions. Our algorithms improve on the $O((\log \log \log n)^2)$ competitive ratio of \cite{DBLP:conf/icalp/GuptaGPW19} for balanced markets in various regimes, and provide the first positive results for unbalanced markets.

翻译：本文研究随机在线度量匹配问题。在该问题中，$m$ 个服务器与 $n$ 个请求位于度量空间中，所有服务器预先就位，请求按序到达。特别地，服务器位置由对抗性方式选定，而请求位置则从已知分布中独立抽取。当新请求到达时，需立即且不可撤销地将其匹配至空闲服务器，产生的成本为两者间的距离。目标是最小化总匹配成本。本文证明，通过承担适度成本，该问题可归约为更易处理的情形——即服务器与请求均从同一分布中抽取。结合现有技术，针对 $[0, 1]^d$ 空间中的多种分布选择，我们在平衡与非平衡市场中均获得了改进的竞争比与近乎最优的遗憾值。特别地，对于 $d \geq 3$ 且具有平滑分布的情形，我们给出了适用于平衡与非平衡市场的 $O(1)$ 竞争比算法。该算法在多类场景下改进了 \cite{DBLP:conf/icalp/GuptaGPW19} 针对平衡市场提出的 $O((\log \log \log n)^2)$ 竞争比，并为非平衡市场提供了首个正向结果。