We consider the online metric matching problem in which we are given a metric space, k of whose points are designated as servers. Over time, up to k requests arrive at an arbitrary subset of points in the metric space, and each request must be matched to a server immediately upon arrival, subject to the constraint that at most one request is matched to any particular server. Matching decisions are irrevocable and the goal is to minimize the sum of distances between the requests and their matched servers. We give an O(log2 k)-competitive randomized algorithm for the online metric matching problem. This improves upon the best known guarantee of O(log3 k) on the competitive factor due to Meyerson, Nanavati and Poplawski (SODA ’06, pp. 954–959, 2006). It is known that for this problem no deterministic algorithm can have a competitive better than 2k-1, and that no randomized algorithm can have a competitive ratio better than lnk.
Bansal, N., Buchbinder, N., Gupta, A., & Naor, J. (2014). A randomized O ( log^2 k )-competitive algorithm for metric bipartite matching. Algorithmica, 68(2), 390-403. https://doi.org/10.1007/s00453-012-9676-9