An O (log²k)-Competitive Algorithm for Metric Bipartite Matching

We consider the online metric matching problem. In this problem, we are given a graph with edge weights satisfying the triangle inequality, and k vertices that are designated as the right side of the matching. Over time up to k requests arrive at an arbitrary subset of vertices in the graph and each vertex must be matched to a right side vertex immediately upon arrival. A vertex cannot be rematched to another vertex once it is matched. The goal is to minimize the total weight of the matching. We give a O(log2 k) competitive randomized algorithm for the problem. This improves upon the best known guarantee of O(log3 k) due to Meyerson, Nanavati and Poplawski [19] . It is well known that no deterministic algorithm can have a competitive less than 2k − 1, and that no randomized algorithm can have a competitive ratio of less than ln k.