Generalized Lower Bounds for Online Matching on the Line

The online matching on the line problem is a notorious special case of the online metric matching problem. The task is to match a set of requests R online to a given set of servers S. The distance metric between all points in R and S is a line metric and the objective for the online algorithm is to minimize the sum of distances between matched server-request pairs.For the more general problem, where the underlying metric is not restricted to the line, the best possible competitive ratio is 2n-1 and there are several different algorithms that achieve this ratio [5, 4, 7]. These algorithms are closely related to the offline algorithm for weighted, bipartite matching and use a scoring function on alternating paths to guide their decision.