An Input Sensitive Online Algorithm for the Metric Bipartite Matching Problem

We present a novel input sensitive analysis of a deterministic online algorithm [1] for the minimum metric bipartite matching problem. We show that, in the adversarial model, for any metric space M and a set of n servers S, the competitive ratio of this algorithm is Omicron((mu M)(S) log(2) n); here (mu M)(S) is the maximum ratio of the traveling salesman tour and the diameter of any subset of S. It is straight-forward to show that any algorithm, even with complete knowledge of M and S, will have a competitive ratio of Omega((mu M)(S)). So, the performance of this algorithm is sensitive to the input and near-optimal for any given S and M. As consequences, we also achieve the following results: If S is a set of points on a line, then (mu M)(S) = Theta(1) and the competitive ratio is Omicron(log(2) n), and, If S is a set of points spanning a subspace with doubling dimension d, then (mu M)(S) = Omicron(n(1-1/d)) and the competitive ratio is Omicron(n(1-1/d) log(2) n). Prior to this result, the previous best-known algorithm for the line metric has a competitive ratio of Omicron(n(0.59)) and requires both S and the request set R to be on a line. There is also an Omicron(log n) competitive algorithm in the weaker oblivious adversary model. To obtain our results, we partition the requests into well-separated clusters and replace each cluster with a small and a large weighted ball; the weight of a ball is the number of requests in the cluster. We show that the cost of the online matching can be expressed as the sum of the weight times radius of the smaller balls. We also show that the cost of edges of the optimal matching inside each larger ball can be shown to be proportional to the weight times the radius of the larger ball. We then use a simple variant of the well-known Vitali's covering lemma to relate the radii of these balls and obtain the competitive ratio.