A Robust and Optimal Online Algorithm for Minimum Metric Bipartite Matching

Sharath Raghvendra
We study the Online Minimum Metric Bipartite Matching Problem. In this problem, we are given point sets S and R which correspond to the server and request locations; here |S| = |R| = n. All these locations are points from some metric space and the cost of matching a server to a request is given by the distance between their locations in this space. In this problem, the request points arrive one at a time. When a request arrives, we must immediately and irrevocably match it to a "free" server.
more » ... e matching obtained after all the requests are processed is the online matching M. The cost of M is the sum of the cost of its edges. The performance of any online algorithm is the worst-case ratio of the cost of its online solution M to the minimum-cost matching. We present a deterministic online algorithm for this problem. Our algorithm is the first to simultaneously achieve optimal performances in the well-known adversarial and the random arrival models. For the adversarial model, we obtain a competitive ratio of 2n − 1 + o(1); it is known that no deterministic algorithm can do better than 2n − 1. In the random arrival model, our algorithm obtains a competitive ratio of 2H n −1+o(1); where H n is the nth Harmonic number. We also prove that any online algorithm will have a competitive ratio of at least 2H n − 1 − o(1) in this model. We use a new variation of the offline primal-dual method for computing minimum cost matching to compute the online matching. Our primal-dual method is based on a relaxed linear-program. Under metric costs, this specific relaxation helps us relate the cost of the online matching with the offline matching leading to its robust properties.