Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line
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In the online metric bipartite matching problem, we are given a set S of server locations in a metric space. Requests arrive one at a time, and on its arrival, we need to immediately and irrevocably match it to a server at a cost which is equal to the distance between these locations. A α-competitive algorithm will assign requests to servers so that the total cost is at most α times the cost of M_OPT where M_OPT is the minimum cost matching between S and R. We consider this problem in the adversarial model for the case where S and R are points on a line and |S|=|R|=n. We improve the analysis of the deterministic Robust Matching Algorithm (RM-Algorithm, Nayyar and Raghvendra FOCS'17) from O(^2 n) to an optimal Θ( n). Previously, only a randomized algorithm under a weaker oblivious adversary achieved a competitive ratio of O( n) (Gupta and Lewi, ICALP'12). The well-known Work Function Algorithm (WFA) has a competitive ratio of O(n) and Ω( n) for this problem. Therefore, WFA cannot achieve an asymptotically better competitive ratio than the RM-Algorithm.
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