We study the following vertex-weighted online bipartite matching problem: G(U, V, E) is a bipartite graph. The vertices in U have weights and are known ahead of time, while the vertices in V arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in U . When all the weights are equal, this reduces to the classic online bipartite matching problem for which Karp, Vazirani and Vazirani gave an optimal 1 −

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We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 − 1/e 'barrier' in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k > 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 − 1 k − 1 k 2 + 1 n -in particular Ranking achieves a factor of 1 − o(1) for graphs with ω(1) disjoint perfect matchings.

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