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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/k2 + 1/n) -- in particular Ranking achieves a factor of 1 - o(1) for graphs with ω(1) disjoint perfect matchings.
Karande et al. (Mon,) studied this question.