Online Matching with Stochastic Rewards: Optimal Competitive Ratio via Path Based Formulation
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The problem of online matching with stochastic rewards is a variant of the online bipartite matching problem where each edge has a probability of "success". When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical and vanishingly small edge probabilities and gave an online algorithm which is 0.567 competitive against a deterministic offline LP. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. We study a generalization of the problem to vertex-weighted graphs and compare against clairvoyant algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches. To the best of our knowledge, no results beating 1/2 were previously known for this setting, even for identical probabilities. By introducing a novel path-based formulation, we show that a natural variant of the RANKING algorithm achieves the best possible competitive ratio of (1-1/e), for heterogeneous but vanishingly small edge probabilities. Our result also holds for non-vanishing probabilities that decompose as a product of two factors, one corresponding to each vertex of the edge. The idea of a path-based program may be of independent interest in other online matching problems with a stochastic component.
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