Online Matching in Sparse Random Graphs: Non-Asymptotic Performances of Greedy Algorithm

Part of Advances in Neural Information Processing Systems 34 pre-proceedings (NeurIPS 2021)

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Nathan Noiry, Vianney Perchet, Flore Sentenac


Motivated by sequential budgeted allocation problems, we investigate online matching problems where connections between vertices are not i.i.d., but they have fixed degree distributions -- the so-called configuration model. We estimate the competitive ratio of the simplest algorithm, GREEDY, by approximating some relevant stochastic discrete processes by their continuous counterparts, that are solutions of an explicit system of partial differential equations. This technique gives precise bounds on the estimation errors, with arbitrarily high probability as the problem size increases. In particular, it allows the formal comparison between different configuration models. We also prove that, quite surprisingly, GREEDY can have better performance guarantees than RANKING, another celebrated algorithm for online matching that usually outperforms the former.