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

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, GREEDYcan have better performance guarantees than RANKING, another celebrated algorithm for online matching that usually outperforms the former. This theoretical setting is particularly well suited for online advertising: U is the set of campaigns/ads that an advertiser can run and users v 1 , v 2 , . . . , v T arrive sequentially [Mehta, 2012 , Manshadi et al., 2012] . Some of them are eligible for a large subset of campaigns, others are not (usually based Preprint. Under review.