Online Matching in Sparse Random Graphs: Non-Asymptotic Performances of Greedy Algorithm
This work addresses sequential budgeted allocation problems for researchers in algorithms and stochastic processes, providing incremental insights by comparing algorithms in non-i.i.d. settings.
The paper tackles the online matching problem in sparse random graphs with fixed degree distributions, showing that the GREEDY algorithm achieves a competitive ratio that can be formally bounded and, surprisingly, outperforms the RANKING algorithm in certain scenarios.
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.