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

TL;DR

This paper analyzes the performance of the GREEDY algorithm in online matching problems on sparse random graphs with fixed degree distributions, providing precise bounds and surprising performance insights.

Contribution

It introduces a PDE-based approximation method to estimate the competitive ratio of GREEDY in configuration models, enabling formal comparison with other algorithms.

Findings

GREEDY can outperform RANKING in certain sparse graph models.

The PDE approach yields high-probability bounds on algorithm performance.

Performance guarantees are derived for large-scale sparse random graphs.

Abstract

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.