Markovian online matching algorithms on large bipartite random graphs

TL;DR

This paper develops a differential equation-based approximation method for analyzing local online matching algorithms on large bipartite graphs, demonstrating accuracy through simulations and comparing algorithm performances.

Contribution

It introduces a novel measure-valued process and ODE system to approximate matching coverage, simplifying analysis of local online algorithms on large bipartite graphs.

Findings

Accurate approximation of matching coverage using ODEs.

Comparison shows the minimal residual degree algorithm outperforms classical greedy matching.

Simulation results validate the theoretical model.

Abstract

In this paper, we present an approximation of the matching coverage on large bipartite graphs, for {\em local} online matching algorithms based on the sole knowledge of the remaining degree of the nodes of the graph at hand. This approximation is obtained by applying the Differential Equation Method to a measure-valued process representing an alternative construction, in which the matching and the graph are constructed simultaneously, by a uniform pairing leading to a realization of the bipartite Configuration Model. The latter auxiliary construction is shown to be equivalent in distribution to the original one. It allows to drastically reduce the complexity of the problem, in that the resulting matching coverage can be written as a simple function of the final value of the process, and in turn, approximated by a simple function of the solution of a system of ODE's. By way of simulations, we illustrate the accuracy of our estimate, and then compare the performance of an algorithm based on the minimal residual degree of the nodes, to the classical greedy matching.