Large graph limits of local matching algorithms on Configuration model graphs

TL;DR

This paper develops a large-graph limit framework for analyzing local matching algorithms on configuration model graphs, providing explicit asymptotic coverage results via a system of ODEs.

Contribution

It introduces a measure-valued CTMC approach and a hydrodynamic limit for local matching algorithms on large random graphs, extending the differential equation method.

Findings

Derived explicit formulas for matching coverage of specific algorithms.

Established convergence of the stochastic process to a deterministic limit.

Analyzed three algorithms: greedy, uni-min, and uni-max.

Abstract

In this work, we propose a large-graph limit estimate of the matching coverage for several matching algorithms, on general graphs generated by the configuration model. For a wide class of {\em local} matching algorithms, namely, algorithms that only use information on the immediate neighborhood of the explored nodes, we propose a joint construction of the graph by the configuration model, and of the resulting matching on the latter graph. This leads to a generalization in infinite dimension of the differential equation method of Wormald: We keep track of the matching algorithm over time by a measure-valued CTMC, for which we prove the convergence, to the large-graph limit, to a deterministic hydrodynamic limit, identified as the unique solution of a system of ODE's in the space of integer measures. Then, the asymptotic proportion of nodes covered by the matching appears as a simple function of that solution. We then make this solution explicit for three particular local algorithms: the classical {\sc greedy} algorithm, and then the {\sc uni-min} and {\sc uni-max} algorithms, two variants of the greedy algorithm that select, as neighbor of any explored node, its neighbor having the least (respectively largest) residual degree.