Random-link matching problems on random regular graphs

TL;DR

This paper investigates the asymptotic behavior and finite-size effects of random-link matching problems on random regular graphs, using the cavity method and numerical simulations to analyze optimal costs and topological influences.

Contribution

It introduces a cavity-based analysis of the matching problem on random regular graphs, including fractional variants and finite-size corrections, expanding understanding beyond fully-connected models.

Findings

Estimated asymptotic average optimal cost

Analyzed finite-size corrections due to topological structures

Compared cavity method results with numerical simulations

Abstract

We study the random-link matching problem on random regular graphs, alongside with two relaxed versions of the problem, namely the fractional matching and the so-called "loopy" fractional matching. We estimated the asymptotic average optimal cost using the cavity method. Moreover, we also study the finite-size corrections due to rare topological structures appearing in the graph at large sizes. We estimate these contributions using the cavity approach, and we compare our results with the output of numerical simulations. The analysis also clarifies the meaning of the finite-size contributions appearing in the fully-connected version of the problem, that has been already analyzed in the literature.