Optimal Unimodular Matching

TL;DR

This paper establishes the convergence of maximum weight matchings in large random graphs to a unique optimal matching on the limiting unimodular tree, using a novel message passing characterization.

Contribution

It introduces the concept of unimodular matchings on weighted trees and proves the convergence and uniqueness of the optimal matching law.

Findings

Maximal weight matchings converge locally to a unique law on the limiting tree.

The optimal matching law is characterized by a message passing equation.

The results hold for atomless weights on unimodular random trees.

Abstract

We consider sequences of finite weighted random graphs that converge locally to unimodular i.i.d. weighted random trees. When the weights are atomless, we prove that the matchings of maximal weight converge locally to a matching on the limiting tree. For this purpose, we introduce and study unimodular matchings on weighted unimodular random trees as well as a notion of optimality for these objects. In this context, we prove that, in law, there is a unique optimal unimodular matching for a given unimodular tree. We then prove that this law is the local limit of the sequence of matchings of maximal weight. Along the way, we also show that this law is characterised by an equation derived from a message passing algorithm.