Optimal matching under size priority

TL;DR

We address the problem of optimal matching under size priority on sparse edge-weighted graphs by introducing optimal unimodular matchings on unimodular BGW trees with i.i.d. weights $\omega$ and a lexicographic objective $\mathrm{perf_E}$. The core method combines belief propagation with a renormalisation that splits edge messages into a macroscopic component taking values in $\{0,\tfrac12,1\}$ and a microscopic component, enabling a rigorous construction of the unique unimodular optimal matching $\mathbb{M}_{\mathrm{opt}}$ and its convergence properties. We prove existence/uniqueness on the limiting tree, show local convergence of finite weighted graphs to the limiting optimal matching, and establish a subcritical regime where edge-state correlations decay exponentially, yielding stronger convergence and explicit densities for mandatory and blocking edges. The results unify and extend classic Karp–Sipser-type phenomena within a rigorous unimodular framework and provide tractable, local-function-based descriptions of the limiting optimal structures. This framework opens avenues for precise density calculations and robust approximations of optimal matchings in large, sparse random graphs with atomless weights.

Abstract

Past studies on the local limit of maximal weight matchings in edge-weighted large random graphs rely fundamentally on the assumption that the weights are atomless, which ensures that the maximal weight matching is unique. This excludes de facto maximal size matchings that correspond to equal edge-weights. In this work, we overcome this difficulty by assigning i.i.d.~atomless weights to edges and choosing the maximal size matching that maximises the weight. We call these doubly constrained matchings \emph{optimal matchings}. The natural generalisation of optimal matchings for infinite unimodular random graphs are unimodular matchings of maximal density at the root that maximise the expected weight at the root when it is matched. For unimodular Bienaymé-Galton-Watson (UBGW) trees and for a broad class of weight distributions, we show existence and uniqueness in law of such matchings. We also prove that if a sequence of finite random weighted graphs converges locally to an UBGW tree with i.i.d.~weights, then there exists a sequence of matchings on the finite graphs that converges locally to the optimal matching on the limiting tree. Finally, we identify a regime, depending only on the offspring distribution of the limiting tree, in which correlations between edge states in the optimal matching decay exponentially with their graph distance. In this regime, we strengthen the previous convergence to the convergence of optimal matchings of the finite graphs. As a by-product, we can explicitly compute the asymptotic densities of edges that belong to all maximal-density matchings, and of edges that belong to none.

Figures (17)

• Figure 1: Plain arrows are valid under general assumptions and dashed arrows are proved in the subcritical regime.
• Figure 2: A 2-Neighbourhood of an edge-rooted UBGW tree with the law of the number of children drawn on every vertex.
• Figure 3: A $3-$Neighbourhood of a vertex-rooted UBGW tree with the law of the number of children drawn on every vertex.
• Figure 4: Definitions of $\bf{Z}$.
• Figure 5: Illustration of the deduction of the recursive equation assuming $v_i$ is matched to $v$.

Theorems & Definitions (100)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Remark 2
• Remark 3
• Remark 4
• Theorem 1.3
• Remark 5
• Theorem 1.4
• Corollary 1.1