Belief propagation for minimum weight many-to-one matchings in the random complete graph

TL;DR

This paper analyzes the minimum weight many-to-one matchings in large random bipartite graphs with exponential weights, proving convergence of the optimal solution and the effectiveness of belief propagation algorithms.

Contribution

It extends the objective method and demonstrates belief propagation's asymptotic optimality for many-to-one matchings in random graphs.

Findings

Minimum weight converges to a constant as n grows large

Belief propagation algorithm converges to the optimal solution asymptotically

Results extend the applicability of the objective method to new combinatorial problems

Abstract

In a complete bipartite graph with vertex sets of cardinalities $n$ and $m$, assign random weights from exponential distribution with mean 1, independently to each edge. We show that, as $n\rightarrow\infty$, with $m = \lceil n/\alpha\rceil$ for any fixed $\alpha>1$, the minimum weight of many-to-one matchings converges to a constant (depending on $\alpha$). Many-to-one matching arises as an optimization step in an algorithm for genome sequencing and as a measure of distance between finite sets. We prove that a belief propagation (BP) algorithm converges asymptotically to the optimal solution. We use the objective method of Aldous to prove our results. We build on previous works on minimum weight matching and minimum weight edge-cover problems to extend the objective method and to further the applicability of belief propagation to random combinatorial optimization problems.