Minimum-cost matching in a random graph with random costs

TL;DR

This paper analyzes the expected minimum-cost perfect matching in Erdős-Rényi and bipartite random graphs with exponential edge costs, extending known results to sparser graphs and providing concentration results.

Contribution

It generalizes the expected minimum-cost matching results to sparser random graphs with $d=np o ext{large}$, including concentration bounds.

Findings

Expected minimum cost in bipartite graphs: $(1+o(1))rac{ ho^2}{6p}$

Expected minimum cost in general graphs: $(1+o(1))rac{ ho^2}{12p}$

Results hold with high probability and include concentration bounds.

Abstract

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$, suppose that each edge $e\in E$ is given an independent uniform exponential rate one cost. Let $C(G)$ denote the random variable equal to the length of the minimum cost perfect matching, assuming that $G$ contains at least one. We show that w.h.p. if $d=np\gg(\log n)^2$ then w.h.p. ${\bf E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}$. This generalises the well-known result for the case $G=K_{n,n}$. We also show that w.h.p. ${\bf E}[C(G_{n,p})] =(1+o(1))\frac{\p^2}{12p}$ along with concentration results for both types of random graph.