Minimum-cost matching in a regular bipartite graph with random costs

Alan Frieze, Tony Johansson

Published 2015-04-01Version 1

Let $G$ be an arbitrary $d$-regular bipartite graph on $2N$ vertices. Suppose that each edge $e$ is given an independent uniform exponential rate one cost. Let $C(G)$ denote the expected length of the minimum cost perfect matching. We show that if $d=d(N)\to\infty$ as $N\to\infty$, then $E(C(G)) =(1+o(1))\frac{N}{d} \frac{\pi^2}{6}$. This generalises the well-known result for the case $G=K_{N,N}$.