Minimum Cost Matching in a Random Graph with Random Costs

Let $G_{n,p}$ be the standard Erdős-Renyi-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.