On the random $n\times m$ assignment problem

Abstract

Consider the random $n\times m$ assignment problem with $m\ge n$. Let $u_{i,j}$ and $t_{i,j}$ be iid uniform random variables on $[0,1]$ and exponential random variables with mean 1, respectively, and let $U_{n,m}$ and $T_{n,m}$ denote the optimal assignment costs corresponding to $u_{i,j}$ and $t_{i,j}$. In this paper we first give a comparison result about the discrepancy $ET_{n,m}-EU_{n,m}$. Using this comparison result with a known lower bound for ${\rm Var}(T_{n,m})$ we obtains a lower bound for ${\rm Var}(U_{n,m})$. Finally, we study the way that $EU_{n,m}$ and $ET_{n,m}$ vary as $m$ does. It turns out that only when $m-n$ is large enough, the cost decreases significantly.