ON THE RANDOM n×n ASSIGNMENT PROBLEM

Abstract

Consider the random n $\times$ m assignment problem with m $\geq$ $_{i,j}$ Let $u_{i,j}$ 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 E $T_{n, m}$ -E $U_{n, m}$. Using this comparison result with a known lower bound for Var( $T_{n, m}$) we obtains a lower bound for Var( $U_{n, m}$). Finally, we study the way that E $U_{n, m}$ and E $T_{n, m}$ vary as m does. It turns out that only when m - n is large enough, the cost decreases significantly.tly.

Keywords

• n${\times}$m assignment problem;
• bipartite matching;
• optimal assignment

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References

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