# ON THE RANDOM n×n ASSIGNMENT PROBLEM

• Lee, Sung-Chul (Department of Mathematics, Yonsei University) ;
• Zhonggen, Su (Department of Mathematics, Zhejiang University and Department of Mathematics, Yonsei University)
• Published : 2002.10.01

#### 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.

#### References

1. Probab. Theory Relat. Fields v.93 Asymptotics in the random assignment problem D.J. Aldous https://doi.org/10.1007/BF01192719
2. Random Struct. Alg., to appear The ?(2) limit in the random assignment problem
3. Preprint Exact expectations and distributions for the random assignment problem S.E. Alm;G.B. Sorkin
4. Random Struct. Alg v.15 Constructive bounds and exact expectations for the random assignment problem D. Coppersmith;G.B. Sorkin https://doi.org/10.1002/(SICI)1098-2418(199909)15:2<113::AID-RSA1>3.0.CO;2-S
5. Proceedings of SODA The probabilistic relationship between the as-signment and asymmetric traveling salesman problems A.M. Frieze;G.B. Sorkin
6. Comm. Korean Math. Soc., to appear On the fluctuation in the random assignment problem Lee S.;Su Z. https://doi.org/10.4134/CKMS.2002.17.2.321
7. Publ. Math. IHES v.81 Concentration of measure and isoperimetric inequalities in product spaces M. Talagrand https://doi.org/10.1007/BF02699376
8. Preprint An assignment problem at high temperature