On the fluctuation in the random assignment problem

Commun. Korean Math. Soc. 2002 Vol. 17, No. 2, 321-330 Printed June 1, 2002

Sungchul Lee, Zhonggen Su Yonsei University, Zhejiang University

Abstract : Consider the random assignment (or bipartite matching) problem with iid uniform edge costs $t(i,j)$. Let $A_n$ be the optimal assignment cost. Just recently does Aldous \cite{A2001} give a rigorous proof that $EA_n \ra \zeta(2)$. In this paper we establish the upper and lower bounds for $\text{\rm Var} A_n$, i.e., there exist two strictly positive but finite constants $C_1$ and $C_2$ such that $C_1 n^{-5/2} (\log n)^{-3/2} \le \text{\rm Var} A_n \le C_2 n^{-1} (\log n)^{2}$.