A new proof of Parisi's conjecture for the finite random assignment problem

Consider the problem of minimizing cost when assigning n jobs to n machines. An assignment is a one-to-one mapping of jobs onto the machines. Assume that the cost of executing job i on machine j is C/sub ij/, i,j = 1,...,n. When the c/sub ij/ are i.i.d. exponentials of mean 1, Parisi conjectured that the average cost of the minimum assignment equals /spl Sigma//sub i=1//sup n/1/i/sup 2/. Recently, the authors, and independently, Linusson and Wastlund, have proved this conjecture. In the above work the authors also made a refined conjecture that, if established, would yield another proof of the Parisi's conjecture. This paper establishes the refined conjecture, thus providing a new proof of Parisi's conjecture.