Completing a (k - 1)-Assignment

Let M be an m× n matrix. A k-assignment is a set of k matrix entries of which no two are in the same row or the same column. The value of a k-assignment is the sum of its entries. A k-assignment is optimal if its value is no larger than the value of any other k-assignment. If the entries of the matrix M are random variables then so is the value of the optimal k-assignment, here denoted by mink(M). The study of the optimal k-assignment has been pursued by researchers from different fields and with different random variables as entries in M. The main focus has been to estimate the size of the expected value of mink(M). For references and more details on the history see [5] or [9]. In 1998 Giorgio Parisi [12] conjectured that if M is a k × k matrix with independent exponential random variables of rate 1, then the expected value of the optimal kassignment is