A proof of Parisi's conjecture on the random assignment problem

Abstract

An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1 + 1/4 + 1/9 + ⋯ + 1/k2 conjectured by G. Parisi for the case k = m = n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.