Proofs of the Parisi and Coppersmith-Sorkin conjectures for the finite random assignment problem

Suppose that there are n jobs and n machines and it costs c_ij to execute job i on machine j. The assignment problem concerns the determination of a one-to-one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the classical random assignment problem has received a lot of interest in the recent literature, mainly due to the following pleasing conjecture of Parisi: The average value of the minimum-cost permutation in an n × n matrix with i.i.d. exp(1) entries equals Σ_i=1ⁿ 1/(i²). D. Coppersmith and G. Sorkin (1999) have generalized Parisi´s conjecture to the average value of the smallest k-assignment when there are n jobs and m machines. We prove both conjectures based on a common set of combinatorial and probabilistic arguments.