An easy proof of the ζ(2) limit in the random assignment problem

Abstract

The edges of the complete bipartite graph Kn, n are given independent exponentially distributed costs. Let Cn be the minimum total cost of a perfect matching. It was conjectured by M. Mézard and G. Parisi in 1985, and proved by D. Aldous in 2000, that Cn converges in probability to π2/6. We give a short proof of this fact, consisting of a proof of the exact formula 1+1/4+1/9+⋯+1/n2 for the expectation of Cn, and a O(1/n) bound on the variance. © 2009 Applied Probability Trust.