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/k 2 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.

The hypergraph assignment problem

“…This was conjectured in [Coppersmith and Sorkin, 1999] and the first proof appeared in [Linusson and Wästlund, 2004]. There is also shown that for m = n = k this term can be written as…”

Weight of a link in a shortest path tree and the Dedekind Eta function

“…which was asymptotically proved by Aldous [2], and for finite N , independently, by Linusson and Wästlund [12] and by Nair, Prabhakar and Sharma [13]. Earlier, Parisi [14] had conjectured (8) based on simulations and Coppersmith and Sorkin [5] have extended the conjecture to partial assignments.…”

Average-Case Analyses of Vickrey Costs

“…Aldous [Ald92,Ald01] made this mathematically rigorous through reasoning about a "Poisson weighted infinite tree". For finite values of n, Parisi [Par98] conjectured the expected cost to be n i=1 i −2 , Coppersmith and Sorkin [CS99] extended the conjecture to cheapest cardinality-k assignments in K m,n , and these results were proved simultaneously, by different methods, by Linusson and Wästlund [LW04] and Nair, Prabhakar and Sharma [NPS05]. A beautiful short proof was later found by Wästlund [Wäs].…”