A Proof of Parisi's Conjecture on the Random Assignment Problem

TL;DR

This paper proves Parisi's conjecture on the expected value of the optimal assignment in a random matrix with some zero entries and others exponentially distributed, confirming a long-standing mathematical hypothesis.

Contribution

It provides a formula for the expected optimal assignment value in a random matrix, proving Parisi's conjecture and generalizing it to arbitrary matrix sizes.

Findings

Confirmed Parisi's conjecture for the case k=m=n.

Extended the formula to matrices with arbitrary dimensions.

Validated the generalized conjecture of Coppersmith and Sorkin.

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.