Probability Theory and Related Fields

, Volume 128, Issue 3, pp 419–440

# A proof of Parisi’s conjecture on the random assignment problem

• Svante Linusson
• Johan Wästlund
Article

## 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.

## Keywords

Real Number Assignment Problem Random Assignment Matrix Entry Nonnegative Real Number
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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