Probability Theory and Related Fields

, Volume 128, Issue 3, pp 419–440 | Cite as

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

  • Svante LinussonEmail author
  • Johan Wästlund


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.


Real Number Assignment Problem Random Assignment Matrix Entry Nonnegative Real Number 
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© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  1. 1.Department of MathematicsLinköpings universitetLinköpingSweden

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