A proof of Parisi’s conjecture on the random assignment problem
A proof of Parisi’s conjecture on the random assignment problem
复制标题
Parisi关于随机分配问题猜想的证明
DOI：
10.1007/s00440-003-0308-9
复制
发表时间：
2003
影响因子：
2
通讯作者：
Johan Wästlund
中科院分区：
文献类型：
--
作者：
Svante Linusson;Johan Wästlund
摘要。分配问题是在非负实数，k条目，同一行或列中没有两个矩阵的n矩阵中找到的优化问题，因此它们的总和很小。如果矩阵条目是随机变量，那么这种优化问题称为随机分配问题。我们给出一个在矩阵中的最佳k分配的预期值的公式，其中某些条目为零，并且所有其他条目都是独立的指数分布的随机变量，其平均值为1。因此，我们证明了公式1+1/4+ 1/9+\ ...+1/k2由G. Parisi猜想的情况K = M = N，以及D. Coppersmith和G. B. Sorkin的广义猜想，用于任意K，M和N。
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/k2 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.