Suppose that there are n jobs and n machines and it costs C{sub}(ij) to execute job i on machine j. The assignment problem concerns the determination of a one-to-one assignment of jobs onto machines so as to minimize the cost of executing all the jobs. The average case analysis of the classical random assignment problem has received a lot of interest in the recent literature, mainly due to the following pleasing conjecture of Parisi: The average value of the minimum-cost permutation in an n × n matrix with i.i.d. exp(l) entries equals Σ(1/I{sup}2)(i from 1 to n) Coppersmith and Sorkin (1999) have generalized Parisi's conjecture to the average value of the smallest k-assignment when there are n jobs and m machines. We prove both conjectures based on a common set of combinatorial and probabilistic arguments.
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机译:假设有n个作业和n个机器,它成本为c {sub}(ij)在机器j上执行作业i。分配问题涉及确定在机器上的一对一分配作业,以便最小化执行所有作业的成本。古典随机分配问题的平均案例分析已在最近的文献中获得了很多兴趣,主要是由于以下令人赏心悦目的Parisi:n×n矩阵中的最小成本置换的平均值。 Exp(l)条目等于σ(1 / i {sup} 2)(i从1到n)Coppersmith和Sorkin(1999)将广义的巴黎的猜想猜测最小k分配的平均值,当有n工作和m时机器。我们证明了基于常见组合和概率论点的猜想。
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