随机矩阵指派问题中$k$ -圈的个数
On the number of $k$-cycles in the assignment problem for random matrices
摘要：
我们继续研究随机费用矩阵的指派问题。我们分析的$k$ -周期的解决方案和它们的随机矩阵的对称性的依赖。我们观察到，对称矩阵的一个和两个周期是占主导地位的最佳解决方案。在反对称的情况下，情况相反，一个和两个循环被抑制。我们解决了一个纯随机矩阵（其条目之间没有相关性）的模型，并给出了分析参数来解释对称和反对称情况下的数值结果。我们表明，结果可以解释到很大的准确性由一个简单的animalent，连接预期数量的$k$ -周期的一个和两个周期。
英文摘要：
We continue the study of the assignment problem for a random cost matrix. We analyse the number of $k$-cycles for the solution and their dependence on the symmetry of the random matrix. We observe that for a symmetric matrix one and two-cycles are dominant in the optimal solution. In the antisymmetric case the situation is the opposite and the one and two-cycles are suppressed. We solve the model for a pure random matrix (without correlations between its entries) and give analytic arguments to explain the numerical results in the symmetric and antisymmetric case. We show that the results can be explained to great accuracy by a simple ansatz that connects the expected number of $k$-cycles to that of one and two cycles.
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