We focus on the all‐pairs minimum cut (APMC) problem, a graph partitioning problem whose solution requires finding the minimum cut for every pair of nodes in a given graph. While it is solved for undirected graphs, a solution for APMC in directed graphs still requires an O (n ²) brute force approach. We show that the empirical number of distinct minimum cuts in randomly generated strongly connected directed graphs is proportional to n rather than the theoretical value of n ², suggesting the possibility of an algorithm which finds all minimum cuts in less than O (n ²) time. We also provide an example of the strict upper bound on the number of cuts in graphs with three nodes. We model the distributions with the Generalized extreme value (GEV) distribution and enable the possibility of using a GEV distribution to predict the probability of achieving a certain number of minimum cuts, given the number of nodes and edges. Finally, we contribute to the notion of symmetric cuts by showing that there can be O (n ²) symmetric cuts in graphs when node replication is allowed.

中文翻译：

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强连通有向随机图上足够多的最小割的实证研究

我们关注所有对的最小割（APMC）问题，这是一种图形分区问题，其解决方案需要找到给定图中每对节点的最小割。虽然可以解决无向图的问题，但有向图的APMC解决方案仍然需要O（n ²）蛮力方法。我们表明，随机生成的强连通有向图中不同最小割的经验数量与n成正比，而不是n ²的理论值，这表明存在一种算法，可以找到所有小于O（n ²）的最小割。时间。我们还提供了一个示例，该示例对具有三个节点的图形中的割数进行严格的上限。我们使用广义极值（GEV）分布对分布进行建模，并在给定节点和边的数量的情况下，允许使用GEV分布来预测实现一定数量的最小割的可能性。最后，我们通过显示在允许节点复制时图上可以存在O（n ²）个对称割，为对称割的概念做出了贡献。