There are many applications of max flow with capacities that depend on one or more parameters. Many of these applications fall into the "Source-Sink Monotone" framework, a special case of Topkis's monotonic optimization framework, which implies that the parametric min cuts are nested. When there is a single parameter, this property implies that the number of distinct min cuts is linear in the number of nodes, which is quite useful for constructing algorithms to identify all possible min cuts. When there are multiple Source-Sink Monotone parameters and the vector of parameters are ordered in the usual vector sense, the resulting min cuts are still nested. However, the number of distinct min cuts was an open question. We show that even with only two parameters, the number of distinct min cuts can be exponential in the number of nodes.

中文翻译：

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Source-Sink Monotone 2-Parameter Min Cut的复杂度

最大流量有许多应用，其容量取决于一个或多个参数。其中许多应用程序都属于“Source-Sink Monotone”框架，这是 Topkis 单调优化框架的一个特例，这意味着参数化最小切割是嵌套的。当只有一个参数时，这个属性意味着不同的最小割的数量与节点的数量呈线性关系，这对于构建识别所有可能的最小割的算法非常有用。当有多个 Source-Sink Monotone 参数并且参数的向量按照通常的向量意义排序时，得到的最小切割仍然是嵌套的。然而，不同的最小削减的数量是一个悬而未决的问题。我们表明，即使只有两个参数，不同最小切割的数量也可以是节点数量的指数。