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.

翻译：有许多最大流量的应用程序,其容量取决于一个或多个参数。许多这些应用程序都属于“源-Sink Monotone”框架,Topkis的单声波优化框架的一个特例,这意味着对准分钟的削减是嵌套的。当有一个单一参数时,这一属性意味着不同的分钟削减数量在节点数量中是线性的,这对构建算法以识别所有可能的分钟削减非常有用。当存在多个源-Sink Monotone参数和参数矢量时,通常的矢量排序时,由此产生的分钟削减仍然是嵌套的。然而,不同的分钟削减数量是一个未决问题。我们显示,即使只有两个参数,不同的分钟削减数量在节点数量上也可能是指数化的。