The k-terminal cut problem, also known as the multiterminal cut problem, is defined on an edge-weighted graph with k distinct vertices called “terminals.” The goal is to remove a minimum weight collection of edges from the graph such that there is no path between any pair of terminals. The problem is APX-hard. Isolating cuts are minimum cuts which separate one terminal from the rest. The union of all the isolating cuts, except the largest, is a \((2-2/k)\)-approximation to the optimal k-terminal cut. An instance of k-terminal cut is \(\gamma \)-stable if edges in the cut can be multiplied by up to \(\gamma \) without changing the unique optimal solution. In this paper, we show that, in any \((k-1)\)-stable instance of k-terminal cut, the source sets of the isolating cuts are the source sets of the unique optimal solution to that k-terminal cut instance. We conclude that the \((2-2/k)\)-approximation algorithm returns the optimal solution on \((k-1)\)-stable instances. Ours is the first result showing that this \((2-2/k)\)-approximation is an exact optimization algorithm on a special class of graphs. We also show that our \((k-1)\)-stability result is tight. We construct \((k-1-\epsilon )\)-stable instances of the k-terminal cut problem which only have trivial isolating cuts: that is, the source set of the isolating cuts for each terminal is just the terminal itself. Thus, the \((2-2/k)\)-approximation does not return an optimal solution.

的K-终端切的问题，也被称为多端切割问题，对与边缘加权图定义ķ不同顶点被称为“终端”。目标是从图中删除最小权重的边集合，使得任何一对终端之间都没有路径。问题是 APX 难的。隔离切口是将一个端子与其他端子分开的最小切口。所有隔离切割的并集，除了最大的，是一个\((2-2/k)\) - 对最优 k 终端切割的近似。如果切割中的边可以乘以高达\(\gamma \)，则k 端切割的实例是\(\gamma \) -stable不改变唯一最优解。在本文中，我们证明，在k 端割的任何\((k-1)\) 稳定实例中，隔离割的源集是该k 端割的唯一最优解的源集实例。我们得出结论，\((2-2/k)\) - 近似算法在\((k-1)\) - 稳定实例上返回最优解。我们的第一个结果表明这个\((2-2/k)\)近似是一种特殊类型图的精确优化算法。我们还表明我们的\((k-1)\) -稳定性结果是严格的。我们构造\((k-1-\epsilon )\) -stable 实例k-终端切割问题只有平凡的孤立切割：也就是说，每个终端的孤立切割的源集就是终端本身。因此，\((2-2/k)\) - 近似不会返回最优解。