Solving $(k-1)$-Stable Instances of k-Terminal Cut with Isolating Cuts

The k-Terminal Cut problem, also known as the Multiway 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 NP-hard. Isolating cuts are minimum cuts that 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. This is the only currently-known approximation algorithm for k-Terminal Cut which does not require solving a linear program. 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 of 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.