Solving $$(k-1)$$ ( k - 1 ) -stable instances of k-terminal cut with isolating cuts

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 $$(2-2/k)$$ ( 2 - 2 / k ) -approximation to the optimal k-terminal cut. An instance of k-terminal cut is $$\gamma $$ γ -stable if edges in the cut can be multiplied by up to $$\gamma $$ γ without changing the unique optimal solution. In this paper, we show that, in any $$(k-1)$$ ( 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 $$(2-2/k)$$ ( 2 - 2 / k ) -approximation algorithm returns the optimal solution on $$(k-1)$$ ( k - 1 ) -stable instances. Ours is the first result showing that this $$(2-2/k)$$ ( 2 - 2 / k ) -approximation is an exact optimization algorithm on a special class of graphs. We also show that our $$(k-1)$$ ( k - 1 ) -stability result is tight. We construct $$(k-1-\epsilon )$$ ( k - 1 - ϵ ) -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 $$(2-2/k)$$ ( 2 - 2 / k ) -approximation does not return an optimal solution.