Computing multiway cut within the given excess over the largest minimum isolating cut

Let (G, T ) be an instance of the (vertex) multiway cut problem where G is a graph and T is a set of terminals. For t ∈ T , a set of nonterminal vertices separating t from T \ {T } is called an isolating cut of t. The largest among all the smallest isolating cuts is a natural lower bound for a multiway cut of (G, T ). Denote this lower bound by m and let k be an integer. In this paper we propose an O(kn) algorithm that computes a multiway cut of (G, T ) of size at most m + k or reports that there is no such multiway cut. The core of the proposed algorithm is the following combinatorial result. Let G be a graph and let X,Y be two disjoint subsets of vertices of G. Let m be the smallest size of a vertex X − Y separator. Then, for the given integer k, the number of important X − Y separators [16] of size at most m+ k is at most ∑ k