On Cutting a Few Vertices from a Graph

We consider the problem of finding in an undirected graph a minimum cut that separates exactly a given number k of vertices. For general k (i.e. k is part of the input and may depend on n) this problem is NP-hard.We present for this problem a randomized approximation algorithm, which is useful when k is relatively small. In particular, for k = O(log n) we obtain a polynomial time approximation scheme, and for k = Ω(log n) we obtain an approximation ratio O(k/log n).