On the Bidirected Relaxation for Multiway Cut Problem

In the MULTIWAY CUT problem, we are given an undirected edge-weighted graph G=(V,E) with ce denoting the cost (weight) of edge e. We are also given a subset S of V, of size k, called the terminals. The objective is to find a minimum cost set of edges whose removal ensures that the terminals are disconnected. In this paper, we study a bidirected linear programming relaxation of MULTIWAY CUT. We resolve an open problem posed by Vazirani [Approximation Algorithms, first ed., Springer, Berlin, Heidelberg, 2001], and show that the integrality gap of this relaxation is not better than that for a geometric linear programming relaxation given by Caˇlinescu et al. [J. Comput. System Sci. 60(3) (2000) 564–574], and may be strictly worse on some instances