The multi-terminal vertex separator problem: polyhedral analysis and branch-and-cut

In this paper we consider a variant of the k-separator problem. Given a graph G=(V∪T,E) with V∪T the set of vertices, where T is a set of k terminals, the multi-terminal vertex separator problem consists in partitioning V∪T into k+1 subsets {S,V1,...,Vk} such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the size of S is minimum. In this paper, we first show that the problem is NP-hard. Then we give two integer programming formulations for the problem. For one of these formulations, we investigate the related polyhedron and discuss its polyhedral structure. We describe some valid inequalities and characterize when these inequalities define facets. We also derive separation algorithms for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem, along with an extensive computational study.