Exact and heuristic methods for the vertex separator problem

•A mixed integer programming model and greedy heuristic are used to solve the VSP.•An efficient inequality is given to strengthen the linear relaxation of the MIP.•Extensive numerical tests have been performed on benchmark instances.•The results indicate that our approach outperforms the state-of-the-art methods. In this paper, we propose a practical and efficient methods to solve the vertex separator problem (VSP for short), based on branch-and-bound procedure, which uses linear programming, and a greedy algorithm. This problem arises in many areas of applications such as graph algorithms, communication networks, solving systems of equations, finite element and finite difference problems. We show, by computational experiments, that our approach is able to solve in short time large-scale instances of VSP from the literature to optimality or near optimality.