• Lagrangian relaxation;
  • cutting planes;
  • Integer Programming;
  • relax-and-cut;
  • vertex separator


In this article, we propose a Lagrangian relaxation framework to solve the vertex separator problem (VSP). This framework is based on the development of relax-and-cut algorithms which embed the separation of valid inequalities for the VSP discussed by Balas and de Souza (Math Program 103 (2005), 583–608) in the subgradient method. These relax-and-cut algorithms are then used as a preprocessing phase in a hybrid algorithm which combines them with branch-and-cut algorithms proposed by de Souza and Balas (Math Program 103 (2005), 609–631). This is done basically by feeding the branch-and-cut algorithms not only with the primal bound but also the cuts separated during the preprocessing phase. Computational results obtained with benchmarks from the literature showed that the hybrid algorithm developed here outperforms the best exact algorithm available for the VSP to date. © 2011 Wiley Periodicals, Inc. NETWORKS, 2011