Solving Graph Bisection Problems with Semidefinite Programming

An exact solution method for the graph bisection problem is presented. We describe a branch-and-bound algorithm which is based on a cutting plane approach combining semidefinite programming and polyhedral relaxations. We report on extensive numerical experiments which were performed for various classes of graphs. The results indicate that the present approach solves general problem instances with 80–90 vertices exactly in reasonable time and provides tight approximations for larger instances. Our approach is particularly well suited for special classes of graphs as planar graphs and graphs based on grid structures.