Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations

We present a method for finding exact solutions of Max-Cut, the problem of finding a cut of maximum weight in a weighted graph. We use a Branch-and-Bound setting, that applies a dynamic version of the bundle method as bounding procedure. This approach uses Lagrangian duality to obtain a “nearly optimal” solution of the basic semidefinite Max-Cut relaxation, strengthened by triangle inequalities. The expensive part of our bounding procedure is solving the basic semidefinite relaxation of the Max-Cut problem, which has to be done several times during the bounding process. We review other solution approaches and compare the numerical results with our method. We also extend our experiments to instances of unconstrained quadratic 0-1 optimization and to instances of the graph equipartition problem. The experiments show, that our method nearly always outperforms all other approaches. In particular, for dense graphs, where linear programming based methods fail, our method performs very well. Exact solutions are obtained in a reasonable time for any instance of size up to n = 100, independent of the density. For some problems of special structure we can solve even larger problem classes. We could prove optimality for several problems of the literature where, to the best of our knowledge, no other method is able to do so. 1 The Max-Cut Problem The Max-Cut problem is one of the basic NP-hard combinatorial optimization problems and has attracted scientific interest from both the discrete (see, e.g., Alpen-Adria-Universität Klagenfurt, Institut für Mathematik, Universitätsstr. 65-67, 9020 Klagenfurt, Austria, ([email protected]) Istituto di Analisi dei Sistemi ed Informatica “Antonio Ruberti” del CNR, viale Manzoni 30, 00185 Roma, Italy, ([email protected]) Alpen-Adria-Universität Klagenfurt, Institut für Mathematik, Universitätsstr. 65-67, 9020 Klagenfurt, Austria, ([email protected])