Acyclic Orientations With Path Constraints

Abstract

Many well-known combinatorial optimization problems can be stated over the set of acyclic orientations of an undirected graph. For example, acyclic orientations with certain diameter constraints are closely related to the optimal solutions of the vertex coloring and frequency assignment problems. In this paper we introduce a linear programming formulation of acyclic orientations with path constraints, and discuss its use in the solution of the vertex coloring problem and some versions of the frequency assignment problem. A study of the polytope associated with the formulation is presented, including proofs of which constraints of the formulation are facet-defining and the introduction of new classes of valid inequalities. © EDP Sciences.424455467Aardal, K., Hipolito, A., van Hoesel, C., Jansen, B., Roos, C., Terlaky, T., (1995) EUCLID CALMA radio link frequency assignment project: A branch-and-cut algorithm for the frequency assignment problem, , Technical report, Delft and Eindhoven Universities of Technology, The NetherlandsBermond, J., Bond, J., Martin, C., Pekec, A., Roberts, F., Optimal orientations of annular networks (2000) J. Interconnection Networks, 1, pp. 21-46Bermond, J., Di Ianni, M., Flammini, M., Perennes, S., Acyclic orientations for deadlock prevention in interconnection networks (1997) Proceedings of the Workshop on Graph-Theoretic Concepts in Computer Science, pp. 52-64Borndörfer, R., Eisenblätter, A., Grötschel, M., Martin, A., The orientation model for frequency assignment problems (1998), Technical Report 98-01, Zuse Institute Berlin, GermanyDeming, R.W., Acyclic orientations of a graph and chromatic and independence numbers (1979) J. Combin. Theory Ser. B, 26, pp. 101-110Gallai, T., On directed paths and circuits (1968) Theory of Graphs, pp. 115-118. , edited by P. Erdös and G. Katona, Academic Press, New York, NYGrötschel, M., Jünger, M., Reinelt, G., Facets of the linear ordering polytope (1985) Math. Program, 33, pp. 43-60Grötschel, M., Jünger, M., Reinelt, G., On the acyclic subgraph polytope (1985) Math. Program, 33, pp. 28-42Maniezzo, V., Carbonaro, A., An ants heuristic for the frequency assignment problem (2000) Future Gener. Comput. Syst, 16, pp. 927-935Roy, B., Nombre chromatique et plus longs chemins d'un graphe (1967) Revue AFIRO, 1, pp. 127-13