Hostname: page-component-848d4c4894-xfwgj Total loading time: 0 Render date: 2024-06-22T08:45:12.258Z Has data issue: false hasContentIssue false
  • English
  • Français

Acyclic Orientations with Path Constraints

Published online by Cambridge University Press:  04 April 2009

Rosa M. V. Figueiredo ,
Valmir C. Barbosa ,
Nelson Maculan  and
Cid C. de Souza
Rosa M. V. Figueiredo
Affiliation:
Universidade do Estado do Rio de Janeiro, Instituto de Matemática e Estatística, 20550-900 Rio de Janeiro - RJ, Brazil; rosa@ime.uerj.br
Valmir C. Barbosa
Affiliation:
Universidade Federal do Rio de Janeiro, Programa de Engenharia de Sistemas e Computação, COPPE, Caixa Postal 68511, 21941-972 Rio de Janeiro - RJ, Brazil; valmir@cos.ufrj.br maculan@cos.ufrj.br
Nelson Maculan
Affiliation:
Universidade Federal do Rio de Janeiro, Programa de Engenharia de Sistemas e Computação, COPPE, Caixa Postal 68511, 21941-972 Rio de Janeiro - RJ, Brazil; valmir@cos.ufrj.br maculan@cos.ufrj.br
Cid C. de Souza
Affiliation:
Universidade Estadual de Campinas, Instituto de Computação, Caixa Postal 6176, 13084-971 Campinas - SP, Brazil; cid@ic.unicamp.br
Get access

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.

Keywords

Acyclic orientationspath constraintscombinatorial optimization problemsfacets of polyhedra.
Type
Research Article
Information
RAIRO - Operations Research , Volume 42 , Issue 4: ALIO/EURO V Conference on Combinatorial Optimization , October 2008 , pp. 455 - 467
Copyright
© EDP Sciences, ROADEF, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

K. Aardal, A. Hipolito, C. van Hoesel, B. Jansen, C. Roos, and T. Terlaky, 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 Netherlands (1995).
Bermond, J., Bond, J., Martin, C., Pekec, A., and Roberts, F., Optimal orientations of annular networks. J. Interconnection Networks 1 (2000) 21–46. CrossRef
J. Bermond, M. Di Ianni, M. Flammini, and S. Perennes, Acyclic orientations for deadlock prevention in interconnection networks, in Proceedings of the Workshop on Graph-Theoretic Concepts in Computer Science (1997) 52–64.
R. Borndörfer, A. Eisenblätter, M. Grötschel, and A. Martin, The orientation model for frequency assignment problems. Technical Report 98-01, Zuse Institute Berlin, Germany (1998).
Deming, R.W., Acyclic orientations of a graph and chromatic and independence numbers. J. Combin. Theory Ser. B 26 (1979) 101–110. CrossRef
T. Gallai, On directed paths and circuits, in Theory of Graphs edited by P. Erdős and G. Katona, Academic Press, New York, NY (1968) 115–118.
Grötschel, M., Jünger, M., and Reinelt, G., Facets of the linear ordering polytope. Math. Program. 33 (1985) 43–60. CrossRef
Grötschel, M., Jünger, M., and Reinelt, G., On the acyclic subgraph polytope. Math. Program. 33 (1985) 28–42. CrossRef
Maniezzo, V. and Carbonaro, A., An ants heuristic for the frequency assignment problem. Future Gener. Comput. Syst. 16 (2000) 927–935. CrossRef
B. Roy, Nombre chromatique et plus longs chemins d'un graphe, Revue AFIRO 1 (1967) 127–132.