# Acyclic orientations with path constraints

• [1] Universidade Estadual de Campinas, Instituto de Computação, Caixa Postal 6176, 13084-971 Campinas - SP, Brazil;
• Volume: 42, Issue: 4, page 455-467
• ISSN: 0399-0559

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## Abstract

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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.

## How to cite

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Rosa M. V. Figueiredo, et al. "Acyclic orientations with path constraints." RAIRO - Operations Research - Recherche Opérationnelle 42.4 (2008): 455-467. <http://eudml.org/doc/245017>.

@article{RosaM2008,
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.},
affiliation = {Universidade Estadual de Campinas, Instituto de Computação, Caixa Postal 6176, 13084-971 Campinas - SP, Brazil;},
author = {Rosa M. V. Figueiredo, Barbosa, Valmir C., Maculan, Nelson, Cid C. de Souza},
journal = {RAIRO - Operations Research - Recherche Opérationnelle},
keywords = {acyclic orientations; path constraints; combinatorial optimization problems; facets of polyhedra},
language = {eng},
number = {4},
pages = {455-467},
publisher = {EDP-Sciences},
title = {Acyclic orientations with path constraints},
url = {http://eudml.org/doc/245017},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Rosa M. V. Figueiredo
AU - Barbosa, Valmir C.
AU - Maculan, Nelson
AU - Cid C. de Souza
TI - Acyclic orientations with path constraints
JO - RAIRO - Operations Research - Recherche Opérationnelle
PY - 2008
PB - EDP-Sciences
VL - 42
IS - 4
SP - 455
EP - 467
AB - 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.
LA - eng
KW - acyclic orientations; path constraints; combinatorial optimization problems; facets of polyhedra
UR - http://eudml.org/doc/245017
ER -

## References

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2. [2] J. Bermond, J. Bond, C. Martin, A. Pekec, and F. Roberts, Optimal orientations of annular networks. J. Interconnection Networks 1 (2000) 21–46.
3. [3] 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. Zbl0895.68100
4. [4] 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). Zbl0895.90090
5. [5] R.W. Deming, Acyclic orientations of a graph and chromatic and independence numbers. J. Combin. Theory Ser. B 26 (1979) 101–110. Zbl0331.05110MR525823
6. [6] 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. Zbl0159.54403MR233733
7. [7] M. Grötschel, M. Jünger, and G. Reinelt, Facets of the linear ordering polytope. Math. Program. 33 (1985) 43–60. Zbl0577.05035MR809748
8. [8] M. Grötschel, M. Jünger, and G. Reinelt, On the acyclic subgraph polytope. Math. Program. 33 (1985) 28–42. Zbl0577.05034MR809747
9. [9] V. Maniezzo and A. Carbonaro, An ants heuristic for the frequency assignment problem. Future Gener. Comput. Syst. 16 (2000) 927–935.
10. [10] B. Roy, Nombre chromatique et plus longs chemins d’un graphe, Revue AFIRO 1 (1967) 127–132. Zbl0157.31302MR225683

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