Skip to main content
Springer Nature Link
Log in

Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

Abstract

It is a long standing open problem to find an explicit description of the stable set polytope of claw-free graphs. Yet more than 20 years after the discovery of a polynomial algorithm for the maximum stable set problem for claw-free graphs, there is even no conjecture at hand today.

Such a conjecture exists for the class of quasi-line graphs. This class of graphs is a proper superclass of line graphs and a proper subclass of claw-free graphs for which it is known that not all facets have 0/1 normal vectors. Ben Rebea’s conjecture states that the stable set polytope of a quasi-line graph is completely described by clique-family inequalities. Chudnovsky and Seymour recently provided a decomposition result for claw-free graphs and proved that Ben Rebea’s conjecture holds, if the quasi-line graph is not a fuzzy circular interval graph.

In this paper, we give a proof of Ben Rebea’s conjecture by showing that it also holds for fuzzy circular interval graphs. Our result builds upon an algorithm of Bartholdi, Orlin and Ratliff which is concerned with integer programs defined by circular ones matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows. Prentice Hall Inc., Englewood Cliffs (1993)

    Google Scholar 

  2. Andersen, K., Cornuéjols, G., Li, Y.: Split closure and intersection cuts. In: Cook, W.J., Schulz, A.S. (eds.) IPCO 2002. LNCS, vol. 2337, pp. 127–144. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  3. Bartholdi, J.J., Orlin, J.B., Ratliff, H.: Cyclic scheduling via integer programs with circular ones. Operations Research 28, 1074–1085 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  4. Caprara, A., Letchford, A.N.: On the separation of split cuts and related inequalities. In: Mathematical Programming. A Publication of the Mathematical Programming Society 94(2-3, Ser. B), 279–294 (2003); The Aussois 2000 Workshop in Combinatorial Optimization

    Google Scholar 

  5. Chudnovsky, M.: Personal communication (2004)

    Google Scholar 

  6. Chudnovsky, M., Seymour, P.: The structure of claw-free graphs. In: Proceedings of the Bristish Combinatorial Conference, Durham (2005) (to appear)

    Google Scholar 

  7. Chvátal, V.: Edmonds polytopes and a hierarchy of combinatorial problems. Discrete Mathematics 4, 305–337 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cook, W., Kannan, R., Schrijver, A.: Chvátal closures for mixed integer programming problems. Mathematical Programming 47, 155–174 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  9. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards 69, 125–130 (1965)

    MATH  MathSciNet  Google Scholar 

  10. Edmonds, J.: Paths, trees and flowers. Canadian Journal of Mathematics 17, 449–467 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  11. Eisenbrand, F.: On the membership problem for the elementary closure of a polyhedron. Combinatorica 19(2), 297–300 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Galluccio, A., Sassano, A.: The rank facets of the stable set polytope for claw-free graphs. Journal on Combinatorial Theory 69, 1–38 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gijswijt, D.: On a packet scheduling problem for smart antennas and polyhedra defined by circular ones matrices. Submitted to Siam Journal of Discrete Mathematics (2003)

    Google Scholar 

  14. Giles, R., jun, L.E.T.: On stable set polyhedra for k (1,3)-free graphs. Journal on Combinatorial Theory 31, 313–326 (1981)

    Article  MATH  Google Scholar 

  15. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bulletin of the American Mathematical Society 64, 275–278 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  16. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1(2), 169–197 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  17. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin (1988)

    MATH  Google Scholar 

  18. Karp, R.M., Papadimitriou, C.H.: On linear characterizations of combinatorial optimization problems. SIAM Journal on Computing 11(4), 620–632 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  19. Khachiyan, L.: A polynomial algorithm in linear programming. Doklady Akademii Nauk SSSR 244, 1093–1097 (1979)

    MATH  MathSciNet  Google Scholar 

  20. Liebling, T.M., Oriolo, G., Spille, B., Stauffer, G.: On the non-rank facets of the stable set polytope of claw-free graphs and circulant graphs. Mathematical Methods of Operations Research 59, 25–35 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  21. Lovász, L., Plummer, M.: Matching Theory. North-Holland, Amsterdam (1986)

    MATH  Google Scholar 

  22. Minty, G.J.: On maximal independent sets of vertices in claw-free graphs. Journal on Combinatorial Theory 28, 284–304 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  23. Nakamura, D., Tamura, A.: A revision of Minty’s algorithm for finding a maximum weighted stable set of a claw-free graph. Journal of the Operations Research Society of Japan 44(2), 194–2004 (2001)

    MATH  MathSciNet  Google Scholar 

  24. Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. John Wiley, Chichester (1988)

    MATH  Google Scholar 

  25. Oriolo, G.: Clique family inequalities for the stable set polytope for quasi-line graphs. Discrete Applied Mathematics 132(3), 185–201 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Padberg, M.W., Rao, M.R.: The Russian method for linear programming III: Bounded integer programming. Technical Report 81-39, New York University, Graduate School of Business and Administration (1981)

    Google Scholar 

  27. Pulleyblank, W.R., Shepherd, F.B.: Formulations for the stable set polytope of a claw-free graph. In: Rinaldi, G., Wolsey, L. (eds.) Proceedings Third IPCO Conference, pp. 267–279 (1993)

    Google Scholar 

  28. Rebea, A.B.: Étude des stables dans les graphes quasi-adjoints. PhD thesis, Université de Grenoble (1981)

    Google Scholar 

  29. Sbihi, N.: Algorithme de recherche d’un stable de cardinalité maximum dans un graphe sans étoile. Discrete Mathematics 29, 53–76 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  30. Schrijver, A.: On cutting planes. Annals of Discrete Mathematics 9, 291–296 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  31. Schrijver, A.: Theory of Linear and Integer Programming. John Wiley, Chichester (1986)

    MATH  Google Scholar 

  32. Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. In: Algorithms and Combinatorics 24, vol. 3, Springer, Berlin (2003)

    Google Scholar 

  33. Shepherd, F.B.: Applying Lehman’s theorems to packing problems. Mathematical Programming 71, 353–367 (1995)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Max-Planck-Institut für Informatik, Stuhlsatzenhausweg 85, 66123, Saarbrücken, Germany

    Friedrich Eisenbrand

  2. Dipartimento di Ingegneria dell’Impresa, Universitá di Tor Vergata, via del Politecnico 1, 00165, Roma, Italy

    Gianpaolo Oriolo

  3. Ecole Polytechnique Fédérale de Lausanne, SB-IMA-ROSO Station 8, CH-1015, Lausanne, Switzerland

    Gautier Stauffer

  4. Istituto di Analisi dei Sistemi ed Informatica del CNR, Viale Manzoni 30, 00185, Roma, Italy

    Paolo Ventura

Authors
  1. Friedrich Eisenbrand
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianpaolo Oriolo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gautier Stauffer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Paolo Ventura
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität zu Köln,  

    Michael Jünger

  2. Fakultät für Mathematik, Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Volker Kaibel

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Eisenbrand, F., Oriolo, G., Stauffer, G., Ventura, P. (2005). Circular Ones Matrices and the Stable Set Polytope of Quasi-Line Graphs. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_22

Download citation

  • DOI: https://doi.org/10.1007/11496915_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics