Skip to main content
Springer Nature Link
Log in

A characterization of the weighted Lovász number based on convex quadratic programming

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

Luz and Schrijver (SIAM J Discrete Math 19(2):382–387, 2005) introduced a characterization of the Lovász number based on convex quadratic programming. This characterization is now extended to the weighted version of that number. In consequence, a class of graphs for which the weighted Lovász number coincides with the weighted stability number is characterized. Several examples of graphs of this class are presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Abello, J., Butenko, S., Pardalos, P.M., Resende, M.G.C.: Finding independent sets in a graph using continuous multivariable polynomial formulations. J. Glob. Optim. 21, 111–137 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. A, pp. 1–74. Kluwer Academic Publishers, Dordrecht, The Netherlands (1999)

    Chapter  Google Scholar 

  3. Cardoso, D.M.: Convex quadratic programming approach to the maximum matching problem. J. Glob. Optim. 21, 91–106 (2001)

    Article  MathSciNet  Google Scholar 

  4. Cardoso, D.M., Luz, C.J.: A simplex like approach based on star set for recognizing convex-\(QP\) adverse graphs. J. Comb. Optim. (2014). doi:10.107/s10870-014-9745-x

  5. De Klerk, E., Pasechnik, D.V.: Approximating the stability number of a graph via copositive programming. SIAM J. Optim. 12, 875–892 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. DIMACS: Cliques, coloring, and satisfiability: second dimacs implementation challenge. http://dimacs.rutgers.edu/Challenges/ (1995)

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

    Article  MathSciNet  MATH  Google Scholar 

  8. Grötschel, M., Lovász, L., Schrijver, A.: Relaxations of vertex packing. J. Comb. Theory Ser. B 40, 330–343 (1986)

    Article  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  10. Knuth, D.E.: The sandwich theorem. Electron. J. Comb. 1, 1–48, Article #A1 (1994)

  11. Jethava, V., Martinsson, A., Bhattacharyya, C., Dubhashi, D.: Lovász \(\vartheta \) function, SVMs and finding dense subgraphs. J. Mach. Learn. Res. 14(1), 3495–3536 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory 25(2), 1–7 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0–1 optimization. SIAM J. Optim. 1(2), 166–190 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Luz, C.J.: An upper bound on the independence number of a graph computable in polynomial time. Oper. Res. Lett. 18, 139–145 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Luz, C.J., Cardoso, D.M.: A generalization of the Hoffman-Lovász upper bound on the independence number of a regular graph. Ann. Oper. Res. 81, 307–319 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Luz, C.J., Cardoso, D.M.: A quadratic programming approach to the determination of an upper bound on the weighted stability number. Eur. J. Oper. Res. 132, 569–581 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Luz, C.J., Schrijver, A.: A convex quadratic characterization of the Lovász theta number. SIAM J. Discrete Math. 19(2), 382–387 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Can. J. Math. 17, 533–540 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  19. Seymour, P.: How the proof of the strong perfect graph conjecture was found. Gazette des Mathématiciens 109, 69–83 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Tütüncü, R.H., Toh, K.C., Todd, M.J.: Solving semidefinite-quadratic-linear programs using SDPT3. Math. Program. Ser. B 95, 189–217 (2003)

    Article  MATH  Google Scholar 

  21. Wu, Q., Hao, J.-K.: A review on algorithms for maximum clique problem. Eur. J. Oper. Res. 242, 693–709 (2015)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author thanks the referees for their helpful comments and suggestions which improved the paper. This research was supported by Portuguese funds through the Center for Research and Development in Mathematics and Applications (CIDMA) and the Portuguese Foundation for Science and Technology (“FCT—Fundação para a Ciência e Tecnologia”), within project UID/MAT/04106/2013.

Author information

Authors and Affiliations

  1. CIDMA, Center for Research and Development in Mathematics and Applications, Aveiro University, Aveiro, Portugal

    Carlos J. Luz

Authors
  1. Carlos J. Luz
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Carlos J. Luz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luz, C.J. A characterization of the weighted Lovász number based on convex quadratic programming. Optim Lett 10, 19–31 (2016). https://doi.org/10.1007/s11590-015-0911-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0911-6

Keywords