Journal of Global Optimization

, Volume 31, Issue 1, pp 61–84 | Cite as

Improving an Upper Bound on the Stability Number of a Graph



In previous works an upper bound on the stability number of a graph based on quadratic programming was introduced and several of its properties were given. The graphs for which this bound is attained has been known as graphs with convex-QP stability number. This paper proposes a new upper bound on the stability number whose determination is also done by quadratic programming. It is proved that the new bound improves the above mentioned bound and several computational tests asserting its interest for large graphs are presented. In addition a necessary and sufficient condition for a graph to attain the new bound is proved. As a consequence a graph with convex-QP stability number also attains the new bound. On the other hand it is shown the existence of graphs attaining the new bound that do not belong to the class of graphs with convex-QP stability number. This allows to assert that the class of graphs with convex-QP stability number is strictly included in the class of graphs that attain the introduced bound. Some conclusions and lines for future work finalize the paper.


combinatorial optimization graph theory maximum stable set quadratic programming 


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  1. Alizadeh, F., Haeberly, J.-P., Nayakkankuppam, M.V., Overton, M.L., Schmieta, S. 1997SDPpack User’s Guide – Version 0.9 beta, Technical Report TR1997-737Courant Institute of Mathematical SciencesNYU, New York.Google Scholar
  2. Berge, C. (1991), Graphs, North-Holland, AmsterdamGoogle Scholar
  3. Bomze, I.M. 1998On standard quadratic optimization problemsJournal of Global Optimization13369387CrossRefMathSciNetMATHGoogle Scholar
  4. Bomze, I.M., Budinich, M., Pardalos, P.M. and Pelillo, M. (1999), The maximum clique problem. In: Du D.Z. and Pardalos, P.M. (eds), Handbook of Combinatorial Optimization, Vol. A, pp. 1–74. Kluwer Academic Publishers.Google Scholar
  5. Cardoso, D.M. 2001Convex quadratic programming approach to the maximum matching problemJournal of Global Optimization19291306CrossRefMathSciNetMATHGoogle Scholar
  6. Cvetkovic, D., Doob, M. and Sachs, H. (1979), Spectra of Graphs, Theory and Applications, VEB Deutscher Verlag der Wissenschaften, Berlin (DDR).Google Scholar
  7. Gibbons, L.E., Hearn, D.W., Pardalos, P.M., Ramana, M.V. 1997Continuous characterizations of the maximum clique problemMathematics of Operations Research22754768CrossRefMathSciNetMATHGoogle Scholar
  8. Goemans, M.X. 1997Semidefinite programming in combinatorial optimizationMathematical Programming79143161MathSciNetMATHGoogle Scholar
  9. Grötschel, M., Lovász, L., Schrijver, A. 1988Geometric Algorithms and Combinatorial OptimizationSpringerBerlinCrossRefMATHGoogle Scholar
  10. Haemers, W.H. 1995Interlacing eigenvalues and graphsLinear Algebra and Its Applications227593616CrossRefMathSciNetGoogle Scholar
  11. Helmberg, C. 2000Semidefinite Programming for Combinatorial OptimizationKonrad-Zuse-Zentrum für InformationstechnikBerlinGoogle Scholar
  12. Horn, R.A. and Johnson, C.R. (1985), Matrix Analysis, Cambridge University Press.Google Scholar
  13. Lovász, L. 1979On the Shannon capacity of a graphIEEE Transactions on Information Theory2517CrossRefMATHGoogle Scholar
  14. Lovász, L., Schrijver, A. 1991Cones of matrices and set-functions and 0–1 optimizationSIAM Journal on Optimization1166190CrossRefMathSciNetMATHGoogle Scholar
  15. Lozin, V.V. and Cardoso, D.M. (1999), On Hereditary Properties of the Class of Graphs with Quadratic Stability Number, Cadernos de Matemática, CM/I-50, Math. Dep. of Aveiro University.Google Scholar
  16. Luz, C.J. 1995An upper bound on the independence number of a graph computable in polynomial timeOperations Research Letters18139145CrossRefMathSciNetMATHGoogle Scholar
  17. Luz, C.J., Cardoso, D.M. 1998A Generalization of the Hoffman-Lovász upper bound on the independence number of a Regular graphAnnals of Operations Research81307319CrossRefMathSciNetMATHGoogle Scholar
  18. Luz, C.J., Cardoso, D.M. 2001A quadratic programming approach to the determination of an upper bound on the weighted stability numberEuropean Journal of Operational Research13291103CrossRefMathSciNetGoogle Scholar
  19. Motzkin, T.S., Straus, E.G. 1965Maxima for graphs and a new proof of a theorem of TuránCanadian Journal of Mathematics17533540CrossRefMathSciNetMATHGoogle Scholar
  20. Peressini, A.L., Sullivan, F.E., Uhl, J.J. 1993The Mathematics of Nonlinear ProgrammingSpringerBerlinGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2005

Authors and Affiliations

  1. 1.Escola Superior de Tecnologia de Setúbal/Instituto Politécnico de SetúbalSetúbalPortugal

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