Mathematical Programming

, Volume 109, Issue 2–3, pp 345–365 | Cite as

Semidefinite programming relaxations for graph coloring and maximal clique problems

  • Igor Dukanovic
  • Franz RendlEmail author


The semidefinite programming formulation of the Lovász theta number does not only give one of the best polynomial simultaneous bounds on the chromatic number χ(G) or the clique number ω(G) of a graph, but also leads to heuristics for graph coloring and extracting large cliques. This semidefinite programming formulation can be tightened toward either χ(G) or ω(G) by adding several types of cutting planes. We explore several such strengthenings, and show that some of them can be computed with the same effort as the theta number. We also investigate computational simplifications for graphs with rich automorphism groups.


Lovász theta number Chromatic number Clique number Cutting planes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benson, S., Ye, Y.: Approximating maximum stable set and minimum graph coloring problems with the positive semidefinite relaxation. In: Applications and Algorithms of Complementarity, pp. 1–18. Kluwer, Dordrecht (2000)Google Scholar
  2. 2.
    Benson S., Ye Y., Zhang X. (2000): Solving large-scale sparse semidefinite programs for combinatorial optimization. SIAM J. Optim. 10, 443–461CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bomze I.M., de Klerk E. (2002): Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24, 163–185CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Burer S., Monteiro R.D.C. (2003): A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program. 95, 329–357CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Burer S., Vandenbussche D. (2006): Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16, 726–750CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Busygin, S., Pasechnik, D.V.: On \(\bar\chi(G)-\alpha(G)>0\) gap recognition and α(G)-upper bounds. Electronic Colloquium on Computational Complexity, Report No. 52 pp. 1–5 (2003)Google Scholar
  7. 7.
    Charikar, M.: On semidefinite programming relaxations for graph coloring and vertex cover. In: Proceedings of the 41th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 616–620 (2002)Google Scholar
  8. 8.
    Dukanovic, I.: Semidefinite programming applied to graph coloring problem. PhD Thesis, University of Klagenfurt, Austria (2006) (forthcoming)Google Scholar
  9. 9.
    Dukanovic, I., Rendl, F.: A semidefinite programming based heuristic for graph coloring. Discrete Appl. Math. (to appear)Google Scholar
  10. 10.
    Dukanovic, I., Rendl, F.: Copositive programming motivated bounds on the clique and the chromatic number of a graph. Preprint (2006)Google Scholar
  11. 11.
    Gatermann K., Parrilo P.A. (2004): Symmetry groups, semidefinite programs, and sums of squares. J. Pure Appl. Algebra 192, 95–128CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Grötschel M., Lovász L., Schrijver A. (1988): Geometric Algorithms and Combinatorial Optimization. Springer, Berlin Heidelberg New YorkzbMATHGoogle Scholar
  13. 13.
    Gruber G., Rendl F. (2003): Computational experience with stable set relaxations. SIAM J. Optim. 13, 1014–1028CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Gvozdenović, N., Laurent, M.: Approximating the chromatic number of a graph by semidefinite programming. Working paper (2005)Google Scholar
  15. 15.
    Helmberg C., Rendl F. (2000): A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673–696CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Hertz A., Werra D.D. (1987): Using tabu search for graph coloring. Computing 39, 345–351CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Karger D., Motwani R., Sudan M. (1998): Approximate graph coloring by semidefinite programming. J. Assoc. Comput. Mach. 45, 246–265MathSciNetzbMATHGoogle Scholar
  18. 18.
    de Klerk, E., Pasechnik, D.V., Schrijver, A.: Reduction of symmetric semidefinite programs using the regular *-representation. Math. Program. (2005)(to appear)Google Scholar
  19. 19.
    de Klerk E., Pasechnik D.V., Warners J.P. (2004): On approximate graph colouring and max-k-cut algorithms based on the θ-function. J. Combinat. Optim. 8, 267–294CrossRefzbMATHGoogle Scholar
  20. 20.
    Knuth D.E. (1994): The sandwich theorem. Elect. J. Combinat. 1, 1–48Google Scholar
  21. 21.
    Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: R.W.e. K. Aardal G. Nemhauser (ed.) Handbook on Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)Google Scholar
  22. 22.
    Lovász L. (1979): On the Shannon capacity of a graph. IEEE Trans. Inf. Theory 25, 1–7CrossRefzbMATHGoogle Scholar
  23. 23.
    Lovász L., Schrijver A. (1991): Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190CrossRefMathSciNetzbMATHGoogle Scholar
  24. 24.
    Malick, J., Povh, J., Rendl, F., Wiegele, A.: A boundary point method to solve semidefinite programs. Working paper (2006)Google Scholar
  25. 25.
    Margot F. (2001): Pruning by isomorphism in branch-and-cut. Lect. Notes Comput. Sci. 2081, 304–317MathSciNetCrossRefGoogle Scholar
  26. 26.
    McEliece J.R., Rodemich E., Rumsey H. (1978): The Lovász bound and some generalizations. J. Combinat. Syst. Sci. 3, 134–152MathSciNetzbMATHGoogle Scholar
  27. 27.
    McKay B.D. (1981): Practical graph isomorphism. Congressus Numer. 30, 45–87MathSciNetGoogle Scholar
  28. 28.
    Meurdesoif P. (2005): Strengthening the Lovász \(\theta(\overline{G})\) bound for graph coloring. Math. Program. 102, 577–588CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Murty K.G., Kabadi S.N. (1987): Some NP-complete problems in quadratic and nonlinear programming. Math. Program. 39, 117–129MathSciNetzbMATHGoogle Scholar
  30. 30.
    Parrilo, P.A.: Structured semidefinite programs and semialgebraic geometry methods in robustness and optimization. PhD Thesis, California Institute of Technology (2000)Google Scholar
  31. 31.
    Parrilo, P.A., Sturmfels, B.: Minimizing polynomial functions. In: S. Basu, L.G.V. (eds.) Algorithmic and quantitative real algebraic geometry. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 60, pp. 83–99. AMS New york (2003)Google Scholar
  32. 32.
    Schrijver A. (1979): A comparison of the Delsarte and Lovász bounds. IEEE Trans. Info. Theory IT-25, 425–429CrossRefMathSciNetzbMATHGoogle Scholar
  33. 33.
    Schrijver A. (2004): New code upper bounds from the Terwilliger algebra. IEEE Trans. Infor. Theory 51, 2859–2866CrossRefMathSciNetGoogle Scholar
  34. 34.
    Szegedy, M.: A note on the theta number of Lovász and the generalized Delsarte bound. In: 35th Annual Symposium on Foundations of Computer Science, pp. 36–39 (1994)Google Scholar
  35. 35.
    Toh K., Kojima M. (2002): Solving some large scale semidefinite programs via the conjugate residual method. SIAM J. Optim. 12, 669–691CrossRefMathSciNetzbMATHGoogle Scholar
  36. 36.
    Vesel A., Žerovnik J. (2002): Improved lower bound on the Shannon capacity of C7. Inf. Process. Lett. 81 (5): 277–282CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Ekonomsko-poslovna fakultetaUniverza v MariboruMariborSlovenia
  2. 2.Institut für MathematikUniversität KlagenfurtKlagenfurtAustria

Personalised recommendations