Randomized graph products, chromatic numbers, and the Lovász ϑ-function

Abstract

For a graphG, let α(G) denote the size of the largest independent set inG, and let ϑ(G) denote the Lovász ϑ-function onG. We prove that for somec>0, there exists an infinite family of graphs such that\(\vartheta (G) > \alpha (G)n/2^{c\sqrt {\log n} }\), wheren denotes the number of vertices in a graph. this disproves a known conjecture regarding the ϑ function.

As part of our proof, we analyse the behavior of the chromatic number in graphs under a randomized version of graph products. This analysis extends earlier work of Linial and Vazirani, and of Berman and Schnitger, and may be of independent interest.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    N. Alon, N. Kahale: Approximating the independence number via the θ-function.Manuscript, November 1994.

  2. [2]

    S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy: Proof verification and hardness of approximation problems.Proc. of 33rd IEEE Symp. on Foundations of Computer Science, 1992, 14–23.

  3. [3]

    S. Arora, S. Safra: Probabilistic checking of proofs: a new characterization of NP.Proc. of 33rd IEEE Symp. on Foundations of Computer Science, 1992, 2–13.

  4. [4]

    M. Bellare, O. Goldreich, M. Sudan: Free bits, PCPs and nonapproximability-towards tight results.Proc. of 36th IEEE Symp. on Foundations of Computer Science, 1995, 422–431.

  5. [5]

    P. Berman, G. Schnitger: On the complexity of approximating the independent set problem,Information and Computation 96 (1992), 77–94.

    Google Scholar 

  6. [6]

    A. Blum: Algorithms for approximate graph coloring, Phd dissertation, MIT, 1991.

  7. [7]

    A. Blum: New approximation algorithms for graph coloring.Journal of the ACM,41 (1994), 470–516.

    Google Scholar 

  8. [8]

    R. Boppana, M. Haldorsson: Approximating maximum independent sets by excluding subgraphs,Proc. of 2nd SWAT, Springer, LNCS 447 (1990), 13–25.

  9. [9]

    U. Feige, S. Goldwasser, L. Lovász, S. Safra, M. Szegedy: Interactive proofs and the hardness of approximating cliques,Journal of the ACM,43(2) (1996), 268–292.

    Google Scholar 

  10. [10]

    U. Feige, J. Kilian: Zero knowledge and the chromatic number,Proc. of Eleventh Annual IEEE Conference on Computational Complexity, 1996, 278–287.

  11. [11]

    P. Frankl, R. Wilson: Intersection theorems with geometric consequences,Combinatorica 1 (1981), 357–368.

    Google Scholar 

  12. [12]

    M. Furer: Improved hardness results for approximating the chromatic number,Proc. of 36th IEEE Symp. on Foundations of Computer Science, (1995), 414–421.

  13. [13]

    M. Goemans, D. Williamson: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,Journal of the ACM,42(6) (1995), 1115–1145.

    Google Scholar 

  14. [14]

    M. Grotschel, L. Lovász, A. Schrijver:Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  15. [15]

    J. Hastad: Clique is hard to approximate withinn 1−ε,Proc. of 37th IEEE Symposium of Foundations of Computer Science, 1996, 627–636.

  16. [16]

    W. Hoeffding: Probability inequalities for sums of bounded random variables,Journal of the American Statistical Association,58 (1963), 13–30.

    Google Scholar 

  17. [17]

    D. Karger, R. Motwani, M. Sudan: Approximate Graph Coloring by Semidefinite Programming,Proc. of 35th IEEE Symp. on Foundations of Computer Science, (1994), 2–13.

  18. [18]

    D. Knuth: The sandwich theorem,Electronic J. Comp.,1 (1994), 1–48.

    Google Scholar 

  19. [19]

    N. Linial, U. Vazirani: Graph products and chromatic numbers,Proc. of 30th IEEE Symp. on Foundations of Computer Science, (1989), 124–128.

  20. [20]

    L. Lovász: On the ratio of the optimal integral and fractional covers,Discrete Mathematics,13 (1975), 383–390.

    Google Scholar 

  21. [21]

    L. Lovász: On the Shannon Capacity of a Graph,IEEE Trans. on Information Theory, Vol. IT-25, No. 1, 1979, 1–7.

    Google Scholar 

  22. [22]

    C. Lund, M. Yannakakis: On the hardness of approximating minimization problems,Journal of the ACM,41(5) (1994), 960–981.

    Google Scholar 

  23. [23]

    J. Moon, L. Moser: On cliques in graphsIsrael J. Math.,3 (1965), 23–28.

    Google Scholar 

  24. [24]

    M. Szegedy: A note on the θ number of Lovász and the generalized Delsarte bound,Proc. of 35th IEEE Symp. on Foundations of Computer Science, (1994), 36–39.

  25. [25]

    A. Wigderson: Improving the performance guarantee for approximate graph coloring,Journal of the ACM,30(4) (1983), 729–735.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Incumbent of the Joseph and Celia Reskin Career Development Chair. Yigal Alon Fellow

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Feige, U. Randomized graph products, chromatic numbers, and the Lovász ϑ-function. Combinatorica 17, 79–90 (1997). https://doi.org/10.1007/BF01196133

Download citation

Mathematics Subject Classification (1991)

  • 05 C 15
  • 90 C 35