Mathematical Programming

, Volume 80, Issue 3, pp 253–264

Approximating the independence number via theϑ-function

  • Noga Alon
  • Nabil Kahale
Article

Abstract

We describe an approximation algorithm for the independence number of a graph. If a graph onn vertices has an independence numbern/k + m for some fixed integerk ⩾ 3 and somem > 0, the algorithm finds, in random polynomial time, an independent set of size\(\tilde \Omega (m^{{3 \mathord{\left/ {\vphantom {3 {(k + 1)}}} \right. \kern-\nulldelimiterspace} {(k + 1)}}} )\), improving the best known previous algorithm of Boppana and Halldorsson that finds an independent set of size Ω(m1/(k−1)) in such a graph. The algorithm is based on semi-definite programming, some properties of the Lovászϑ-function of a graph and the recent algorithm of Karger, Motwani and Sudan for approximating the chromatic number of a graph. If theϑ-function of ann vertex graph is at leastMn1−2/k for some absolute constantM, we describe another, related, efficient algorithm that finds an independent set of sizek. Several examples show the limitations of the approach and the analysis together with some related arguments supply new results on the problem of estimating the largest possible ratio between theϑ-function and the independence number of a graph onn vertices. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.

Keywords

Independence number of a graph Semi-definite programming Approximation algorithms 

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References

  1. [1]
    N. Alon, Explicit Ramsey graphs and orthonormal labelings,The Electronic Journal of Combinatorics 1 (1994) R12.Google Scholar
  2. [2]
    N. Alon and Y. Peres, Euclidean Ramsey theory and a construction of Bourgain,Acta Mathematica Hungarica 57 (1991) 61–64.Google Scholar
  3. [3]
    S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, Proof verification and intractability of approximation problems, in:Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1992) 14–23.Google Scholar
  4. [4]
    S. Arora and S. Safra, Probabilistic checking of proofs; a new characterization of NP, in:Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1992) 2–13.Google Scholar
  5. [5]
    B. Berger and J. Rompel, A better performance guarantee for approximate graph coloring,Algorithmica 5 (1990) 459–466.Google Scholar
  6. [6]
    P. Berman and G. Schnitger, On the complexity of approximating the independent set problem,Information and Computation 96 (1992) 77–94.Google Scholar
  7. [7]
    R. Boppana and M.M. Halldorsson, Approximating maximum independent sets by excluding subgraphs,BIT 32 (1992) 180–196.Google Scholar
  8. [8]
    P. Erdös, Some remarks on chromatic graphs,Colloquium Mathematicum 16 (1967) 253–256.Google Scholar
  9. [9]
    P. Erdös and G. Szekeres, A combinatorial problem in geometry,Compositio Mathematica 2 (1935) 463–470.Google Scholar
  10. [10]
    U. Feige, Randomized graph products, chromatic numbers, and the Lovászϑ-function, in:27th Annual ACM Symposium on Theory of Computing (ACM Press, New York, 1995) 635–640.Google Scholar
  11. [11]
    U. Feige, S. Goldwasser, L. Lovász, S. Safra and M. Szegedy, Approximating Clique is almost NP-complete, in:Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1991) 2–12.Google Scholar
  12. [12]
    P. Frankl and V. Rödl, Forbidden intersections,Transactions AMS 300 (1987) 259–286.Google Scholar
  13. [13]
    M. Goemans and D. Williamson, Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming,Journal ACM 42 (1995) 1115–1145.Google Scholar
  14. [14]
    G.H. Golub and C.F. Van Loan,Matrix Computations (The Johns Hopkins University Press, Baltimore, 1989).Google Scholar
  15. [15]
    M.M. Halldorsson, A still better performance guarantee for approximate graph coloring,Information Processing Letters 45 (1993) 19–23.Google Scholar
  16. [16]
    J. Håstad, Clique is hard to approximate withinn 1-ε,Proc. 37th IEEE FOCS (IEEE, 1996) 627–636.Google Scholar
  17. [17]
    M. Grötschel, L. Lovász and A. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981) 169–197.Google Scholar
  18. [18]
    D. Karger, R. Motwani and M. Sudan, Approximate graph coloring by semi-definite programming, in:35th Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994) 2–13.Google Scholar
  19. [19]
    R. Karp,Reducibility among combinatorial problems, eds. Miller and Thatcher (Plenum Press, New York, 1972).Google Scholar
  20. [20]
    B.S. Kashin and S.V. Konyagin, On systems of vectors in a Hilbert space,Trudy Mat. Inst. imeni V.A. Steklova 157 (1981) 64–67. English translation in:Proceedings of the Steklov Institute of Mathematics (AMS, 1983) 67–70.Google Scholar
  21. [21]
    D.E. Knuth, The sandwich theorem,The Electronic Journal of Combinatorics 1 (A1) (1994).Google Scholar
  22. [22]
    S.V. Konyagin, Systems of vectors in Euclidean space and an extremal problem for polynomials,Mat. Zametki 29 (1981) 63–74. English translation in:Mathematical Notes of the Academy of the USSR 29 (1981) 33–39.Google Scholar
  23. [23]
    L. Lovász, On the Shannon capacity of a graph,IEEE Transactions on Information Theory 25 (1979) 1–7.Google Scholar
  24. [24]
    M. Szegedy, A note on theϑ number of Lovász and the generalized Delsarte bound, in:35th Symposium on Foundations of Computer Science (IEEE Computer Society Press, Los Alamitos, CA, 1994) 36–39.Google Scholar

Copyright information

© The Mathematical Programming Society, Inc. 1998

Authors and Affiliations

  • Noga Alon
    • 1
    • 2
  • Nabil Kahale
    • 2
  1. 1.Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.AT & T Bell LaboratoriesMurray HillUSA

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