Clique Is Hard to Approximate within n1-o(1)

  • Lars Engebretsen
  • Jonas Holmerin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1853)


It was previously known that Max Clique cannot be approximated in polynomial time within n1-ε, for any constant ε > 0, unless NP = ZPP. In this paper, we extend the reductions used to prove this result and combine the extended reductions with a recent result of Samorodnitsky and Trevisan to show that clique cannot be approximated within \( n^{1 - O} \left( {1/\sqrt {\log \log n} } \right) \) unless NPZPTIMENP ⊆ ZPTIME(2O(log n(log log n )3/2)).


Polynomial Time Proof System Hardness Result Random String Probabilistic Polynomial Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Arora, C. Lund, R. Motwani, M. Sudhan, and M. Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501–555, May 1998.Google Scholar
  2. 2.
    S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70–122, Jan. 1998.Google Scholar
  3. 3.
    M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs and non-approximability—towards tight results. SIAM J. Comput., 27(3):804–915, June 1998.Google Scholar
  4. 4.
    M. Bellare and M. Sudan. Improved non-approximability results. In Proc. Twenty-sixth Ann. ACM Symp. on Theory of Comp., pages 184–193, Montréal, Québec, May 1994. ACM Press.Google Scholar
  5. 5.
    R. Boppana and M. M. Halldórsson. Approximating maximum independent sets by excluding subgraphs. Bit, 32(2):180–196, June1992.Google Scholar
  6. 6.
    U. Feige. Randomized graph products, chromatic numbers, and the Lovasz ϑ-function. In Proc. Twenty-seventh Ann. ACM Symp. on Theory of Comp., pages 635–640, Las Vegas, Nevada, May 1995. ACM Press.Google Scholar
  7. 7.
    U. Feige. A threshold of ln n for approximating set cover. J. ACM, 45(4):634–652, July1998.Google Scholar
  8. 8.
    U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. J. ACM, 43(2):268–292, Mar. 1996.Google Scholar
  9. 9.
    J. Hästad. Clique is hard to approximate within n1-ε. In Proc. 37th Ann. IEEE Symp. on Foundations of Comput. Sci., pages 627–636, Burlington, Vermont, Oct. 1996. IEEE Computer Society.Google Scholar
  10. 10.
    R. M. Karp. Reducibility among combinatorial problems. In R.E. Miller and J. W. Thatcher, editors, Complexity of Computer Computations, pages 85–103. Plenum Press, New York, 1972.Google Scholar
  11. 11.
    A. Samorodnitsky and L. Trevisan. Notes on a PCP characterization of NP with optimal amortized query complexity. Manuscript, May 1999.Google Scholar
  12. 12.
    D. Zuckerman. On unapproximable versions of NP-complete problems. SIAM J. Comput., 25(6):1293–1304, Dec. 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Lars Engebretsen
    • 1
  • Jonas Holmerin
    • 1
  1. 1.Department of Numerical Analysis and Computing ScienceRoyal Institute of TechnologyStockholmSweden

Personalised recommendations