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Abstract

We consider the problem of finding a maximum independent set in a random graph. The random graph G is modelled as follows. Every edge is included independently with probability \(\frac{d}{n}\), where d is some sufficiently large constant. Thereafter, for some constant α, a subset I of αn vertices is chosen at random, and all edges within this subset are removed. In this model, the planted independent set I is a good approximation for the maximum independent set I max , but both II max and I max I are likely to be nonempty. We present a polynomial time algorithms that with high probability (over the random choice of random graph G, and without being given the planted independent set I) finds a maximum independent set in G when \(\alpha \geq \sqrt{c_0 \log d /d}\), where c 0 is some sufficiently large constant independent of d.

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References

  1. 1.
    Alon, N., Kahale, N.: A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Computing 26(6), 1733–1748 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. Random Structures and Algorithms 13(3-4), 457–466 (1988)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Chen, H., Frieze, A.: Coloring bipartite hypergraphs. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) IPCO 1996. LNCS, vol. 1084, pp. 345–358. Springer, Heidelberg (1996)Google Scholar
  4. 4.
    Coja-Oghlan, A.: A spectral heuristic for bisecting random graphs. In: SODA 2005, pp. 850–859 (2005)Google Scholar
  5. 5.
    Coja-Oghlan, A.: Finding Large Independent Sets in Polynomial Expected Time. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 511–522. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  6. 6.
    Feige, U.: Approximating maximum clique by removing subgraphs. Siam J. on Discrete Math. 18(2), 219–225 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Feige, U., Kilian, J.: Heuristics for semirandom graph problems. Journal of Computing and System Sciences 63(4), 639–671 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Feige, U., Krauthgamer, R.: Finding and certifying a large hidden clique in a semirandom graph. Random Structures and Algorithms 16(2), 195–208 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Flaxman, A.: A spectral technique for random satisfiable 3cnf formulas. In: SODA 2003, 357– 363 (2003)Google Scholar
  10. 10.
    Goerdt, A., Lanka, A.: On the hardness and easiness of random 4-SAT formulas. In: Fleischer, R., Trippen, G. (eds.) ISAAC 2004. LNCS, vol. 3341, pp. 470–483. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  11. 11.
    Grimmet, G., McDiarmid, C.: On colouring random graphs. Math. Proc. Cam. Phil. Soc. 77, 313–324 (1975)CrossRefGoogle Scholar
  12. 12.
    Håstad, J.: Clique is hard to approximate within n1 − ε. Acta Mathematica 182(1), 105–142 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jerrum, M.: Large clique elude the metropolis process. Random Structures and Algorithms 3(4), 347–359 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Karp, R.M.: The probabilistic analysis of some combinatorial search algorithms. In: Traub, J.F. (ed.) Algorithms and Complexity: New Directions and Recent Results, pp. 1–19. Academic Press, New York (1976)Google Scholar
  15. 15.
    Kučera, L.: Expected complexity of graph partitioning problems. Discrete Appl. Math. 57(2-3), 193–212 (1995)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Uriel Feige
    • 1
  • Eran Ofek
    • 1
  1. 1.Department of Computer Science and Applied MathematicsWeizmann Institute of ScienceRehovotIsrael

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