Finding Large Independent Sets in Polynomial Expected Time

  • Amin Coja-Oghlan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2607)


We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. First, let G n,p be a random graph, and let S be a set consisting of k vertices, chosen uniformly at random. Then, let G 0 be the graph obtained by deleting all edges connecting two vertices in S. Adding to G 0 further edges that do not connect two vertices in S, an adversary completes the instance G = G. n,p,k . We propose an algorithm that in the case k ≥C(n/p) 1/2 on input G within polynomial expected time finds an independent set of size ≥ k.


Polynomial Time Random Graph Polynomial Time Algorithm Input Graph Greedy Heuristic 
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.


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  1. 1.
    Alon, N., Kahale, N.: Approximating the independence number via the φ-function. Math. Programming 80 (1998) 253–264.MathSciNetGoogle Scholar
  2. 2.
    Alon, N., Krivelevich, M., Sudakov, B.: Finding a large hidden clique in a random graph. Random Structures & Algorithms 13 (1998) 457–466zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alon, N., Krivelevich, M., Vu, V.H.: On the concentration of the eigenvalues of random symmetric matrices. to appear in Israel J. of Math.Google Scholar
  4. 4.
    Blum, A., Spencer, J.: Coloring random and semirandom k-colorable graphs. J. of Algorithms 19(2) (1995) 203–234MathSciNetGoogle Scholar
  5. 5.
    Bollobás, B.: Random graphs, 2nd edition. Cambridge University Press (2001)Google Scholar
  6. 6.
    Coja-Oghlan, A.: Finding sparse induced subgraphs of semirandom graphs. Proc. 6. Int. Workshop RANDOM (2002) 139–148Google Scholar
  7. 7.
    Coja-Oghlan, A.: Coloring k-colorable semirandom graphs in polynomial expected time via semidefinite programming, Proc. 27th Int. Symp. on Math. Found. of Comp. Sci. (2002) 201–211Google Scholar
  8. 8.
    Coja-Oghlan, A., Taraz, A.: Colouring random graphs in expected polynomial time. To appear in STACS 2003.Google Scholar
  9. 9.
    Feige, U., Kilian, J.: Heuristics for semirandom graph problems. J. Comput. and System Sci. 63 (2001) 639–671zbMATHMathSciNetGoogle Scholar
  10. 10.
    Feige, U., Krauthgamer, J.: Finding and certifying a large hidden clique in a semirandom graph. Random Structures & Algorithms 16 (2000) 195–208zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Frieze, A., McDiarmid, C.: Algorithmic theory of random graphs. Random Structures & Algorithms 10 (1997) 5–42CrossRefMathSciNetGoogle Scholar
  12. 12.
    Füredi, Z., Komloś, J.: The eigenvalues of random symmetric matrices, Combinatorica 1 (1981) 233–241zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer (1988) ai]14._Håstad, J.: Clique is hard to approximate within n 1-. Proc. 37th Annual Symp. on Foundations of Computer Science (1996) 627–636Google Scholar
  14. 15.
    Janson, S., Luczak, T., Ruciński, A.: Random Graphs. Wiley (2000)Google Scholar
  15. 16.
    Jerrum, M.: Large cliques elude the metropolis process. Random Structures & Algorithms 3 (1992) 347–359zbMATHCrossRefMathSciNetGoogle Scholar
  16. 17.
    Juhász, F.: The asymptotic behaviour of Lovász. function for random graphs. Combinatorica 2 (1982) 269–280Google Scholar
  17. 18.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semide finite programming. J. Assoc. Comput. Mach. 45 (1998) 246–265zbMATHMathSciNetGoogle Scholar
  18. 19.
    Karp, R.: Reducibility among combinatorial problems. Miller, R.E., Thatcher, J.W. (eds.): Complexity of Computer Computations. Plenum Press (1972) 85–103Google Scholar
  19. 20.
    Karp, R.: Probabilistic analysis of some combinatorial search problems. Traub, J.F. (ed.): Algorithms and complexity: New Directions and Recent Results. Academic Press (1976) 1–19Google Scholar
  20. 21.
    Knuth, D.: The sandwich theorem, Electron. J. Combin. 1 (1994)Google Scholar
  21. 22.
    Kuĉera, L.: Expected complexity of graph partitioning problems. Discrete Applied Math. 57 (1995) 193–212CrossRefzbMATHGoogle Scholar
  22. 23.
    Krivelevich, M., Vu, V.H.: Approximating the independence number and the chromaticnumber in expected polynomial time. J. of Combinatorial Optimization 6 (2002) 143–155zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Amin Coja-Oghlan
    • 1
  1. 1.Institut für InformatikHumboldt-Universität zu BerlinBerlinGermany

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