Colouring Random Graphs in Expected Polynomial Time

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


We investigate the problem of colouring random graphs G ε G(n, p) in polynomial expected time. For the case p < 1.01/n, we present an algorithm that finds an optimal colouring in linear expected time. For suficiently large values of p, we give algorithms which approximate the chromatic number within a factor of O(√np). As a byproduct, we obtain an O(√np/ ln(np))-approximation algorithm for the independence number which runs in polynomial expected time provided p ln6 n/n.


Approximation Algorithm Polynomial Time Greedy Algorithm Random Graph Approximation Ratio 
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.
    Achlioptas, D., Molloy, M.: The analysis of a list-coloring algorithm on a random graph, Proc. 38th. IEEE Symp. Found. of Comp. Sci. (1997) 204–212Google Scholar
  2. 2.
    Beck, L.L., Matula, D.W.: Smallest-last ordering and clustering and graph coloring algorithms, J. ACM 30 (1983) 417–427zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beigel, R., Eppstein, D.: 3-coloring in time O(1.3446n): a no-MIS algorithm, Proc. 36th. IEEE Symp. Found. of Comp. Sci. (1995) 444–453Google Scholar
  4. 4.
    Bollobás, B.: Random graphs, 2nd edition, Cambridge University Press 2001Google Scholar
  5. 5.
    Coja-Oghlan, A.: Finding sparse induced subgraphs of semirandom graphs. Proc. 6th. Int. Workshop Randomization and Approximation Techniques in Comp. Sci. (2002) 139–148Google Scholar
  6. 6.
    Coja-Oghlan, A.: Finding large independent sets in expected polynomial time. To appear in Proc. STACS 2003Google Scholar
  7. 7.
    Dyer, M., Frieze, A.: The solution of some NP-hard problems in polynomial expected time, J. Algorithms 10 (1989) 451–489zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. To appear in J. Graph Algorithms and ApplicationsGoogle Scholar
  9. 9.
    Feige, U., Kilian, J.: Zero knowledge and the chromatic number. Proc. 11. IEEE Conf. Comput. Complexity (1996) 278–287Google Scholar
  10. 10.
    Feige, U., Kilian, J.: Heuristics for semirandom graph problems. J. Comput. and System Sci. 63 (2001) 639–671zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Füredi, Z., Komloś, J.: The eigenvalues of random symmetric matrices, Combinatorica 1 (1981) 233–241zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Fürer, M., Subramanian, C.R., Veni Madhavan, C.E.: Coloring random graphs in polynomial expected time. Algorithms and Comput. (Hong Kong 1993), Springer LNCS 762, 31–37Google Scholar
  13. 13.
    Frieze, A., McDiarmid, C.: Algorithmic theory of random graphs. Random Structures and Algorithms 10 (1997) 5–42CrossRefMathSciNetGoogle Scholar
  14. 14.
    Grimmett, G., McDiarmid, C.: On colouring random graphs. Math. Proc. Cam. Phil. Soc 77 (1975) 313–324Google Scholar
  15. 15.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization. Springer 1988Google Scholar
  16. 16.
    Janson, S., Luczak, T., Ruciński, A.: Random Graphs. Wiley 2000Google Scholar
  17. 17.
    Juhász, F.: The asymptotic behaviour of Lovász. function for random graphs, Combinatorica 2 (1982) 269–280Google Scholar
  18. 18.
    Karger, D., Motwani, R., Sudan, M.: Approximate graph coloring by semidefinite programming. Proc. of the 35th. IEEE Symp. on Foundations of Computer Science (1994) 2–13Google Scholar
  19. 19.
    Karp, R.: Reducibility among combinatorial problems. In: Complexity of computer computations. Plenum Press (1972) 85–103.Google Scholar
  20. 20.
    Karp, R.: The probabilistic analysis of combinatorial optimization algorithms. Proc. Int. Congress of Mathematicians (1984) 1601–1609.Google Scholar
  21. 21.
    Knuth, D.: The sandwich theorem, Electron. J. Combin. 1 (1994)Google Scholar
  22. 22.
    Krivelevich, M., Vu, V.H.: Approximating the independence number and the chromatic number in expected polynomial time. J. of Combinatorial Optimization 6 (2002) 143–155zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Krivelevich, M.: Deciding k-colorability in expected polynomial time, Information Processing Letters 81 (2002) 1–6zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Krivelevich, M.: Coloring random graphs-an algorithmic perspective, Proc. 2nd Coll. on Mathematics and Computer Science, B. Chauvin et al. Eds., Birkhauser, Basel (2002) 175–195.Google Scholar
  25. 25.
    Krivelevich, M., Sudakov, B.: Coloring random graphs. Informat. Proc. Letters 67 (1998) 71–74Google Scholar
  26. 26.
    Kuĉera, L.: The greedy coloring is a bad probabilistic algorithm. J. Algorithms 12 (1991) 674–684zbMATHGoogle Scholar
  27. 27.
    Lawler, E.L.: A note on the complexity of the chromatic number problem, Information Processing Letters 5 (1976) 66–67zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Pittel, B., Spencer, J., Wormald, N.: Sudden emergence of a giant k-core in a random graph. JCTB 67 (1996) 111–151zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Prömel, H.J., Steger, A.: Coloring clique-free graphs in polynomial expected time, Random Str. Alg. 3 (1992) 275–302Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

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

Personalised recommendations