Linear-Time Enumeration of Isolated Cliques

  • Hiro Ito
  • Kazuo Iwama
  • Tsuyoshi Osumi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


For a given graph G of n vertices and m edges, a clique S of size k is said to be c-isolated if there are at most ck outgoing edges from S. It is shown that this parameter c is an interesting measure which governs the complexity of finding cliques. In particular, if c is a constant, then we can enumerate all c-isolated maximal cliques in linear time, and if c = O(log n), then we can enumerate all c-isolated maximal cliques in polynomial time. Note that there is a graph which has a superlinear number of c-isolated cliques if c is not a constant, and there is a graph which has a superpolynomial number of c-isolated cliques if c = ω(log n). In this sense our algorithm is optimal for the linear-time and polynomial-time enumeration of c-isolated cliques.


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  1. 1.
    Abello, J., Buchsbaum, A.L., Westbrook, J.R.: A functional approach to external graph algorithms. Algorithmica 32, 437–458 (2002)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arora, S., Karger, D.R., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. In: Proceedings of the 27th ACM Symposium on Theory of Computing, pp. 284–293 (1995)Google Scholar
  3. 3.
    Asahiro, Y., Hassin, R., Iwama, K.: Complexity of finding dense subgraph. Discrete Applied Mathematics 121(1-3), 15–26 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. J. Algorithms 34, 203–221 (2000)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bomze, I.: The Maximum Clique Problem. In: Handbook of Combinatorial Optimization (Supple. vol. A). Kluwer, Dordrecht (1999)Google Scholar
  6. 6.
    Downey, R., Fellows, M.: Fixed-Parameter Tractability and Completeness II: On Completeness for W[1]. Theor. Comput. Sci. 141(1,2), 109–131 (1995)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parametrized Complexity. Springer, Heidelberg (1999)Google Scholar
  8. 8.
    Flake, G.W., Lawrence, S., Giles, C.L.: Efficient identification of web communities. In: Proceedings of the Sixth International Conference on Knowledge Discovery and Data Mining (ACM SIGKDD-2000), pp. 150–160. ACM Press, Boston (2000)CrossRefGoogle Scholar
  9. 9.
    Gibson, D., Kleinberg, J.M., Raghavan, P.: Inferring web communities from link topology. In: UK Conference on Hypertext, pp. 225–234 (1998)Google Scholar
  10. 10.
    Håstad, J.: Clique is hard to approximate within n 1 − ε. Acta Mathematica 182, 105–142 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    He, X., Zha, H., Ding, C., Simon, H.: Web document clustering using hyperlink structures, Tech. Rep. CSE-01-006, Department of Computer Science and Engineering, Pennsylvania State University (2001)Google Scholar
  12. 12.
    Johnson, D., Trick, M. (eds.): Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. American Mathematical Society, Providence (1996)MATHGoogle Scholar
  13. 13.
    Kortsarz, G., Peleg, D.: On choosing a dense subgraph. In: Proceedings of the 34th Annual IEEE Symposium on Foundation of Computer Science, pp. 692–701 (1993)Google Scholar
  14. 14.
    Moon, J., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3, 23–28 (1965)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Reddy, P.K., Kitsuregawa, M.: An approach to relate the web communities through bipartite graphs. In: Proceedings of The Second International Conference on Web Information Systems Engineering, pp. 301–310 (2001)Google Scholar
  16. 16.
    Simon, H.U.: On approximate solutions for combinatorial optimization problems. SIAM J. Disc. Math. 3, 294–310 (1990)MATHCrossRefGoogle Scholar
  17. 17.
    Makino, K., Uno, T.: New algorithms for enumerating all maximal cliques. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 260–272. Springer, Heidelberg (2004)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hiro Ito
    • 1
  • Kazuo Iwama
    • 1
  • Tsuyoshi Osumi
    • 1
  1. 1.Department of Communications and Computer Engineering, School of InformaticsKyoto UniversityKyotoJapan

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