New Algorithms for Enumerating All Maximal Cliques

  • Kazuhisa Makino
  • Takeaki Uno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3111)


In this paper, we consider the problems of generating all maximal (bipartite) cliques in a given (bipartite) graph G=(V,E) with n vertices and m edges. We propose two algorithms for enumerating all maximal cliques. One runs with O(M(n)) time delay and in O(n 2) space and the other runs with O(Δ4) time delay and in O(n+m) space, where Δ denotes the maximum degree of G, M(n) denotes the time needed to multiply two n × n matrices, and the latter one requires O(nm) time as a preprocessing.

For a given bipartite graph G, we propose three algorithms for enumerating all maximal bipartite cliques. The first algorithm runs with O(M(n)) time delay and in O(n 2) space, which immediately follows from the algorithm for the non-bipartite case. The second one runs with O(Δ3) time delay and in O(n+m) space, and the last one runs with O(Δ2) time delay and in O(n+m+NΔ) space, where N denotes the number of all maximal bipartite cliques in G and both algorithms require O(nm) time as a preprocessing.

Our algorithms improve upon all the existing algorithms, when G is either dense or sparse. Furthermore, computational experiments show that our algorithms for sparse graphs have significantly good performance for graphs which are generated randomly and appear in real-world problems.


Time Delay Bipartite Graph Association Rule Maximal Clique Recursive Call 
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.
    Barabasi, A.-L.: LINKED – The New Science of Networks. Perseus Publishing ,Cambridge(2002)Google Scholar
  2. 2.
    Agrawal, R., Srikant, R.: Fast algorithms for mining association rules in large databases. In: Proc. VLDB 1994, pp. 487–499 (1994)Google Scholar
  3. 3.
    Agrawal, R., Mannila, H., Srikant, R., Toivonen, H., Verkamo, A.I.: Fast discovery of association rules. In: Advances in Knowledge Discovery and Data Mining, pp. 307–328. MIT Press, Cambridge (1996)Google Scholar
  4. 4.
    Avis, D., Fukuda, K.: Reverse search for enumeration. Discrete App. Math. 65, 21–46 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Boros, E., Gurvich, V., Khachiyan, L., Makino, K.: On the complexity of generating maximal frequent and minimal infrequent sets. Annals of Math. and Artif. Int. 39, 211–221 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Boros, E., Elbassioni, K., Gurvich, V., Khachiyan, L., Makino, K.: Dual-bounded generating problems: All minimal integer solutions for a monotone system of linear inequalities. SIAM J. Comput. 31, 1624–1643 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM J. Comput. 14, 210–223 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Coppersmith, D., Winograd, S.: Matrix multiplication via arithmetic progression. Journal of Symbolic Computation 9, 251–280 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Eiter, T., Gottlob, G., Makino, K.: New results on monotone dualization and generating hypergraph transversals. SIAM J. Comput. 32, 514–537 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Eiter, T., Makino, K.: On computing all abductive explanations. In: Proc. AAAI 2002, pp. 62–67. AAAI Press, Menlo Park (2002)Google Scholar
  11. 11.
    Eppstein, D.: Arboricity and bipartite subgraph listing algorithms, Info. Proc. Lett. 51, 207–211 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ganter, B., Wille, R.: Formal Concept Analysis. Springer, Heidelberg (1996)zbMATHGoogle Scholar
  13. 13.
    Goldberg, L.A.: Efficient algorithms for listing combinatorial structures. Cambridge University Press, NewYork (1993)zbMATHCrossRefGoogle Scholar
  14. 14.
    Johnson, D.S., Yanakakis, M., Papadimitriou, C.H.: On generating all maximal independent sets. Info. Proc. Lett. 27, 119–123 (1998)CrossRefGoogle Scholar
  15. 15.
    Kumar, S.R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the web for emerging cyber-communities. In: Proc. the Eighth International World Wide Web Conference, Toronto, Canada (1999)Google Scholar
  16. 16.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Generating all maximal independent sets, NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9, 558–565 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Discovering frequent closed itemsets for association rules. In: Beeri, C., Bruneman, P. (eds.) ICDT 1999. LNCS, vol. 1540, pp. 398–416. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  18. 18.
    Pasquier, N., Bastide, Y., Taouil, R., Lakhal, L.: Closed set based discovery of small covers for association rules. In: Proc. 15emes Journees Bases de Donnees Avancees, pp. 361–381 (1999)Google Scholar
  19. 19.
    Read, R.C., Tarjan, R.E.: Bounds on backtrack algorithms for listing cycles, paths, and spanning trees. Networks 5, 237–252 (1975)zbMATHMathSciNetGoogle Scholar
  20. 20.
    Selman, B., Levesque, H.J.: Support set selection for abductive and default reasoning. Artif. Int. 82, 259–272 (1996)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Tsukiyama, S., Ide, M., Ariyoshi, H., Shirakawa, I.: A new algorithm for generating all the maximal independent sets. SIAM J. Comput. 6, 505–517 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Uno, T.: Fast algorithms for computing web communities and frequent sets by using maximal clique generations (in preparation)Google Scholar
  23. 23.
    Uno, T.: Two general methods to reduce delay and change of enumeration algorithms,Technical Report of National Institute of Informatics, Japan (2003)Google Scholar
  24. 24.
    Uno, T., Asai, T., Arimura, H., Uchida, Y.: LCM: An efficient algorithm for enumerating frequent closed item sets In:Workshop on Frequent Itemset Mining Implementations (FIMI 2003) (2003)Google Scholar
  25. 25.
    Zaki, M.J., Ogihara, M.: Theoretical foundations of association rules .In: 3rd SIGMOD Workshop on Research Issues in Data Mining and Knowledge Discovery (June 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Kazuhisa Makino
    • 1
  • Takeaki Uno
    • 2
  1. 1.Division of Mathematical Science for Social Systems, Graduate School of Engineering ScienceOsaka UniversityToyonaka, OsakaJapan
  2. 2.National Institute or InformaticsTokyoJapan

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