Solving the Maximal Clique Problem on Compressed Graphs

  • Jocelyn Bernard
  • Hamida SebaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11177)


The Maximal Clique Enumeration problem (MCE) is a graph problem encountered in many applications such as social network analysis and computational biology. However, this problem is difficult and requires exponential time. Consequently, appropriate solutions must be proposed in the case of massive graph databases. In this paper, we investigate and evaluate an approach that deals with this problem on a compressed version of the graphs. This approach is interesting because compression is a staple of massive data processing. We mainly show, through extensive experimentations, that besides reducing the size of the graphs, this approach enhances the efficiency of existing algorithms.


Maximal clique enumeration Graph compression Modular decomposition 


  1. 1.
  2. 2.
  3. 3.
    Stanford large networks.
  4. 4.
    Batagelj, V., Zaversnik, M.: An o (m) algorithm for cores decomposition of networks. arXiv preprint cs/0310049 (2003)Google Scholar
  5. 5.
    Boldi, P., Vigna, S.: The webgraph framework i: compression techniques. In: Proceedings of the 13th International Conference on World Wide Web, pp. 595–602. ACM (2004)Google Scholar
  6. 6.
    Bonnici, V., Giugno, R., Pulvirenti, A., Shasha, D., Ferro, A.: A subgraph isomorphism algorithm and its application to biochemical data. BMC Bioinf. 14(Suppl 7), (S13) (2013)CrossRefGoogle Scholar
  7. 7.
    Bron, C., Kerbosch, J.: Algorithm 457: finding all cliques of an undirected graph. Commun. ACM 16(9), 575–577 (1973)CrossRefGoogle Scholar
  8. 8.
    Conte, A., Virgilio, R.D., Maccioni, A.: Finding all maximal cliques in very large social networks. In: 19th International Conference on Extending Database Technology (EDB), 15–18 March, Bordeaux, France (2016)Google Scholar
  9. 9.
    Du, N., Wu, B., Pei, X., Wang, B., Xu, L.: Community detection in large-scale social networks. In: Proceedings of the 9th WebKDD and 1st SNA-KDD 2007 Workshop on Web Mining and Social Network Analysis, WebKDD/SNA-KDD 2007, pp. 16–25. ACM, New York (2007)Google Scholar
  10. 10.
    Eppstein, D., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. In: Pardalos, P.M., Rebennack, S. (eds.) SEA 2011. LNCS, vol. 6630, pp. 364–375. Springer, Heidelberg (2011). Scholar
  11. 11.
    Fan, W., Li, J., Wang, X., Wu, Y.: Query preserving graph compression. In: Proceedings of the 2012 ACM SIGMOD International Conference on Management of Data, pp. 157–168 (2012)Google Scholar
  12. 12.
    Gallai, T.: Transitiv orientierbare graphen. Acta Mathematica Hungarica 18(1), 25–66 (1967)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Habib, M., de Montgolfier, F., Paul, C.: A simple linear-time modular decomposition algorithm for graphs, using order extension. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 187–198. Springer, Heidelberg (2004). Scholar
  14. 14.
    Lagraa, S., Seba, H., Khennoufa, R., M’Baya, A., Kheddouci, H.: A distance measure for large graphs based on prime graphs. Pattern Recognit. 47(9), 2993–3005 (2014)CrossRefGoogle Scholar
  15. 15.
    Lawler, E., Lenstra, J., Rinnooy Kan, A.: Generating all maximal independent sets: NP-hardness and polynomial-time algorithms. SIAM J. Comput. 9(3), 558–565 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lessley, B., Perciano, T., Mathai, M., Childs, H., Bethel, E.W.: Maximal clique enumeration with data-parallel primitives. In: 2017 IEEE 7th Symposium on Large Data Analysis and Visualization (LDAV), pp. 16–25, October 2017Google Scholar
  17. 17.
    Liu, Y., Dighe, A., Safavi, T., Koutra, D.: A graph summarization: A survey. CoRR abs/1612.04883 (2016).
  18. 18.
    Möhring, R., Radermacher, F.: Substitution decomposition and connection with combinatorial optimization. Ann. Discrete Math. 19, 257–356 (1984)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Navlakha, S., Rastogi, R., Shrivastava, N.: Graph summarization with bounded error. In: Proceedings of the 2008 ACM SIGMOD International Conference on Management of Data, SIGMOD 2008, pp. 419–432 (2008)Google Scholar
  20. 20.
    Segundo, P.S., Artieda, J., Strash, D.: Efficiently enumerating all maximal cliques with bit-parallelism. Comput. Oper. Res. 92, 37–46 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tedder, M., Corneil, D.G., Habib, M., Paul, C.: Simpler linear-time modular decomposition via recursive factorizing permutations. In: 35th International Colloquium on Automata, Languages and Programming, Iceland, 7–11 July 2008, pp. 634–645 (2008)CrossRefGoogle Scholar
  22. 22.
    Toivonen, H., Zhou, F., Hartikainen, A., Hinkka, A.: Compression of weighted graphs. In: Proceedings of the 17th International Conference on Knowledge Discovery and Data Mining, KDD 2011, pp. 965–973 (2011)Google Scholar
  23. 23.
    Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theor. Comput. Sci. 363(1), 28–42 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Wang, J., Cheng, J., Fu, A.W.C.: Redundancy-aware maximal cliques. In: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2013, pp. 122–130 (2013)Google Scholar
  25. 25.
    Wasserman, S., Faust, K.: Social Network Analysis Methods and Applications, vol. 8, January 1993Google Scholar
  26. 26.
    Zhang, B., Park, B.H., Karpinets, T.V., Samatova, N.F.: From pull-down data to protein interaction networks and complexes with biological relevance. Bioinformatics 24(7), 979–86 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Université de Lyon, Universié Lyon 1, CNRS, LIRIS, UMR5205VilleurbanneFrance

Personalised recommendations