Skip to main content
SpringerLink
Log in

Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications

  • Conference paper
  • First Online:
WALCOM: Algorithms and Computation (WALCOM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10167))

Included in the following conference series:

Abstract

The problem of finding a maximum clique or enumerating all maximal cliques is very important and has been explored in several excellent survey papers. Here, we focus our attention on the step-by-step examination of a series of branch-and-bound depth-first search algorithms: Basics, MCQ, MCR, MCS, and MCT. Subsequently, as with the depth-first search as above, we present our algorithm, CLIQUES, for enumerating all maximal cliques. Finally, we describe some of the applications of the algorithms and their variants in bioinformatics, data mining, and other fields.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akutsu, T., Hayashida, M., Bahadur, D.K.C., Tomita, E., Suzuki, J., Horimoto, K.: Dynamic programming and clique based approaches for protein threading with profiles and constraints. IEICE Trans. Fundam. Electron. Commun. Comput. Sci. E89–A, 1215–1222 (2006)

    Article  Google Scholar 

  2. Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. J. Heuristics 18, 525–547 (2012)

    Article  Google Scholar 

  3. Bahadur, D.K.C., Akutsu, T., Tomita, E., Seki, T., Fujiyama, A.: Point matching under non-uniform distortions and protein side chain packing based on an efficient maximum clique algorithm. Genome Inf. 13, 143–152 (2002)

    Google Scholar 

  4. Bahadur, D.K.C., Tomita, E., Suzuki, J., Akutsu, T.: Protein side-chain packing problem: a maximum edge-weight clique algorithmic approach. J. Bioinform. Comput. Biol. 3, 103–126 (2005)

    Article  Google Scholar 

  5. Bahadur, D.K.C., Tomita, E., Suzuki, J., Horimoto, K., Akutsu, T.: Protein threading with profiles and distance constraints using clique based algorithms. J. Bioinform. Comput. Biol. 4, 19–42 (2006)

    Article  Google Scholar 

  6. Batsyn, M., Goldengorin, B., Maslov, E., Pardalos, P.M.: Improvements to MCS algorithm for the maximum clique problem. J. Comb. Optim. 27, 397–416 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.-Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, Supplement vol. A, pp. 1–74. Kluwer Academic Publishers (1999)

    Google Scholar 

  8. Bron, C., Kerbosch, J.: Algorithm 457, finding all cliques of an undirected graph. Commun. ACM 16, 575–577 (1973)

    Article  MATH  Google Scholar 

  9. Brown, J.B., Bahadur, D.K.C., Tomita, E., Akutsu, T.: Multiple methods for protein side chain packing using maximum weight cliques. Genome Inf. 17, 3–12 (2006)

    Google Scholar 

  10. Butenko, S., Wilhelm, W.E.: Clique-detection models in computational biochemistry and genomics - invited review-. Eur. J. Oper. Res. 173, 1–17 (2006)

    Article  MATH  Google Scholar 

  11. Carraghan, R., Pardalos, P.M.: An exact algorithm for the maximum clique problem. Oper. Res. Lett. 9, 375–382 (1990)

    Article  MATH  Google Scholar 

  12. Eppstein, D., Löffler, M., Strash, D.: Listing all maximal cliques in large sparse real-world graphs. J. Exp. Algorithmics 18, 3.1:1–3.1:21 (2013)

    Article  Google Scholar 

  13. Fujii, T., Tomita, E.: On efficient algorithms for finding a maximum clique. Technical report IECE, AL81-113, pp. 25–34 (1982)

    Google Scholar 

  14. Fukagawa, D., Tamura, T., Takasu, A., Tomita, E., Akutsu, T.: A clique-based method for the edit distance between unordered trees and its application to analysis of glycan structure. BMC Bioinform. 12(S–1), S:13 (2011)

    Article  Google Scholar 

  15. Hatta, T., Tomita, E., Ito, H., Wakatsuki, M.: An improved branch-and-bound algorithm for finding a maximum clique. In: Proceedings of the Summer LA Symposium, no. 9, pp. 1–8 (2015)

    Google Scholar 

  16. Hotta, K., Tomita, E., Takahashi, H.: A view-invariant human face detection method based on maximum cliques. Trans. IPSJ 44(SIG14(TOM9)), 57–70 (2003)

    Google Scholar 

  17. Johnson, D.S., Trick, M.A. (eds.): Cliques, Coloring, and Satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. American Mathematical Society (1996)

    Google Scholar 

  18. Karp, R.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Comlexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  19. Katayama, K., Hamamoto, A., Narihisa, H.: An effective local search for the maximum clique problem. Inf. Process. Lett. 95, 503–511 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kobayashi, S., Kondo, T., Okuda, K., Tomita, E.: Extracting globally structure free sequences by local structure freeness. In: Chen, J., Reif, J. (eds.) Proceedings of Ninth International Meeting on DNA Based Computers, p. 206 (2003)

    Google Scholar 

  21. Kohata, Y., Nishijima, T., Tomita, E., Fujihashi, C., Takahashi, H.: Efficient algorithms for finding a maximum clique. Technical report IEICE, COM89-113, pp. 1–8 (1990)

    Google Scholar 

  22. Konc, J., Janežič, D.: An improved branch and bound algorithm for the maximum clique problem. MATCH Commun. Math. Comput. Chem. 58, 569–590 (2007)

    MathSciNet  MATH  Google Scholar 

  23. Li, C.M., Quan, Z.: Combining graph structure exploitation and propositional reasoning for the maximum clique problem. In: Proceedings of IEEE ICTAI, pp. 344–351 (2010)

    Google Scholar 

  24. 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). doi:10.1007/978-3-540-27810-8_23

    Chapter  Google Scholar 

  25. Maslov, E., Batsyn, M., Pardalos, P.M.: Speeding up branch and bound algorithms for solving the maximum clique problem. J. Glob. Optim. 59, 1–21 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. Matsunaga, T., Yonemori, C., Tomita, E., Muramatsu, M.: Clique-based data mining for related genes in a biomedical database. BMC Bioinform. 10, 205 (2009)

    Article  Google Scholar 

  27. Moon, J.W., Moser, L.: On cliques in graphs. Israel J. Math. 3, 23–28 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mori, T., Tamura, T., Fukagawa, D., Takasu, A., Tomita, E., Akutsu, T.: A clique-based method using dynamic programming for computing edit distance between unordered trees. J. Comput. Biol. 19, 1089–1104 (2012)

    Article  MathSciNet  Google Scholar 

  29. Nagai, M., Tabuchi, T., Tomita, E., Takahashi, H.: An experimental evaluation of some algorithms for finding a maximum clique. In: Conference Records of the National Convention of IEICE 1988, p. D-348 (1988)

    Google Scholar 

  30. Nakui, Y., Nishino, T., Tomita, E., Nakamura, T.: On the minimization of the quantum circuit depth based on a maximum clique with maximum vertex weight. Technical report RIMS, 1325, Kyoto University, pp. 45–50 (2003)

    Google Scholar 

  31. Okubo, Y., Haraguchi, M., Tomita, E.: Structural change pattern mining based on constrained maximal k-plex search. In: Ganascia, J.-G., Lenca, P., Petit, J.-M. (eds.) DS 2012. LNCS (LNAI), vol. 7569, pp. 284–298. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33492-4_23

    Chapter  Google Scholar 

  32. Okubo, Y., Haraguchi, M., Tomita, E.: Relational change pattern mining based on modularity difference. In: Ramanna, S., Lingras, P., Sombattheera, C., Krishna, A. (eds.) MIWAI 2013. LNCS (LNAI), vol. 8271, pp. 187–198. Springer, Heidelberg (2013). doi:10.1007/978-3-642-44949-9_18

    Chapter  Google Scholar 

  33. Okubo, Y., Haraguchi, M., Tomita, E.: Enumerating maximal isolated cliques based on vertex-dependent connection lower bound. In: Perner, P. (ed.) MLDM 2016. LNCS (LNAI), vol. 9729, pp. 569–583. Springer, Heidelberg (2016)

    Chapter  Google Scholar 

  34. Pardalos, P.M., Xue, J.: The maximum clique problem. J. Glob. Optim. 4, 301–328 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  35. Pattillo, J., Youssef, N., Butenko, S.: Clique relaxation models in social network analysis. In: Thai, M.T., Pardalos, P.M. (eds.) Handbook of Optimization in Complex Networks: Communication and Social Networks. Springer Optimization and Its Applications, vol. 58, pp. 143–162. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  36. Segundo, P.S., Nikolaev, A., Batsyn, M.: Infra-chromatic bound for exact maximum clique search. Comput. Oper. Res. 64, 293–303 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shindo, M., Tomita, E., Maruyama, Y.: An efficient algorithm for finding a maximum clique. Technical report IECE, CAS86-5, pp. 33–40 (1986)

    Google Scholar 

  38. Shindo, M., Tomita, E.: A simple algorithm for finding a maximum clique and its worst-case time complexity. Syst. Comput. Jpn. 21, 1–13 (1990). Wiley

    Article  MathSciNet  Google Scholar 

  39. Sutani, Y., Higashi, T., Tomita, E., Takahashi, S., Nakatani, H.: A faster branch-and-bound algorithm for finding a maximum clique. Technical report IPSJ, 2006-AL-108, pp. 79–86 (2006)

    Google Scholar 

  40. Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. SIAM J. Comput. 6(3), 537–546 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  41. Tomita, E., Yamada, M.: An algorithm for finding a maximum complete subgraph. In: Conference Records of the National Convention of IECE 1978, p. 8 (1978)

    Google Scholar 

  42. Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for finding all the cliques. Technical report, University of Electro-Communications, UEC-TR-C5(2) (1988). (Reference [238] in [34], Reference [308] in [7]). http://id.nii.ac.jp/1438/00001898/

  43. Tomita, E., Kohata, Y., Takahashi, H.: A simple algorithm for finding a maximum clique. Technical report, University of Electro-Communications, UEC-TR-C5(1) (1988). (Reference [239] in [34], Reference [309] in [7]). http://id.nii.ac.jp/1438/00001899/

  44. Tomita, E., Mitsuma, S., Takahashi, H.: Two algorithms for finding a near-maximum clique. Technical report, University of Electro-Communications, UEC-TR-C1 (1988). (Reference [240] in [34], Reference [310] in [7]). http://id.nii.ac.jp/1438/00001900/

  45. Tomita, E., Seki, T.: An efficient branch-and-bound algorithm for finding a maximum clique. In: Calude, C.S., Dinneen, M.J., Vajnovszki, V. (eds.) DMTCS 2003. LNCS, vol. 2731, pp. 278–289. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  46. Tomita, E., Tanaka, A., Takahashi, H.: The worst-case time complexity for generating all maximal cliques and computational experiments. Theoret. Comput. Sci. 363, 28–42 (2006). (Special Issue on COCOON 2004)

    Article  MathSciNet  MATH  Google Scholar 

  47. Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Glob. Optim. 37, 95–111 (2007). J. Glob. Optim. 44, 311 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  48. Tomita, E., Sutani, Y., Higashi, T., Takahashi, S., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique. In: Rahman, M.S., Fujita, S. (eds.) WALCOM 2010. LNCS, vol. 5942, pp. 191–203. Springer, Heidelberg (2010). doi:10.1007/978-3-642-11440-3_18

    Chapter  Google Scholar 

  49. Tomita, E., Akutsu, T., Matsunaga, T.: Efficient algorithms for finding maximum and maximal cliques: Effective tools for bioinformatics. In: Laskovski, A.N. (ed.) Biomedical Engineering, Trends in Electronics, Communications and Software, pp. 625–640. InTech, Rijeka (2011). http://cdn.intechopen.com/pdfs-wm/12929.pdf

    Google Scholar 

  50. Tomita, E., Sutani, Y., Higashi, T., Wakatsuki, M.: A simple and faster branch-and-bound algorithm for finding a maximum clique with computational experiments. IEICE Trans. Inf. Syst. E96–D, 1286–1298 (2013). http://id.nii.ac.jp/1438/00000287/

    Article  MATH  Google Scholar 

  51. Tomita, E.: Clique enumeration. In: Kao, M.-Y. (ed.) Encyclopedia of Algorithms, 2nd edn, pp. 313–317. Springer, Heidelberg (2016)

    Google Scholar 

  52. Tomita, E., Yoshida, K., Hatta, T., Nagao, A., Ito, H., Wakatsuki, M.: A much faster branch-and-bound algorithm for finding a maximum clique. In: Zhu, D., Bereg, S. (eds.) FAW 2016. LNCS, vol. 9711, pp. 215–226. Springer, Heidelberg (2016)

    Chapter  Google Scholar 

  53. 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)

    Article  MathSciNet  MATH  Google Scholar 

  54. Wu, Q., Hao, J.K.: A review on algorithms for maximum clique problems - invited review-. Eur. J. Oper. Res. 242, 693–709 (2015)

    Article  MATH  Google Scholar 

  55. Yonemori, C., Matsunaga, T., Sekine, J., Tomita, E.: A structural analysis of enterprise relationship using cliques. DBSJ J. 7, 55–60 (2009)

    Google Scholar 

  56. Zhai, H., Haraguchi, M., Okubo, Y., Tomita, E.: Enumerating maximal clique sets with pseudo-clique constraint. In: Japkowicz, N., Matwin, S. (eds.) DS 2015. LNCS (LNAI), vol. 9356, pp. 324–339. Springer, Heidelberg (2015). doi:10.1007/978-3-319-24282-8_28

    Chapter  Google Scholar 

  57. Zhai, H., Haraguchi, M., Okubo, Y., Tomita, E.: A fast and complete algorithm for enumerating pseudo-cliques in large graphs. Int. J. Data Sci. Anal. 2, 145–158 (2016). Springer

    Article  Google Scholar 

Download references

Acknowledgments

The author would like to express his sincere gratitude to H. Ito, T. Akutsu, M. Haraguchi, Y. Okubo, T. Nishino, H. Takahashi and many others for their fruitful joint work and kind help. This work was supported by JSPS KAKENHI Grant Numbers JP16300001, JP19300040, JP19500010, JP21300047, JP22500009, JP25330009, Kayamori Foundation of Informational Science Advancement, Funai Foundation for Information Technologies, and others.

Author information

Authors and Affiliations

  1. The Advanced Algorithms Research Laboratory, The University of Electro-Communications, Chofugaoka 1-5-1, Chofu, Tokyo, 182-8585, Japan

    Etsuji Tomita

Authors
  1. Etsuji Tomita
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Etsuji Tomita .

Editor information

Editors and Affiliations

  1. Universiti Teknologi Brunei, Bandar Seri Begawan, Brunei Darussalam

    Sheung-Hung Poon

  2. Bangladesh University of Engineering and Technology, Dhaka, Bangladesh

    Md. Saidur Rahman

  3. National Taiwan University, Taipei, Taiwan

    Hsu-Chun Yen

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Tomita, E. (2017). Efficient Algorithms for Finding Maximum and Maximal Cliques and Their Applications. In: Poon, SH., Rahman, M., Yen, HC. (eds) WALCOM: Algorithms and Computation. WALCOM 2017. Lecture Notes in Computer Science(), vol 10167. Springer, Cham. https://doi.org/10.1007/978-3-319-53925-6_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-53925-6_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-53924-9

  • Online ISBN: 978-3-319-53925-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation