Skip to main content

Computational Challenges with Cliques, Quasi-cliques and Clique Partitions in Graphs

  • Conference paper
Experimental Algorithms (SEA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6049))

Included in the following conference series:

Abstract

During the last decade, many problems in social, biological, and financial networks require finding cliques, or quasi-cliques. Cliques or clique partitions have also been used as clustering or classification tools in data sets represented by networks. These networks can be very large and often massive and therefore external (or semi-external) memory algorithms are needed. We discuss four applications where we identify computational challenges which are both of practical and theoretical interest.

This research is partially supported by DTRA and Air Force grants.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Diestel, R.: Graph Theory. Electronic Edition 2000. Springer, New York (2000)

    Book  Google Scholar 

  2. West, D.B.: Introduction to Graph Theory, 2nd edn. Prentice-Hall, Englewood Cliffs (2000)

    Google Scholar 

  3. Karp, R.: Reducibility Among Combinatorial Problems. In: Miller, R.E., Thatcher, J. (eds.) Proceedings of a Symposium on the Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Google Scholar 

  4. Garey, M.R., Johnson, D.S.: Computers and Intractability, A guide to the Theory of NP-Completeness. In: Klee, V. (ed.) A series of books in the mathematical sciences. W. H. Freeman and Company, New York (1979)

    Google Scholar 

  5. Arora, S., Safra, S.: Probabilistic Checking of Proofs; a new Characterization of NP. In: Proceedings 33rd IEEE Symposium on Foundations of Computer Science, pp. 2–13. IEEE Computer Society, Los Angeles (1992)

    Chapter  Google Scholar 

  6. Lund, C., Yannakakis, M.: On the hardness of approximating minimization problmes. JACM 41, 960–981 (1994)

    Article  MATH  MathSciNet  Google Scholar 

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

    Google Scholar 

  8. Rebennack, S.: Stable Set Problem: Branch & Cut Algorithms. In: Floudas, C.A., Pardalos, P.M. (eds.) Encyclopedia of Optimization, 2nd edn., pp. 3676–3688. Springer, Heidelberg (2008)

    Google Scholar 

  9. Rebennack, S., Oswald, M., Theis, D., Seitz, H., Reinelt, G., Pardalos, P.: A Branch and Cut solver for the maximum stable set problem. Journal of Combinatorial Optimization, doi:10.1007/s10878-009-9264-3

    Google Scholar 

  10. Pardalos, P., Mavridou, T., Xue, J.: The graph coloring problem: a bibliographic survey. In: Du, D.Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, vol. 2, pp. 331–395. Kluwer Academic Publishers, Dordrecht (1990)

    Google Scholar 

  11. Brunato, M., Hoos, H., Battiti, R.: On Effectively Finding Maximal Quasi-cliques in Graphs. In: Maniezzo, V., Battiti, R., Watson, J.-P. (eds.) LION 2007 II. LNCS, vol. 5313, pp. 41–55. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  12. Hayes, B.: Graph Theory in Practice: Part I. American Scientist 88(1), 9 (2000)

    Google Scholar 

  13. Cipra, B.: Massive graphs pose big problems. Technical report, SIAM NEWS, April 22 (1999)

    Google Scholar 

  14. Abello, J., Resende, M., Sudarsky, S.: Massive Quasi-Clique Detection. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, p. 598. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Ye, Q., Wu, B., Suo, L., Zhu, T., Han, C., Wang, B.: TeleComVis: Exploring Temporal Communities in Telecom Networks. In: Buntine, W., Grobelnik, M., Mladenić, D., Shawe-Taylor, J. (eds.) ECML PKDD 2009. LNCS, vol. 5782, pp. 755–758. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  16. Abello, J., Pardalos, P., Resende, M.: On Maximum Clique Problems in Very Lagre Graphs. In: External Memory Algorithms. DIMACS Series, pp. 119–130. American Mathematical Society, Providence (1999)

    Google Scholar 

  17. Nanavati, A., Singh, R., Chakraborty, D., Dasgupta, K., Mukherjea, S., Das, G., Gurumurthy, S., Joshi, A.: Analyzing the Structure and Evolution of Massive Telecom Graphs. IEEE Transactions on Knowledge and Data Engineering 20(5), 703–718 (2008)

    Article  Google Scholar 

  18. Narasimhamurthy, A., Greene, D., Hurley, N., Cunningham, P.: Community Finding in Large Social Networks Through Problem Decomposition. Technical report, UCD School of Computer Science and Informatics (2008)

    Google Scholar 

  19. Faloutsos, M., Faloutsos, P., Faloutsos, C.: On Power-law Relationships of the Internet Topology. In: Proceedings of the ACM SIGCOMM Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, pp. 251–262 (1999)

    Google Scholar 

  20. Hayes, B.: Connecting the Dots: Can the tools of graph theory and social-network studies unravel the next big plot? American Scientist 94(5), 400 (2006)

    Google Scholar 

  21. Schintler, L., Gorman, S., Reggiani, A., Patuelli, R., Nijkamp, P.: Small-World Phenomena in Communications Networks: A Cross-Atlantic Comparison. Advances in Spatial Science. In: Methods and Models in Transport and Telecommunications, pp. 201–220. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  22. Butenko, S., Chaovalitwongse, W., Pardalos, P. (eds.): Clustering Challenges in Biological Networks. World Scientific, Singapore (2009)

    Google Scholar 

  23. Butenko, S., Pardalos, P., Sergieko, I., Shylo, V., Stetsyuk, P.: Estimating the size of correcting codes using extremal graph problems. In: Optimization: Structure and Applications. Springer Optimization and Its Applications, vol. 32, pp. 227–243. Springer, Heidelberg (2009)

    Google Scholar 

  24. van Pul, C., Etzion, T.: New lower bounds for constatn weight codes. IEEE Trans. Inform. Theory 35, 1324–1329 (1989)

    Article  MATH  Google Scholar 

  25. Gendreau, M., Laporte, G., Semet, F.: Solving an ambulance location model by tabu search. Location Science 5, 75–88 (1997)

    Article  MATH  Google Scholar 

  26. Brotcorne, L., Laporte, G., Semet, F.: Fast heuristics for large scale covering ocation problems. Computers and Operations Research 29, 651–665 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mitchell, E., Artymiuk, P., Rice, D., Willett, P.: Use of techniques derived from graph theory to compare secondary structure motifs in proteins. J. Mol. Biol. 212, 151 (1990)

    Article  Google Scholar 

  28. Brint, A., Willett, P.: Algorithms for the Identification of Three-Dimensional Maximal Common Substructures. J. Chem. ZnJ Comput. Sci. 27, 152–158 (1987)

    Google Scholar 

  29. Gardiner, E., Artymiuk, P., Willett, P.: Clique-detection algorithms for matching three-dimensional molecular structures. Journal of Molecular Graphics and Modelling 15, 245–253 (1997)

    Article  Google Scholar 

  30. Raymond, J., Willett, P.: Maximum common subgraph isomorphism algorithms for the matching of chemical structures. Journal of Computer-Aided Molecular Design 16, 521–533 (2002)

    Article  Google Scholar 

  31. Gardiner, E., Willett, P., Artymiuk, P.: Graph-theoretic techniques for macromolecular docking. J. Chem. Inf. Comput. 40, 273–279 (2000)

    Google Scholar 

  32. Butenko, S., Wilhelm, W.: Clique-detection models in computational biochemistry and genomics. Euorpean Journal of Operational Research 173, 1–17 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Keller, O.: Über die lückenlose Einfüllung des Raumes mit Würfeln. J. Reine Angew. Math. 163, 231–248 (1930)

    MATH  Google Scholar 

  34. Minkowski, H.: Diophantische Approximationen. Teubner, Leipzig

    Google Scholar 

  35. Stein, S., Szabó, S.: Algebra and Tiling: Homomorphisms in the Service of Geometry. The Carus Mathematical Monographs, vol. 25. The Mathematical Associtaion of America (1994)

    Google Scholar 

  36. Hajós, G.: Sur la factorisation des abeliens. Casopis 50, 189–196

    Google Scholar 

  37. Perron, O.: Über lückenlose Ausfüllung des n-dimensioanlen Raumes durch kongruente Würfel. Math. Z. 46, 161–180 (1940)

    Article  MATH  MathSciNet  Google Scholar 

  38. Lagarias, J., Shor, P.: Keller’s Cube-Tiling Conjecture is False in High Dimensions. Bulletin AMS 27, 279–283 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  39. Mackey, J.: A Cube Tiling of Dimension Eight with No Facesharing. Discrete Comput. Geom. 28, 275–279 (2002)

    MATH  MathSciNet  Google Scholar 

  40. Corrádi, K., Szabó, S.: A Combinatorial Approach for Keller’s Conjecture. Periodica Math. Hung. 21(2), 95–100 (1990)

    Article  MATH  Google Scholar 

  41. Hasselberg, J., Pardalos, P., Vairaktarakis, G.: Test Case Generators and Computational Results for the Maximum Clique Problem. Journal of Global Optimization 3, 463–482 (1993)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pardalos, P.M., Rebennack, S. (2010). Computational Challenges with Cliques, Quasi-cliques and Clique Partitions in Graphs. In: Festa, P. (eds) Experimental Algorithms. SEA 2010. Lecture Notes in Computer Science, vol 6049. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13193-6_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-13193-6_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-13192-9

  • Online ISBN: 978-3-642-13193-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics