Skip to main content
SpringerLink
Log in

A New Polynomial Algorithm for Cluster Deletion Problem

  • Chapter
  • First Online:
Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD 2018)

Part of the book series: Studies in Computational Intelligence ((SCI,volume 790))

  • 496 Accesses

Abstract

Given a simple graph G, the cluster deletion problem asks for transforming G into a disjoint union of cliques by removing as few edges as possible. This optimization problem is known as the Cluster Deletion (CD) problem and, for general graphs, it belongs to the NP-hard computational complexity class. In the present paper, we propose graph reduction that enable the identification of new polynomially solvable CD sub-problem. Specifically, we show that if a graph is (butterfly,diamond)-free then a cluster deletion solution can be found in polynomial time on that graph.

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 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover 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. Le, V.B.: Gallai graphs and anti-Gallai graphs. Discrete Math. 159(1–3), 179–189 (1996)

    Article  MathSciNet  Google Scholar 

  2. Mamalis, B., Gavalas, D., Konstantopoulos, C., Pantziou, G.: Chapter 12: Clustering in Wireless Sensor Networks, RFID and Sensor Networks: Architectures, Protocols, Security, and Integrations (2009)

    Google Scholar 

  3. Zahn, C.T.: Approximating symmetric relations by equivalence relations. J. Soc. Ind. Appl. Math. 12, 840–847 (1964)

    Article  MathSciNet  Google Scholar 

  4. Pipenbacher, P., Schliep, A., Schneckener, S., Schnhuth, A., Schomburg, D., Schrader, R.: ProClust: improved clustering of protein sequences with an extended graph-based approach. Bioinformatics 18(Suppl. 2), 182–191 (1964)

    Google Scholar 

  5. Guo, J.: A more effective linear kernelization for cluster editing. Theoret. Comput. Sci. 410, 718–726 (2009)

    Article  MathSciNet  Google Scholar 

  6. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144, 173–182 (2002)

    Article  MathSciNet  Google Scholar 

  7. Vadim, V.L.: Conic reduction of graphs for the stable set problem. Discrete Math. 222(1–3), 199–211 (2000)

    MathSciNet  Google Scholar 

  8. Lovász, L., Plummer, M.D.: Matching Theory. Akadémiai Kiadó, vol. 121 of the North-Holland Mathematics Studies, North-Holland Publishing, Amsterdam (1986)

    Google Scholar 

  9. Brandstädt, A.L., Hammer, P.: On the stability number of claw-free P5-free and more general graphs. Discrete Appl. Math. 95(1–3), 163–167 (1999)

    Article  MathSciNet  Google Scholar 

  10. Catlin, P.A.: A reduction method for graphs. Congressus Numerantium 65, 159–170 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Goldman, A., Ngoko, Y.: On graph reduction for QoS prediction of very large web service compositions. In: International Conference on Service Computing, Honolulu, US (2012)

    Google Scholar 

  12. Cardoso, J., Sheth, A., Miller, J., Arnold, J., Kochut, K.: Quality of service for workflows and web service processes. Web Semant. Sci. Serv. Agents World Wide Web 1(3), 281–308 (2004)

    Article  Google Scholar 

  13. Gao, Y., Hare, D.R., Nastos, J.: The cluster deletion problem for cographs. Discrete Math. 313(23), 2763–2771 (2013)

    Article  MathSciNet  Google Scholar 

  14. Bonomo, F., Durn, G., Napoli, A., Valencia-Pabon, M.: A one-to-one correspondence between potential solutions of the cluster deletion problem and the minimum sum coloring problem, and its application to \(P_4\)-sparse graphs. Inform. Process. Lett. 115, 600–603 (2015)

    Google Scholar 

  15. Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. J. Res. Natl. Bur. Stand. B Math. Math. Phys. 69B, 125–130 (1965)

    Article  MathSciNet  Google Scholar 

  16. Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113, 109–128 (2001)

    Article  MathSciNet  Google Scholar 

  17. Bron, C., Kerbosch, J.: Finding all cliques of an undirected graph (Algorithm 457). Commun. ACM 16(9), 575–576 (1973)

    Article  Google Scholar 

  18. Cazals, F., Karande, C.: A note on the problem of reporting maximal cliques. Theor. Comput. Sci. 407(1–3), 564–568 (2008). https://doi.org/10.1016/j.tcs.2008.05.010

    Article  MathSciNet  MATH  Google Scholar 

  19. Tomita, E., Kameda, T.: An efficient branch-and-bound algorithm for finding a maximum clique with computational experiments. J. Global Optim. 37(1), 95–111 (2007)

    Article  MathSciNet  Google Scholar 

  20. Malek, S., Naanaa, W.: A new polynomial cluster deletion subproblem. In: 13th IEEE/ACS International Conference of Computer Systems and Applications, pp. 1–8 (2016)

    Google Scholar 

  21. Ur Rehman, S., Ullah, K.A., Fong, S.: Graph mining: a survey of graph mining techniques. In: International Conference on Digital Information Management, ICDIM, pp. 88–92 (2012). https://doi.org/10.1109/ICDIM.2012.6360146

  22. Fellows, M.R., Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Graph-based data clustering with overlaps. Discrete Optim. 8, 2–17 (2011)

    Article  MathSciNet  Google Scholar 

  23. Bossard, A.: The decycling problem in hierarchical cubic networks. J. Supercomputing 69(1), 293–305 (2014)

    Article  Google Scholar 

  24. David. E., Maarten, L., Darren, S.: Listing all maximal cliques in sparse graphs in near-optimal time. In: Exact Complexity of NP-hard Problems, 31 October–05 November 2010 (2010)

    Google Scholar 

  25. Alessio C., Review of the Bron-Kerbosch algorithm and variations, report, School of Computing Science, Sir Alwyn Williams Building, University of Glasgow (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Economics and Management of Sfax, Tunis, Tunisia

    Sabrine Malek

  2. National Engineering School of Tunis, Tunis, Tunisia

    Wady Naanaa

Authors
  1. Sabrine Malek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wady Naanaa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sabrine Malek .

Editor information

Editors and Affiliations

  1. Software Engineering and Information Technology Institute, Central Michigan University, Mount Pleasant, MI, USA

    Roger Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Malek, S., Naanaa, W. (2019). A New Polynomial Algorithm for Cluster Deletion Problem. In: Lee, R. (eds) Software Engineering, Artificial Intelligence, Networking and Parallel/Distributed Computing. SNPD 2018. Studies in Computational Intelligence, vol 790. Springer, Cham. https://doi.org/10.1007/978-3-319-98367-7_7

Download citation

Publish with us

Policies and ethics

Search

Navigation