Skip to main content
SpringerLink
Log in

Editing Graphs into Disjoint Unions of Dense Clusters

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

In the Π-Cluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k≥0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K=(V K ,E K ) satisfies Π. The well-studied Cluster Editing problem is a special case of this problem with Π:=“being a clique”. In this work, we consider three other density measures that generalize cliques: (1) having at most s missing edges (s-defective cliques), (2) having average degree at least |V K |−s (average-s-plexes), and (3) having average degree at least μ⋅(|V K |−1) (μ-cliques), where s and μ are a fixed integer and a fixed rational number, respectively. We first show that the Π-Cluster Editing problem is NP-complete for all three density measures. Then, we study the fixed-parameter tractability of the three clustering problems, showing that the first two problems are fixed-parameter tractable with respect to the parameter (s,k) and that the third problem is W[1]-hard with respect to the parameter k for 0<μ<1.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abello, J., Resende, M.G.C., Sudarsky, S.: Massive quasi-clique detection. In: Proceedings of the 5th Latin American Symposium on Theoretical Informatics (LATIN ’02). Lecture Notes in Computer Science, vol. 2286, pp. 598–612. Springer, Berlin (2002)

    Chapter  Google Scholar 

  2. Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. J. ACM 55(5), 23 (2008), 27 pp.

    Article  MathSciNet  Google Scholar 

  3. Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1–3), 89–113 (2004)

    Article  MATH  Google Scholar 

  4. Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: parameterized algorithms for cluster editing. Theor. Comput. Sci. 410(52), 5467–5480 (2009)

    Article  MATH  Google Scholar 

  5. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    Article  MATH  Google Scholar 

  6. Cao, Y., Chen, J.: Weighted cluster editing: kernelization based on edge-cuts. In: Proceedings of the 5th International Symposium on Parameterized and Exact Computation (IPEC ’10). Lecture Notes in Computer Science. Springer, Berlin (2010)

    Google Scholar 

  7. Chen, J., Meng, J.: A 2k kernel for the cluster editing problem. In: Proceedings of the 16th Annual International Conference on Computing and Combinatorics (COCOON ’10). Lecture Notes in Computer Science, vol. 6196, pp. 459–468. Springer, Berlin (2010)

    Google Scholar 

  8. Chesler, E.J., Lu, L., Shou, S., Qu, Y., Gu, J., Wang, J., Hsu, H.C., Mountz, J.D., Baldwin, N.E., Langston, M.A., Threadgill, D.W., Manly, K.F., Williams, R.W.: Complex trait analysis of gene expression uncovers polygenic and pleiotropic networks that modulate nervous system function. Nat. Genet. 37(3), 233–242 (2005)

    Article  Google Scholar 

  9. Dehne, F.K.H.A., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: Implementations and experiments. In: Proceedings of the 2nd International Workshop on Parameterized and Exact Computation (IWPEC ’06). Lecture Notes in Computer Science, vol. 4169, pp. 13–24. Springer, Berlin (2006)

    Chapter  Google Scholar 

  10. Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999)

    Book  Google Scholar 

  11. Fellows, M.R., Langston, M.A., Rosamond, F.A., Shaw, P.: Efficient parameterized preprocessing for Cluster Editing. In: Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT ’07). Lecture Notes in Computer Science, vol. 4639, pp. 312–321. Springer, Berlin (2007)

    Chapter  Google Scholar 

  12. Fellows, M.R., Hermelin, D., Rosamond, F.A., Vialette, S.: On the parameterized complexity of multiple-interval graph problems. Theor. Comput. Sci. 410(1), 53–61 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  14. Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: exact algorithms for clique generation. Theory Comput. Syst. 38(4), 373–392 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Greenwell, D.L., Hemminger, R.L., Klerlein, J.B.: Forbidden subgraphs. In: Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, pp. 389–394 (1973)

    Google Scholar 

  16. Guo, J.: A more effective linear kernelization for Cluster Editing. Theor. Comput. Sci. 410(8–10), 718–726 (2009)

    Article  MATH  Google Scholar 

  17. Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: A more relaxed model for graph-based data clustering: s-plex cluster editing. SIAM J. Discrete Math. 24(4), 1662–1683 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Harary, F.: The maximum connectivity of a graph. Proc. Natl. Acad. Sci. USA 48(7), 1142–1146 (1962)

    Article  MATH  Google Scholar 

  19. Kosub, S.: Local density. In: Network Analysis. Lecture Notes in Computer Science, vol. 3418, pp. 112–142. Springer, Berlin (2004)

    Chapter  Google Scholar 

  20. Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Inform. 23(3), 311–323 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  21. Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, London (2006)

    Book  MATH  Google Scholar 

  22. Seidman, S.B., Foster, B.L.: A graph-theoretic generalization of the clique concept. J. Math. Sociol. 6, 139–154 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  23. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yu, H., Paccanaro, A., Trifonov, V., Gerstein, M.: Predicting interactions in protein networks by completing defective cliques. Bioinformatics 22(7), 823–829 (2006)

    Article  Google Scholar 

  25. van Zuylen, A., Williamson, D.P.: Deterministic pivoting algorithms for constrained ranking and clustering problems. Math. Oper. Res. 34(3), 594–620 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität des Saarlandes, Campus E 1.7, 66123, Saarbrücken, Germany

    Jiong Guo

  2. School of Computing, DePaul University, 243. S. Wabash Avenue, Chicago, IL, 60604, USA

    Iyad A. Kanj

  3. Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Christian Komusiewicz & Johannes Uhlmann

Authors
  1. Jiong Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Iyad A. Kanj
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Komusiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Johannes Uhlmann
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johannes Uhlmann.

Additional information

Research of J. Guo was supported by the Excellence Cluster on Multimodal Computing and Interaction (MMCI). Main work was done while the author was with the Friedrich-Schiller-Universität Jena.

Part of I.A. Kanj’s work was done while the author was visiting the Friedrich-Schiller-Universität Jena.

Research of C. Komusiewicz was supported by a PhD fellowship of the Carl-Zeiss-Stiftung and the DFG, research project PABI, NI 369/7.

Research of J. Uhlmann was supported by the DFG, research project PABI, NI 369/7.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, J., Kanj, I.A., Komusiewicz, C. et al. Editing Graphs into Disjoint Unions of Dense Clusters. Algorithmica 61, 949–970 (2011). https://doi.org/10.1007/s00453-011-9487-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9487-4

Keywords

Search

Navigation