On the Min-Max 2-Cluster Editing Problem

  • Li-Hsuan Chen
  • Maw-Shang Chang
  • Chun-Chieh Wang
  • Bang Ye Wu
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 20)


In this paper, we study the problem Min-Max 2-Cluster Editing which asks for a modification of a given graph into two maximal cliques by inserting or deleting edges such that the maximum number k of the editing edges incident to any vertex is minimized. We show the NP-hardness of the problem and present a polynomial-time algorithm when k < n/4, in which n is number of vertices. In addition, we design a 2-approximation algorithm and a branching algorithm for finding an optimal solution. By experiments on random graphs, we show that the exact algorithm is much more efficient than a trivial one.


algorithm clustering editing graph modification problem NP-hard approximation algorithm 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. J. ACM 55(5), 1–27 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial optimization problems and their approximability properties. Springer (1999)Google Scholar
  3. 3.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning, Special Issue on Clustering 56, 89–113 (2004)MATHGoogle Scholar
  4. 4.
    Böcker, S., Damaschke, P.: Even faster parameterized cluster deletion and cluster editing. Information Processing Letters 111(14), 717–721 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Böcker, S., Briesemeister, S., Bui, Q.B.A., Truss, A.: Going weighted: Parameterized algorithm for cluster editing. Theor. Comput. Sci. 410(52), 5467–5480 (2009)MATHCrossRefGoogle Scholar
  6. 6.
    Bonizzoni, P., Della Vedova, G., Dondi, R.: A ptas for the minimum consensus clustering problem with a fixed number of clusters. In: Proc. Eleventh Italian Conference on Theoretical Computer Science (2009)Google Scholar
  7. 7.
    Bonizzoni, P., Della Vedova, G., Dondi, R., Jiang, T.: On the approximation of correlation clustering and consensus clustering. Journal of Computer and System Sciences 74(5), 671–696 (2008)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Chen, J., Meng, J.: A 2k Kernel for the Cluster Editing Problem. In: Thai, M.T., Sahni, S. (eds.) COCOON 2010. LNCS, vol. 6196, pp. 459–468. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  9. 9.
    Damaschke, P.: Bounded-Degree Techniques Accelerate Some Parameterized Graph Algorithms. In: Chen, J., Fomin, F.V. (eds.) IWPEC 2009. LNCS, vol. 5917, pp. 98–109. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Damaschke, P.: Fixed-parameter enumerability of cluster editing and related problems. Theory Computing Syst. 46, 261–283 (2010)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fellows, M.R., Guo, J., Komusiewicz, C., Niedermeier, R., Uhlmann, J.: Graph-based data clustering with overlaps. Discrete Optimization 8(1), 2–17 (2011)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Filkov, V., Skiena, S.: Integrating microarray data by consensus clustering. International Journal on Artificial Intelligence Tools 13(4), 863–880 (2004)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, NewYork (1979)MATHGoogle Scholar
  14. 14.
    Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. Theory Comput. 2, 249–266 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Fixedparameter algorithms for clique generation. Theory Computing Syst. 38, 373–392 (2005)MATHCrossRefGoogle Scholar
  16. 16.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39, 321–347 (2004)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Guo, J.: A more effective linear kernelization for cluster editing. Theor. Comput. Sci. 410, 718–726 (2009)MATHCrossRefGoogle Scholar
  18. 18.
    Harary, F.: On the notion of balance of a signed graph. Michigan Mathematical Journal 2(2), 143–146 (1953)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hüffner, F., Komusiewicz, C., Moser, H., Niedermeier, R.: Fixed-parameter algorithms for cluster vertex deletion. Theory of Computing Systems 47(1), 196–217 (2010)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|}|E|)\) algorithm for finding maximum matching in general graphs. In: FOCS, pp. 17–27 (1980)Google Scholar
  21. 21.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discr. Appl. Math. 144(1-2), 173–182 (2004)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Talmaciu, M., Nechita, E.: Recognition algorithm for diamond-free graphs. Informatica 18(3), 457–462 (2007)MathSciNetMATHGoogle Scholar
  23. 23.
    Wasserman, S., Faust, K.: Social Network Analysis. Cambridge University Press, Cambridge (1994)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Li-Hsuan Chen
    • 1
  • Maw-Shang Chang
    • 2
  • Chun-Chieh Wang
    • 1
  • Bang Ye Wu
    • 1
  1. 1.National Chung Cheng UniversityChiaYiTaiwan, R.O.C.
  2. 2.Hungkung UniversityTaichungTaiwan, R.O.C.

Personalised recommendations