On the Parameterized Complexity of Consensus Clustering

  • Martin Dörnfelder
  • Jiong Guo
  • Christian Komusiewicz
  • Mathias Weller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7074)


Given a collection \({\mathcal{C}}\) of partitions of a base set S, the NP-hard Consensus Clustering problem asks for a partition of S which has a total Mirkin distance of at most t to the partitions in \({\mathcal{C}}\), where t is a nonnegative integer. We present a parameterized algorithm for Consensus Clustering with running time \(O(4.24^k\cdot k^3+|{\mathcal C}|\cdot |S|^2)\), where \(k:=t/|{\mathcal{C}}|\) is the average Mirkin distance of the solution partition to the partitions of \({\mathcal{C}}\). Furthermore, we strengthen previous hardness results for Consensus Clustering, showing that Consensus Clustering remains NP-hard even when all input partitions contain at most two subsets. Finally, we study a local search variant of Consensus Clustering, showing W[1]-hardness for the parameter “radius of the Mirkin-distance neighborhood”. In the process, we also consider a local search variant of the related Cluster Editing problem, showing W[1]-hardness for the parameter “radius of the edge modification neighborhood”.


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  1. 1.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Mach. Learn. 56(1), 89–113 (2004)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bertolacci, M., Wirth, A.: Are approximation algorithms for consensus clustering worthwhile? In: Proc. 7th SDM, pp. 437–442. SIAM (2007)Google Scholar
  3. 3.
    Betzler, N., Guo, J., Komusiewicz, C., Niedermeier, R.: Average parameterization and partial kernelization for computing medians. J. Comput. Syst. Sci. 77(4), 774–789 (2011)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bonizzoni, P., Vedova, G.D., Dondi, R.: A PTAS for the minimum consensus clustering problem with a fixed number of clusters. In: Proc. 11th ICTCS (2009)Google Scholar
  5. 5.
    Bonizzoni, P., Vedova, G.D., Dondi, R., Jiang, T.: On the approximation of correlation clustering and consensus clustering. J. Comput. Syst. Sci. 74(5), 671–696 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Coleman, T., Wirth, A.: A polynomial time approximation scheme for k-consensus clustering. In: Proc. 21st SODA, pp. 729–740. SIAM (2010)Google Scholar
  7. 7.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefMATHGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)MATHGoogle Scholar
  9. 9.
    Gionis, A., Mannila, H., Tsaparas, P.: Clustering aggregation. ACM Trans. Knowl. Discov. Data 1(1) (2007)Google Scholar
  10. 10.
    Goder, A., Filkov, V.: Consensus clustering algorithms: Comparison and refinement. In: Proc. 10th ALENEX, pp. 109–117. SIAM (2008)Google Scholar
  11. 11.
    Karpinski, M., Schudy, W.: Linear time approximation schemes for the Gale-Berlekamp game and related minimization problems. In: Proc. 41st STOC, pp. 313–322. ACM (2009)Google Scholar
  12. 12.
    Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Inform. 23(3), 311–323 (1986)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Monti, S., Tamayo, P., Mesirov, J.P., Golub, T.R.: Consensus clustering: A resampling-based method for class discovery and visualization of gene expression microarray data. Mach. Learn. 52(1-2), 91–118 (2003)CrossRefMATHGoogle Scholar
  14. 14.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press (2006)Google Scholar
  15. 15.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1-2), 173–182 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wakabayashi, Y.: The complexity of computing medians of relations. Resenhas 3(3), 323–350 (1998)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Martin Dörnfelder
    • 1
  • Jiong Guo
    • 1
  • Christian Komusiewicz
    • 2
  • Mathias Weller
    • 2
  1. 1.Universität des SaarlandesSaarbrückenGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTechnische Universität BerlinBerlinGermany

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