In the NP-hard Cluster Editing problem, we have as input an undirected graph G and an integer k ≥ 0. The question is whether we can transform G, by inserting and deleting at most k edges, into a cluster graph, that is, a union of disjoint cliques. We first confirm a conjecture by Michael Fellows [IWPEC 2006] that there is a polynomial-time kernelization for Cluster Editing that leads to a problem kernel with at most 6k vertices. More precisely, we present a cubic-time algorithm that, given a graph G and an integer k ≥ 0, finds a graph G′ and an integer k′ ≤ k such that G can be transformed into a cluster graph by at most k edge modifications iff G′ can be transformed into a cluster graph by at most k′ edge modifications, and the problem kernel G′ has at most 6k vertices. So far, only a problem kernel of 24k vertices was known. Second, we show that this bound for the number of vertices of G′ can be further improved to 4k. Finally, we consider the variant of Cluster Editing where the number of cliques that the cluster graph can contain is stipulated to be a constant d > 0. We present a simple kernelization for this variant leaving a problem kernel of at most (d + 2)k + d vertices.


Domination Number Reduction Rule Cluster Graph Proximation Algorithm Edge Insertion 
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  1. 1.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: ranking and clustering. In: Proc. 37th ACM STOC, pp. 684–693. ACM Press, New York (2005)Google Scholar
  2. 2.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial time data reduction for Dominating Set. Journal of the ACM 51(3), 363–384 (2004)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning 56(1), 89–113 (2004)zbMATHCrossRefGoogle Scholar
  4. 4.
    Ben-Dor, A., Shamir, R., Yakhini, Z.: Clustering gene expression patterns. Journal of Computational Biology 6(3/4), 281–297 (1999)CrossRefGoogle Scholar
  5. 5.
    Charikar, M., Guruswami, V., Wirth, A.: Clustering with qualitative information. Journal of Computer and System Sciences 71(3), 360–383 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: Lower bounds and upper bounds on kernel size. In: Diekert, V., Durand, B. (eds.) STACS 2005. LNCS, vol. 3404, pp. 269–280. Springer, Heidelberg (2005)Google Scholar
  7. 7.
    Chen, Z.-Z., Jiang, T., Lin, G.: Computing phylogenetic roots with bounded degrees and errors. SIAM Journal on Computing 32(4), 864–879 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dehne, F., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The Cluster Editing problem: Implementations and experiments. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 13–24. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Extending the tractability border for closest leaf powers. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 397–408. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R.: Error compensation in leaf power problems. Algorithmica 44(4), 363–381 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  12. 12.
    Fellows, M.R.: The lost continent of polynomial time: Preprocessing and kernelization. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 276–277. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  13. 13.
    Fellows, M.R., Langston, M.A., Rosamond, F., Shaw, P.: Polynomial-time linear kernelization for Cluster Editing. Manuscript (2006)Google Scholar
  14. 14.
    Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. In: Proc. 17th ACM-SIAM SODA, pp. 1167–1176. ACM Press, New York (2006)CrossRefGoogle Scholar
  15. 15.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Graph-modeled data clustering: Exact algorithms for clique generation. Theory of Computing Systems 38(4), 373–392 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  17. 17.
    Hsu, W., Ma, T.: Substitution decomposition on chordal graphs and applications. In: Hsu, W.-L., Lee, R.C.T. (eds.) ISA 1991. LNCS, vol. 557, pp. 52–60. Springer, Heidelberg (1991)Google Scholar
  18. 18.
    Křivánek, M., Morávek, J.: NP-hard problems in hierarchical-tree clustering. Acta Informatica 23(3), 311–323 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lin, G., Kearney, P.E., Jiang, T.: Phylogenetic k-root and Steiner k-root. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 539–551. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  20. 20.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  21. 21.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized bicluster editing. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 1–12. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  22. 22.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Applied Mathematics 144, 173–182 (2004)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Jiong Guo
    • 1
  1. 1.Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 JenaGermany

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