Advertisement

Theory of Computing Systems

, Volume 47, Issue 1, pp 196–217 | Cite as

Fixed-Parameter Algorithms for Cluster Vertex Deletion

  • Falk Hüffner
  • Christian Komusiewicz
  • Hannes Moser
  • Rolf Niedermeier
Article

Abstract

We initiate the first systematic study of the NP-hard Cluster Vertex Deletion (CVD) problem (unweighted and weighted) in terms of fixed-parameter algorithmics. In the unweighted case, one searches for a minimum number of vertex deletions to transform a graph into a collection of disjoint cliques. The parameter is the number of vertex deletions. We present efficient fixed-parameter algorithms for CVD applying the fairly new iterative compression technique. Moreover, we study the variant of CVD where the maximum number of cliques to be generated is prespecified. Here, we exploit connections to fixed-parameter algorithms for (weighted) Vertex Cover.

Keywords

Parameterized complexity Iterative compression NP-hard problems Graph algorithms Clustering 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abu-Khzam, F.N.: Kernelization algorithms for d-hitting set problems. In: Proc. 10th WADS. Lecture Notes in Computer Science, vol. 4619, pp. 434–445. Springer, Berlin (2007) Google Scholar
  2. 2.
    Abu-Khzam, F.N., Fernau, H.: Kernels: Annotated, proper and induced. In: Proc. 2nd IWPEC. Lecture Notes in Computer Science, vol. 4169, pp. 264–275. Springer, Berlin (2006) Google Scholar
  3. 3.
    Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. In: Proc. 37th STOC, pp. 684–693. Assoc. Comput. Math., New York (2005) Google Scholar
  4. 4.
    Ailon, N., Charikar, M., Newman, A.: Proofs of conjectures in “Aggregating inconsistent information: Ranking and clustering”. Technical Report TR-719-05, Department of Computer Science, Princeton University (2005) Google Scholar
  5. 5.
    Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: A fixed-parameter approach for weighted cluster editing. In: Proc. 6th APBC. Series on Advances in Bioinformatics and Computational Biology, vol. 5, pp. 211–220. Imperial College Press, London (2008) Google Scholar
  6. 6.
    Böcker, S., Briesemeister, S., Bui, Q.B.A., Truß, A.: Going weighted: Parameterized algorithms for cluster editing. In: Proc. 2nd COCOA. Lecture Notes in Computer Science, vol. 5165, pp. 1–12. Springer, Berlin (2008) Google Scholar
  7. 7.
    Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996) zbMATHCrossRefGoogle Scholar
  8. 8.
    Cai, M.-C., Deng, X., Zang, W.: An approximation algorithm for feedback vertex sets in tournaments. SIAM J. Comput. 30(6), 1993–2007 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Chen, J., Kanj, I.A., Xia, G.: Improved parameterized upper bounds for vertex cover. In: Proc. 31st MFCS. Lecture Notes in Computer Science, vol. 4162, pp. 238–249. Springer, Berlin (2006) Google Scholar
  10. 10.
    Dehne, F., Langston, M.A., Luo, X., Pitre, S., Shaw, P., Zhang, Y.: The cluster editing problem: Implementations and experiments. In: Proc. 2nd IWPEC. Lecture Notes in Computer Science, vol. 4169, pp. 13–24. Springer, Berlin (2006) Google Scholar
  11. 11.
    Dom, M., Guo, J., Hüffner, F., Niedermeier, R., Truß, A.: Fixed-parameter tractability results for feedback set problems in tournaments. In: Proc. 6th CIAC. Lecture Notes in Computer Science, vol. 3998, pp. 320–331. Springer, Berlin (2006) Google Scholar
  12. 12.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Berlin (1999) Google Scholar
  13. 13.
    Fellows, M.R., Langston, M.A., Rosamond, F.A., Shaw, P.: Efficient parameterized preprocessing for cluster editing. In: Proc. 16th FCT. Lecture Notes in Computer Science, vol. 4639, pp. 312–321. Springer, Berlin (2007) Google Scholar
  14. 14.
    Fernau, H.: Parameterized algorithms for hitting set: The weighted case. In: Proc. 6th CIAC. Lecture Notes in Computer Science, vol. 3998, pp. 332–343. Springer, Berlin (2006) Google Scholar
  15. 15.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  16. 16.
    Fomin, F.V., Gaspers, S., Kratsch, D., Liedloff, M., Saurabh, S.: Iterative compression and exact algorithms. In: Proc. 33rd MFCS. Lecture Notes in Computer Science, vol. 5162, pp. 335–346. Springer, Berlin (2008) Google Scholar
  17. 17.
    Fredman, M.L., Tarjan, R.E.: Fibonacci heaps and their uses in improved network optimization algorithms. J. ACM 34(3), 596–615 (1987) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Gabow, H.N., Tarjan, R.E.: Faster scaling algorithms for network problems. SIAM J. Comput. 18(5), 1013–1036 (1989) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. Theory Comput. 2, 249–266 (2006) CrossRefMathSciNetGoogle Scholar
  20. 20.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Automated generation of search tree algorithms for hard graph modification problems. Algorithmica 39(4), 321–347 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    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) zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Guo, J.: A more effective linear kernelization for cluster editing. In: Proc. 1st ESCAPE. Lecture Notes in Computer Science, vol. 4614, pp. 36–47. Springer, Berlin (2007). Long version to appear in Theoretical Computer Science Google Scholar
  23. 23.
    Guo, J., Moser, H., Niedermeier, R.: Iterative compression for exactly solving NP-hard minimization problems. In: Algorithmics of Large and Complex Networks. Lecture Notes in Computer Science. Springer, Berlin (2008, to appear) Google Scholar
  24. 24.
    Hüffner, F.: Algorithms and experiments for parameterized approaches to hard graph problems. Ph.D. thesis, Institut für Informatik, Friedrich-Schiller-Universität Jena (2007) Google Scholar
  25. 25.
    Hüffner, F., Niedermeier, R., Wernicke, S.: Techniques for practical fixed-parameter algorithms. Comput. J. 51(1), 7–25 (2008) CrossRefGoogle Scholar
  26. 26.
    Jansen, K., Scheffler, P., Woeginger, G.: The disjoint cliques problem. RAIRO Rech. Opér. 31(1), 45–66 (1997) zbMATHMathSciNetGoogle Scholar
  27. 27.
    Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. Syst. Sci. 20(2), 219–230 (1980) zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Lund, C., Yannakakis, M.: The approximation of maximum subgraph problems. In: Proc. 20th ICALP. Lecture Notes in Computer Science, vol. 700, pp. 40–51. Springer, Berlin (1993) Google Scholar
  29. 29.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and Its Applications, vol. 31. Oxford University Press, London (2006) zbMATHGoogle Scholar
  30. 30.
    Niedermeier, R., Rossmanith, P.: On efficient fixed-parameter algorithms for weighted vertex cover. J. Algorithms 47(2), 320–331 (2003) MathSciNetGoogle Scholar
  31. 31.
    Niedermeier, R., Rossmanith, P.: A general method to speed up fixed-parameter-tractable algorithms. Inf. Process. Lett. 73, 125–129 (2006) CrossRefMathSciNetGoogle Scholar
  32. 32.
    Protti, F., da Silva, M.D., Szwarcfiter, J.L.: Applying modular decomposition to parameterized cluster editing problems. Theory Comput. Syst. (2008, to appear) Google Scholar
  33. 33.
    Rahmann, S., Wittkop, T., Baumbach, J., Martin, M., Truß, A., Böcker, S.: Exact and heuristic algorithms for weighted cluster editing. In: Proc. 6th CSB. Computational Systems Bioinformatics, vol. 6, pp. 391–401. Imperial College Press, London (2007) Google Scholar
  34. 34.
    Raman, V., Saurabh, S.: Parameterized algorithms for feedback set problems and their duals in tournaments. Theor. Comput. Sci. 351(3), 446–458 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Reed, B., Smith, K., Vetta, A.: Finding odd cycle transversals. Oper. Res. Lett. 32(4), 299–301 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004) zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Wahlström, M.: Algorithms, measures and upper bounds for satisfiability and related problems. Ph.D. thesis, Department of Computer and Information Science, Linköpings universitet (2007) Google Scholar
  38. 38.
    van Zuylen, A., Williamson, D.P.: Deterministic algorithms for rank aggregation and other ranking and clustering problems. In: Proc. 5th WAOA. Lecture Notes in Computer Science, vol. 4927, pp. 260–273. Springer, Berlin (2008) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Falk Hüffner
    • 1
  • Christian Komusiewicz
    • 1
  • Hannes Moser
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Institut für InformatikFriedrich-Schiller-Universität JenaJenaGermany

Personalised recommendations