Advertisement

Graph-Modeled Data Clustering: Fixed-Parameter Algorithms for Clique Generation

  • Jens Gramm
  • Jiong Guo
  • Falk Hüffner
  • Rolf Niedermeier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2653)

Abstract

We present efficient fixed-parameter algorithms for the NP-complete edge modification problems Cluster Editing and Cluster Deletion. Here, the goal is to make the fewest changes to the edge set of an input graph such that the new graph is a vertex-disjoint union of cliques. Allowing up to k edge additions and deletions (Cluster Editing), we solve this problem in O(2.27k + |V|3) time; allowing only up to k edge deletions (Cluster Deletion), we solve this problem in O(1.77k + |V|3) time. The key ingredients of our algorithms are two easy to implement bounded search tree algorithms and a reduction to a problem kernel of size O(k 3). This improves and complements previous work.

Keywords

NP-complete problems edge modification problems data clustering fixed-parameter tractability exact algorithms 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Alber, J. Gramm, and R. Niedermeier. Faster exact solutions for hard problems: a parameterized point of view. Discrete Mathematics, 229:3–27, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    A. Ben-Dor, R. Shamir, and Z. Yakhini. Clustering gene expression patterns. Journal of Computational Biology, 6(3/4):281–297, 1999.CrossRefGoogle Scholar
  3. 3.
    Leizhen Cai. Fixed-parameter tractability of graph modification problems for hereditary properties. Information Processing Letters, 58:171–176, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Chen, I. Kanj, and W. Jia Vertex cover: further observations and further improvements. Journal of Algorithms, 41:280–301, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    R. G. Downey and M. R. Fellows. Parameterized Complexity. Springer. 1999.Google Scholar
  6. 6.
    M. R. Fellows. Parameterized complexity: the main ideas and connections to practical computing. In Experimental Algorithmics, number 2547 in LNCS, pages 51–77, 2002. Springer.CrossRefGoogle Scholar
  7. 7.
    J. Gramm, J. Guo, F. Hüffner, and R. Niedermeier. Automated generation of search tree algorithms for graph modification problems. Manuscript in preparation, March 2003.Google Scholar
  8. 8.
    P. Hansen and B. Jaumard. Cluster analysis and mathematical programming. Mathematical Programming, 79:191–215, 1997.MathSciNetGoogle Scholar
  9. 9.
    A. K. Jain and R. C. Dubes. Algorithms for clustering data. Prentice Hall, 1988.Google Scholar
  10. 10.
    H. Kaplan, R. Shamir, and R. E. Tarjan. Tractability of parameterized completion problems on chordal, strongly chordal, and proper interval graphs. SIAM Journal on Computing, 28(5):1906–1922, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    S. Khot and V. Raman. Parameterized complexity of finding subgraphs with hereditary properties. Theoretical Computer Science, 289:997–1008, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    O. Kullmann. New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science, 223(1–2):1–72, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Mahajan and V. Raman. Parameterizing above guaranteed values: MaxSat and MaxCut. Journal of Algorithms, 31:335–354, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Natanzon, R. Shamir, and R. Sharan. Complexity classification of some edge modification problems. Discrete Applied Mathematics, 113:109–128, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    R. Niedermeier and P. Rossmanith. A general method to speed up fixed parameter-tractable algorithms. Information Processing Letters, 73:125–129, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    R. Niedermeier and P. Rossmanith. On efficient fixed-parameter algorithms for Weighted Vertex Cover. Journal of Algorithms, to appear, 2003.Google Scholar
  17. 17.
    R. Shamir, R. Sharan, and D. Tsur. Cluster graph modification problems. In Proc. of 28th WG, number 2573 in LNCS, pages 379–390, 2002, Springer.Google Scholar
  18. 18.
    R. Sharan and R. Shamir. CLICK: A clustering algorithm with applications to gene expression analysis. In Proc. of 8th ISMB, pp. 307–316, 2000. AAAI Press.Google Scholar
  19. 19.
    R. Sharan and R. Shamir. Algorithmic approaches to clustering gene expression data. In T. Jiang et al. (eds): Current Topics in Computational Molecular Biology, pages 269–300, The MIT Press. 2002.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Jens Gramm
    • 1
  • Jiong Guo
    • 1
  • Falk Hüffner
    • 1
  • Rolf Niedermeier
    • 1
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany

Personalised recommendations