Approximation Algorithms for Some Parameterized Counting Problems

  • V. Arvind
  • Venkatesh Raman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2518)


We give a randomized fixed parameter tractable algorithm to approximately count the number of copies of a k-vertex graph with bounded treewidth in an n vertex graph. As a consequence, we get randomized algorithms with running time k O (k) n O(1), approximation ratio 1/k O(k), and error probability 2-n o(1) for (a) approximately counting the number of matchings of size k in an n vertex graph and (b) approximately counting the number of paths of length k in an n vertex graph. Our algorithm is based on the Karp-Luby approximate counting technique [8] applied to fixed parameter tractable problems, and the color-coding technique of Alon, Yuster and Zwick [1]. We also show some W-hardness results for parameterized exact counting problems.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    N. Alon, R. Yuster and U. Zwick, “Color-Coding”, Journal of the Association for Computing Machinery, 42(4) (1995) 844–856.zbMATHMathSciNetGoogle Scholar
  2. 2.
    H. Bodlaender, “A Linear Time Algorithm for Finding Tree-Decompositions of Small Treewidth”, SI AM J. Computing 25 (1996) 1305–1317.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. G. Downey and M. R. Fellows, Parameterized Complexity, Springer-Verlag, 1998.Google Scholar
  4. 4.
    U. Feige and J. Kilian, “On Limited versus Polynomial Nondeterminism”, Chicago Journal of Theoretical Computer Science, March (1997).Google Scholar
  5. 5.
    H. Fernau, “Parameterized Enumeration”, to appear in the Proceedings of COCOON 2002.Google Scholar
  6. 6.
    J. Flum and M. Grohe, “The Parameterized Complexity of Counting Problems”, To appear in 43rd IEEE Symposium on Foundations of Computer Science 2002.Google Scholar
  7. 7.
    T. Johnson, N. Robertson, P. D. Seymour, R. Thomas, “Directed Tree-Width”, preprint (1998) (available at
  8. 8.
    R. M. Karp and M. Luby, “Monte-Carlo Algorithms for Enumeration and Reliability Problems”, In Proceedings of the 24th Annual IEEE Symposium on Foundations of Computer Science (1983) 56–64.Google Scholar
  9. 9.
    R. M. Karp, M. Luby and N. Madras, “Monte-Carlo Approximation Algorithms for Enumeration Problems”, Journal of Algorithms 10 (1989) 429–448.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    S. Khot and V. Raman, “Parameterized Complexity of Finding Subgraphs with Hereditary Properties”, Proceedings of the Sixth Annual International Computing and Combinatorics Conference (COCOON, July 2000, Sydney, Australia, Lecture Notes in Computer Science, Springer Verlag 1858 (2000) 137–147. Full version to appear in Theoretical Computer Science.Google Scholar
  11. 11.
    T. Kloks, “Treewidth: Computations and Approximations”, Lecture Notes in Computer Science, Springer-Verlag 842 1994.Google Scholar
  12. 12.
    M. Mahajan and V. Raman, “Parameterizing Above Guaranteed Values: MaxSat and MaxCut”, Journal of Algorithms 31 (1999) 335–354.zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    C. McCartin, “Parameterized Counting Problems”, to appear in the Proceedings of MFCS 2002 conference.Google Scholar
  14. 14.
    R. Motwani and P. Raghavan, Randomized Algorithms, Cambridge University Press, 1995.Google Scholar
  15. 15.
    N. Nisan, Using Hard Problems to Create Pseudorandom Generators, MIT Press (1992).Google Scholar
  16. 16.
    P. Rossmanith, private communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • V. Arvind
    • 1
  • Venkatesh Raman
    • 1
  1. 1.The Institute of Mathematical SciencesChennaiIndia

Personalised recommendations