Skip to main content
SpringerLink
Log in

On Counting 3-D Matchings of Size k

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

The computational complexity of counting the number of matchings of size k in a given triple set has been open. It is conjectured that the problem is not fixed parameter tractable. In this paper, we present a fixed parameter tractable randomized approximation scheme (FPTRAS) for the problem. More precisely, we develop a randomized algorithm that, on given positive real numbers ε and δ, and a given set S of n triples and an integer k, produces a number h in time O(5.483k n 2ln (2/δ)/ε 2) such that

$$\mathop{\mathrm{Prob}}[(1-\epsilon)h_{0}\leq h\leq(1+\epsilon)h_{0}]\geq1-\delta,$$

where h 0 is the total number of matchings of size k in the triple set S. Our algorithm is based on the recent improved color-coding techniques and the Monte-Carlo self-adjusting coverage algorithm developed by Karp, Luby and Madras.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42, 844–856 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arvind, V., Raman, V.: Approximation algorithms for some parameterized counting problems. In: Proc. 13th International Symposium on Algorithms and Computation (ISAAC 02), Lecture Notes in Computer Science, vol. 2518, pp. 453–464. Springer, Berlin (2002)

    Google Scholar 

  3. Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proc. 39th Symposium on Theory of Computing (STOC 07), pp. 122–127 (2007)

  4. Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: Proc. 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 07), pp. 298–307 (2007)

  5. Chien, S.: A determinant-based algorithm for counting perferct matchings in a general graph. In: Proc. 15th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 04), pp. 728–735 (2004)

  6. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33, 892–922 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fellows, M., Knauer, C., Nishimura, N., Ragde, P., Rosamond, F., Stege, U., Thilikos, D., Whitesides, S.: Faster fixed-parameter tractable algorithms for matching and packing problems. In: Proc. 12th Annual European Symposium on Algorithms (ESA 04). Lecture Notes in Computer Science, vol. 3221, pp. 311–322. Springer, Berlin (2004)

    Google Scholar 

  8. Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)

    MATH  Google Scholar 

  9. Koutis, I.: A faster parameterized algorithm for set packing. Inf. Process. Lett. 94, 7–9 (2005)

    Article  MathSciNet  Google Scholar 

  10. Karp, R., Luby, M., Madras, N.: Monte-Carlo approximation algorithms for enumeration problems. J. Algorithms 10, 429–448 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sankowski, P.: Alternative algorithms for counting all matchings in graph. In: Proc. 20th Annual Symposium on Theoretical Aspects of Computer Science (STACS 03). Lecture Notes in Computer Science, vol. 2607, pp. 427–438. Springer, Berlin (2003)

    Google Scholar 

  12. Liu, Y., Lu, S., Chen, J., Sze, S.-H.: Greedy localization and color-coding: improved matching and packing algorithms. In: Proc. 2nd International Workshop on Parameterized and Exact Computation (IWPEC 06). Lecture Notes in Computer Science, vol. 4169, pp. 84–95. Springer, Berlin (2006)

    Chapter  Google Scholar 

  13. Vadhan, S.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31, 398–427 (2002)

    Article  MathSciNet  Google Scholar 

  14. Valiant, L.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8, 410–421 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Valiant, L.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Central South University, Changsha, 410083, China

    Yunlong Liu, Jianer Chen & Jianxin Wang

  2. School of Further Education, Hunan Normal University, Changsha, 410012, China

    Yunlong Liu

  3. Department of Computer Science, Texas A&M University, College Station, TX, 77843, USA

    Jianer Chen

Authors
  1. Yunlong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jianer Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxin Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianxin Wang.

Additional information

A preliminary version of this paper was presented at The 13th Annual International Computing and Combinatorics Conference (COCOON 2007), July 16–19, 2007, Banff, Alberta, Canada. This work is supported by the National Natural Science Foundation of China (No. 60433020 and No. 60773111), by the National Basic Research 973 Program of China (No. 2008CB317107), and by the China Program for New Century Excellent Talents in University (NCET-05-0683).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Chen, J. & Wang, J. On Counting 3-D Matchings of Size k . Algorithmica 54, 530–543 (2009). https://doi.org/10.1007/s00453-008-9207-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-008-9207-x

Keywords

Search

Navigation