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A Randomized Approximation Algorithm for Parameterized 3-D Matching Counting Problem

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Computing and Combinatorics (COCOON 2007)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4598))

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Abstract

The computational complexity of counting the number of matchings of size k in a given triple set remains open, and it is conjectured that the problem is infeasible. 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 2 ln (2/δ)/ε 2) such that

$$ prob[(1-\epsilon)h_0 \leq h \leq (1+\epsilon)h_0] \geq 1-\delta $$

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 and Luby.

This work is supported by the National Natural Science Foundation of China (60433020) and the Program for New Century Excellent Talents in University (NCET-05-0683)

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References

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

    Article  MATH  MathSciNet  Google Scholar 

  2. Arvind, V., Raman, V.: Approximation algorithms for some parameterized counting problems. In: Bose, P., Morin, P. (eds.) ISAAC 2002. LNCS, vol. 2518, pp. 453–464. Springer, Heidelberg (2002)

    Google Scholar 

  3. Bayati, M., Gamarnik, D., Katz, D., Nair, C., Tetali, P.: Simple deterministic approximation algorithms for counting matchings. In: Proc. 39th Symp. on Theory of Computation (STOC 2007) (to appear)

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  6. Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM J. Comput. 33(4), 892–922 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  7. Fellows, M.R., 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: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 311–322. Springer, Heidelberg (2004)

    Google Scholar 

  8. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, NewYork (1979)

    MATH  Google Scholar 

  9. Koutis, I.: A faster parameterized algorithm for set packing. Information processing letters 94, 7–9 (2005)

    Article  MathSciNet  Google Scholar 

  10. Karp, R., Luby, M., Madras, N.: Monte-Carlo Approximation Algorithms for Enumerartion Problems. Journal of Algorithms 10, 429–448 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Sankowski, P.: Alternative algorithms for counting all matchings in graph. In: Alt, H., Habib, M. (eds.) STACS 2003. LNCS, vol. 2607, pp. 427–438. Springer, Heidelberg (2003)

    Google Scholar 

  12. Liu, Y., Lu, S., Chen, J., Sze, S.-H.: Greedy localization and color-coding: improved matching and packing algorithms. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 84–95. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  13. Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(22), 398–427 (2002)

    MathSciNet  Google Scholar 

  14. Valiant, L.: The complexity of enumeration and reliability problems. SIAM J. Comput. 8(3), 410–421 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Valiant, L.: The complexity of computing the permanent. Theoretical Computer Science (8), 189–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. College of Information Science and Engineering, Central South University, Changsha 410083, P.R. China

    Yunlong Liu, Jianer Chen & Jianxin Wang

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

    Yunlong Liu

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

    Jianer Chen

Authors
  1. Yunlong Liu
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  2. Jianer Chen
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  3. Jianxin Wang
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Guohui Lin

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Liu, Y., Chen, J., Wang, J. (2007). A Randomized Approximation Algorithm for Parameterized 3-D Matching Counting Problem. In: Lin, G. (eds) Computing and Combinatorics. COCOON 2007. Lecture Notes in Computer Science, vol 4598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73545-8_35

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  • DOI: https://doi.org/10.1007/978-3-540-73545-8_35

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-73544-1

  • Online ISBN: 978-3-540-73545-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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