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On Counting Parameterized Matching and Packing

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Frontiers in Algorithmics (FAW 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9711))

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Abstract

In this paper, we first show that the complexity of parameterized m-set packing (resp. m-d matching) counting is \(\sharp \)W[1]-hard by a reduction from parameterized graph (resp. bipartite graph) matching counting \((m\ge 3)\). Subsequently, based on the algorithm for 3-d matching counting, we develop fixed-parameter tractable randomized approximation schemes (FPTRAS) for m-set packing counting, m-d matching counting, and bipartite graph matching counting, respectively. Our results indicate that parameterized m-set packing counting and m-d matching counting are typical examples that are \(\sharp \)W[1]-hard but admit FPTRAS. Furthermore, we show that edge disjoint subgraph packing counting, i.e., a special subgraph counting problem parameterized by the size of the packing, admits FPTRAS even if some of the counted subgraphs don’t have bounded treewidth.

This research was supported in part by the National Natural Science Foundation of China under Grants 61572190, 61232001, and 61420106009.

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Notes

  1. 1.

    Note that in these problems, m is a constant rather than a parameter.

References

  1. 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)

    Chapter  Google Scholar 

  2. Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Counting paths and packings in halves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 578–586. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2007), pp. 298–307 (2007)

    Google Scholar 

  4. Curticapean, R.: Counting matchings of size k is \(\sharp \)W[1]-hard. In: Fomin, F.V., Freivalds, R., Kwiatkowska, M., Peleg, D. (eds.) ICALP 2013, Part I. LNCS, vol. 7965, pp. 352–363. Springer, Heidelberg (2013)

    Google Scholar 

  5. Curticapean, R., Marx, D.: Complexity of counting subgraphs: only the boundedness of the vertex-cover number counts. In: Proceedings of the 55th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2014), pp. 130–139 (2014)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Jerrum, M., Meeks, K.: The parameterised complexity of counting connected subgraphs and graph motifs. J. Comput. Syst. Sci. 81(4), 702–716 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  10. Liu, Y., Chen, J., Wang, J.: On counting 3-D matchings of size \(k\). Algorithmica 54, 530–543 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Mathieson, L., Prieto, E., Shaw, P.: Packing edge disjoint triangles: a parameterized view. In: Downey, R.G., Fellows, M.R., Dehne, F. (eds.) IWPEC 2004. LNCS, vol. 3162, pp. 127–137. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. McCartin, C.: Parameterized counting problems. Ann. Pure Appl. Logic 138(13), 147–182 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meeks, K.: The challenges of unbounded treewidth in parameterised subgraph counting problems. Discrete Appl. Math. 198, 170–194 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yu, D., Wang, Y., Hua, Q.-S., Lau, F.C.M.: Exact parameterized multilinear monomial counting via k-layer subset convolution and k-disjoint sum. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 74–85. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  15. Zhang, C., Chen, Y.: Counting problems in parameterized complexity. TsingHua Sci. Technol. 19(4), 410–420 (2014)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We are grateful to the anonymous referees for helpful suggestions that improve the presentation of this paper.

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Authors and Affiliations

  1. Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry of Education of China), College of Mathematics and Computer Science, Hunan Normal University, Changsha, 410081, People’s Republic of China

    Yunlong Liu

  2. School of Information Science and Engineering, Central South University, Changsha, 410083, People’s Republic of China

    Jianxin Wang

Authors
  1. Yunlong Liu
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  2. Jianxin Wang
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Correspondence to Jianxin Wang .

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Editors and Affiliations

  1. Shandong University, Jinan, China

    Daming Zhu

  2. MS EC-31, University of Texas at Dallas, Richardson, Texas, USA

    Sergey Bereg

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Liu, Y., Wang, J. (2016). On Counting Parameterized Matching and Packing. In: Zhu, D., Bereg, S. (eds) Frontiers in Algorithmics. FAW 2016. Lecture Notes in Computer Science(), vol 9711. Springer, Cham. https://doi.org/10.1007/978-3-319-39817-4_13

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  • DOI: https://doi.org/10.1007/978-3-319-39817-4_13

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-39816-7

  • Online ISBN: 978-3-319-39817-4

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