Counting Matchings of Size k Is \(\sharp\)W[1]-Hard

  • Radu Curticapean
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

Abstract

We prove \(\sharp\)W[1]-hardness of the following parameterized counting problem: Given a simple undirected graph G and a parameter k ∈ ℕ, compute the number of matchings of size k in G.

It is known from [1] that, given an edge-weighted graph G, computing a particular weighted sum over the matchings in G is \(\sharp\)W[1]-hard. In the present paper, we exhibit a reduction that does not require weights.

This solves an open problem from [5] and adds a natural parameterized counting problem to the scarce list of \(\sharp\)W[1]-hard problems. Since the classical version of this problem is well-studied, we believe that our result facilitates future \(\sharp\)W[1]-hardness proofs for other problems.

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References

  1. 1.
    Bläser, M., Curticapean, R.: Weighted counting of k-matchings is #W[1]-hard. In: Thilikos, D.M., Woeginger, G.J. (eds.) IPEC 2012. LNCS, vol. 7535, pp. 171–181. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation 85(1), 12–75 (1990)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ehrenborg, R., Rota, G.-C.: Apolarity and canonical forms for homogeneous polynomials. European Journal of Combinatorics 14(3), 157–181 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM Journal on Computing, 538–547 (2002)Google Scholar
  5. 5.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer-Verlag New York, Inc., Secaucus (2006)Google Scholar
  6. 6.
    Hartshorne, R.: Algebraic geometry. Springer (1977)Google Scholar
  7. 7.
    Makowsky, J.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126, 159–213 (2004)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Schwartz, J.: Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27(4), 701–717 (1980)MATHCrossRefGoogle Scholar
  9. 9.
    Temperley, H., Fisher, M.: Dimer problem in statistical mechanics - an exact result. Philosophical Magazine 6(68) (1961) 1478–6435Google Scholar
  10. 10.
    Valiant, L.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Valiant, L.: The complexity of enumeration and reliability problems. SIAM Journal on Computing 8(3), 410–421 (1979)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Valiant, L.: Holographic algorithms. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004, pp. 306–315 (2004)Google Scholar
  13. 13.
    Vassilevska, V., Williams, R.: Finding, minimizing, and counting weighted subgraphs. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 455–464. ACM, New York (2009)Google Scholar
  14. 14.
    Xia, M., Zhang, P., Zhao, W.: Computational complexity of counting problems on 3-regular planar graphs. Theor. Comp. Sc. 384(1), 111–125 (2007)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Ng, K.W. (ed.) EUROSAM 1979 and ISSAC 1979. LNCS, vol. 72, pp. 216–226. Springer, Heidelberg (1979)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Radu Curticapean
    • 1
  1. 1.Dept. of Computer ScienceSaarland UniversityGermany

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