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)


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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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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