Counting Matchings of Size k Is \(\sharp\)W-Hard
We prove \(\sharp\) W-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  that, given an edge-weighted graph G, computing a particular weighted sum over the matchings in G is \(\sharp\) W-hard. In the present paper, we exhibit a reduction that does not require weights.
This solves an open problem from  and adds a natural parameterized counting problem to the scarce list of \(\sharp\) W-hard problems. Since the classical version of this problem is well-studied, we believe that our result facilitates future \(\sharp\) W-hardness proofs for other problems.
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