Counting matchings of size k Is #W[1]-hard

Abstract

We prove #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 #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 #W[1]-hard problems. Since the classical version of this problem is well-studied, we believe that our result facilitates future #W[1]-hardness proofs for other problems. © 2013 Springer-Verlag.