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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- 4.Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM Journal on Computing, 538–547 (2002)Google Scholar
- 5.Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer-Verlag New York, Inc., Secaucus (2006)Google Scholar
- 6.Hartshorne, R.: Algebraic geometry. Springer (1977)Google Scholar
- 9.Temperley, H., Fisher, M.: Dimer problem in statistical mechanics - an exact result. Philosophical Magazine 6(68) (1961) 1478–6435Google Scholar
- 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.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