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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(More)
For a class H of graphs, #Sub(H) is the counting problem that, given a graph H ∈ H and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if H has bounded vertex-cover number (equivalently, the size of the maximum matching in H is bounded), then #Sub(H) is polynomial-time solvable. We complement this result with a(More)
We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis #ETH introduced by Dell et al. (ACM Transactions on Algorithms, 2014). Our framework allows to convert many known #P-hardness results for counting problems into tight lower bounds of the following type: Assuming #ETH, the given problem admits no algorithm(More)
Let P be a set of $n$ points in the plane. A crossing-free structure on P is a straight-edge planar graph with vertex set in P. Examples of crossing-free structures include triangulations of P, and spanning cycles of P, also known as polygonalizations of P, among others. There has been a large amount of research trying to bound the number of such(More)
We identify and study relevant structural parameters for the problem PerfMatch of counting perfect matchings in a given input graph G. These generalize the well-known tractable planar case, and they include the genus of G, its apex number (the minimum number of vertices whose removal renders G planar), and its Hadwiger number (the size of a largest clique(More)