SIAM Journal on Computing
The monadic second-order logic of graphs. I. recognizable sets of finite graphs
Information and Computation
On the cover polynomial of a digraph
Journal of Combinatorial Theory Series B
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
Generalized Model-Checking over Locally Tree-Decomposable Classes
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Finding, minimizing, and counting weighted subgraphs
Proceedings of the forty-first annual ACM symposium on Theory of computing
The complexity of the cover polynomials for planar graphs of bounded degree
MFCS'11 Proceedings of the 36th international conference on Mathematical foundations of computer science
Complexity of the cover polynomial
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Counting matchings of size k is # W[1]-hard
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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In the seminal paper for parameterized counting complexity [1], the following problem is conjectured to be #W[1]-hard: Given a bipartite graph G and a number k∈ℕ, which is considered as a parameter, count the number of matchings of size k in G. We prove hardness for a natural weighted generalization of this problem: Let G=(V,E,w) be an edge-weighted graph and define the weight of a matching as the product of weights of all edges in the matching. We show that exact evaluation of the sum over all such weighted matchings of size k is #W[1]-hard for bipartite graphs G. As an intermediate step in our reduction, we also prove #W[1]- hardness of the problem of counting k-partial cycle covers, which are vertex-disjoint unions of cycles including k edges in total. This hardness result even holds for unweighted graphs.