Abstract
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.
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References
Flum, J., Grohe, M.: The parameterized complexity of counting problems. SIAM Journal on Computing, 538–547 (2002)
Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)
Temperley, H.N.V., Fisher, M.E.: Dimer problem in statistical mechanics - an exact result. Philosophical Magazine 68(6), 1478–6435 (1961)
Kasteleyn, P.: The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice. Physica 27(12), 1209–1225 (1961)
Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001)
Jerrum, M., Sinclair, A.: Approximating the permanent. SIAM J. Comput. 18(6), 1149–1178 (1989)
Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004); Provinces of logic determined. Essays in the memory of Alfred Tarski. Parts I, II and III
Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation 85(1), 12–75 (1990)
Galluccio, A., Loebl, M.: On the theory of Pfaffian orientations. I. Perfect matchings and permanents. Electronic Journal of Combinatorics 6 (1998)
Frick, M.: Generalized Model-Checking over Locally Tree-Decomposable Classes. In: Alt, H., Ferreira, A. (eds.) STACS 2002. LNCS, vol. 2285, pp. 632–644. Springer, Heidelberg (2002)
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)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Counting Paths and Packings in Halves. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 578–586. Springer, Heidelberg (2009)
Bläser, M., Curticapean, R.: The Complexity of the Cover Polynomials for Planar Graphs of Bounded Degree. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 96–107. Springer, Heidelberg (2011)
Bläser, M., Dell, H.: Complexity of the Cover Polynomial. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 801–812. Springer, Heidelberg (2007)
Chung, F.R.K., Graham, R.L.: On the cover polynomial of a digraph. J. Combin. Theory Ser. B 65, 273–290 (1995)
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Bläser, M., Curticapean, R. (2012). Weighted Counting of k-Matchings Is #W[1]-Hard. In: Thilikos, D.M., Woeginger, G.J. (eds) Parameterized and Exact Computation. IPEC 2012. Lecture Notes in Computer Science, vol 7535. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33293-7_17
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DOI: https://doi.org/10.1007/978-3-642-33293-7_17
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