Abstract
We give a randomized fixed parameter tractable algorithm to approximately count the number of copies of a k-vertex graph with bounded treewidth in an n vertex graph. As a consequence, we get randomized algorithms with running time k O(k) n O(1), approximation ratio 1/k O(k), and error probability 2-n o(1) for (a) approximately counting the number of matchings of size k in an n vertex graph and (b) approximately counting the number of paths of length k in an n vertex graph. Our algorithm is based on the Karp-Luby approximate counting technique [8] applied to fixed parameter tractable problems, and the color-coding technique of Alon, Yuster and Zwick [1]. We also show some W-hardness results for parameterized exact counting problems.
Work supported by a DST-DAAD project (Indo-German Personnel Exchange Programme 2000).
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Arvind, V., Raman, V. (2002). Approximation Algorithms for Some Parameterized Counting Problems. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_40
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DOI: https://doi.org/10.1007/3-540-36136-7_40
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