Approximation Algorithms for Some Parameterized Counting Problems
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  applied to fixed parameter tractable problems, and the color-coding technique of Alon, Yuster and Zwick . We also show some W-hardness results for parameterized exact counting problems.
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