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 kO(k)nO(1), approximation ratio 1/kO(k), and error probability 2-no(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.