  In this paper we give a fully polynomial randomized approximation scheme
(FPRAS) for the number of matchings in k-uniform hypergraphs whose intersection
graphs contain few claws. Our method gives a generalization of the canonical
path method of Jerrum and Sinclair to hypergraphs satisfying a local
restriction. Our proof method depends on an application of the Euler tour
technique for the canonical paths of the underlying Markov chains. On the other
hand, we prove that it is NP-hard to approximate the number of matchings even
for the class of k-uniform, 2-regular and linear hypergraphs, for all k >= 6,
without the above restriction.
