Approximately counting Hamilton paths and cycles in dense graphs

Abstract: "We describe fully polynomial randomized approximation schemes for the problems of determining the number of Hamilton paths and cycles in an n-vertex graph with minimum degree (1/2 + [epsilon])n, for any fixed [epsilon] > 0. We show that the exact counting problems are #P-complete. We also describe fully polynomial randomized approximation schemes for counting paths and cycles of all sizes in such graphs."