  We investigate properties of Ehrhart polynomials for matroid polytopes,
independence matroid polytopes, and polymatroids. In the first half of the
paper we prove that for fixed rank their Ehrhart polynomials are computable in
polynomial time. The proof relies on the geometry of these polytopes as well as
a new refined analysis of the evaluation of Todd polynomials. In the second
half we discuss two conjectures about the h^*-vector and the coefficients of
Ehrhart polynomials of matroid polytopes; we provide theoretical and
computational evidence for their validity.
