Toric degenerations, tropical curves and Gromov-Witten invariants of Fano manifolds

In this paper, we give a tropical method for computing Gromov-Witten type invariants of Fano manifolds of special type. This method applies to those Fano manifolds which admit toric degenerations to toric Fano varieties with singularities allowing small resolutions. Examples include (generalized) flag manifolds of type A, and some moduli space of rank two bundles on a genus two curve.