  In this paper, we compute the open Gromov-Witten invariants for every compact
toric surface X which is semi-Fano (i.e. the anticanonical line bundle is nef).
Unlike the Fano case, this involves non-trivial obstructions in the
corresponding moduli problem. As a consequence, an explicit formula for the
Lagrangian Floer superpotential W is obtained, which in turn gives an explicit
presentation of the small quantum cohomology ring of X. We also provide a
computational verification of the conjectural ring isomorphism between the
small quantum cohomology of X and the Jacobian ring of W.
