It is known that the open topological A-model amplitudes encode the information about Gromov-Witten invariants of maps from Riemann surfaces into a manifold (Marino, Chern-Simons Theory and Topological Strings). The simplest case of these amplitudes is disc amplitudes. There are essentially two ways to compute them -- mirror symmetry with the B-model on the mirror manifold, and virtual localization on the moduli space of maps, which can be used if the image of the worldsheet in the manifold is fixed under the action of some symmetry (usually, toric).
There is a more or less well defined procedure for computing disc amplitudes via localization if the source curve is irreducible, being a genuine disc (see Enumerative Geometry of Stable Maps with Lagrangian Boundary Conditions and Multiple Covers of the Disc), while this disc may also be reducible, \(D=D_0 \cup S^2\), and get mapped to the submanifold of the same topology. Sometimes, it is possible to compute such amplitudes using mirror symmetry (Mirror Symmetry, D-branes and Counting Holomorphic Discs).
So my question is whether there is a way to compute such amplitudes using localization?