Recently we proposed a new, complete formalism, inspired by matrix models, to compute B-model open and closed topological string amplitudes in local Calabi-Yau geometries, including the mirrors of toric Calabi-Yau threefolds. The formalism is non-perturbative in the moduli, hence can be used to study stringy phase transitions in the open/closed moduli space. In this talk I would like to summarize our B-model formalism, and focus on its implications for the study of large N dualities, such as 't Hooft resummations at strong coupling of perturbative Chern-Simons amplitudes on lens spaces.