  In this paper, we investigate the optimal (partial) transport problem between
non-convex polygonal domains in \(\mathbb{R}^2\). In the case of the complete
optimal transport problem, we demonstrate that the singular set is either a
finite set, or, except for a finite number of points, is locally a
1-dimensional smooth curve. As for the optimal partial transport, we establish
that the free boundary is smooth except for a finite number of singular points.
