Given a digraph with two terminal vertices s and t as well as a conservative cost function and several not necessarily disjoint color classes on its arc set, our goal is to find a minimum-cost subset of the arcs such that its intersection with each color class contains an s-t dipath. Problems of this type arise naturally in multi-commodity network design settings where each commodity is restricted to use links of its own color only. We study several variants of the problem, deriving strong hardness results even for restricted cases, but we also identify cases that can be solved in polynomial time. The latter ones include the cases where the color classes form a laminar family, or where the underlying digraph is acyclic and the number of color classes is constant. We also present an FPT algorithm for the general case parameterized by the number of multi-colored arcs.
Kakimura et al. (Sat,) studied this question.