Extracting a spanning tree from a directed acyclic graph with minimum total distance between terminal nodes
Your problem is similar to the directed Steiner tree problem. In this problem we are given a directed graph, a set of terminal vertices and a specified root. The goal is to find a minimum cost tree that connects all terminal vertices to the root.

This is not exactly what you are asking for, but perhaps a solution to such a problem (or a suitable variant) is good enough.