I am looking for an algorithm with polynomial complexity where, given a strongly connected edge-weighted digraph I can find the minimal subgraph which connects some root vertex v to a known set of other vertices.


As an example, given a strongly connected edge-weighted digraph with vertices labeled a-z, I want to find the minimal subgraph rooted at node j that includes nodes b, f, g, and p.


In my case I am going to be using this to determine an optimal pipeline for doing image manipulation (I already have a strongly connected edge-weighted digraph for this application).



Minimum edge-weighted directed subgraph in polynomial time
This is an instance of the Directed Steiner Network Problem and as such it's solvable in time $|V(G)|^{O(|T|)}$ as proved by [Feldman & Ruhl (2006)](https://doi.org/10.1137/S0097539704441241), where $G$ is the given graph and $T\subset V(G)$ is the given subset of vertices (terminals).


