Given a directed graph $G=(V,E)$ and a node $r\in V$, I need to grow a tree $T$ rooted at $r$ that has a minimum weight and spans all reachable nodes in $G$.The weight function assigns a non-negative weight to each node, which depends on the node's ancestors in $T$.  Specifically, for some fixed sets of nodes $S_1, S_2, \dots, S_k \subseteq V$, the weight of node $v$ is the number of sets $S_i$ that contain $v$ and all its ancestors in $T$.Any suggestion how to approach this problem?
Minimum vertex-weight directed spanning tree where the weight function depends on the tree
algorithms;graph theory;optimization;spanning trees