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Given a directed graph G=(V,E)$$G=(V,E)$$ and a node r∈V$$r\in V$$, I need to grow a tree T$$T$$ rooted at r$$r$$ that has a minimum weight and spans all reachable nodes in G$$G$$.

The weight function assigns a non-negative weight to aeach node and, which depends on the node's antecedentsancestors in T$$T$$. More precisely Specifically, I havefor some fixed sets of nodes S1,S2,...$$S_1, S_2, \dots, S_k \subseteq V$$,Sk and my the weight function simply returns in how many different sets,of node $$v$$ is the number of sets $$S_i$$ that I consider adding to Tcontain $$v$$ and all of its antecedentsancestors in T reside$$T$$.

Any suggestion how to approach this problem?

Given a directed graph G=(V,E) and a node r∈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 a node and depends on node's antecedents in T. More precisely, I have some sets of nodes S1,S2,...,Sk and my weight function simply returns in how many different sets, node that I consider adding to T and all of its antecedents in T reside. Any suggestion how to approach this problem?

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?