I have a connected graph $G$ and a real-valued function $f$ on sub-graphs $G' \subseteq G$. Given a node $n \in G$ and a positive integer $s$, I am looking for the connected subgraph $G' \subseteq G$ that minimizes $f(G')$ from all the subgraphs that contain the node $n$ and exactly $s$ nodes. More formally, I am looking for the following:
$$\DeclareMathOperator*{\argmin}{arg\,min} \argmin_{g\in\{G' \subseteq G | G' \text{ is connected},\ n\in G',\ |G'|=s\}}\ {f(g)}\enspace .$$
Is there any efficient exact solution to this problem or is this NP-hard? If the latter is the case, is there any polynomial-time approximation to the exact solution?