I have been trying to find an easy-to-implement approximation algorithm on the problem of Prize collecting Steiner tree on node-weighted graph without weights on the edges. The closest I have come is the one in The prize collecting Steiner tree problem: Theory and Practice by Johnson, Minkoff and Phillips, but the GW algorithm provided won't work without weights on edges (I think).
The problem I'm trying to solve is:
Given a graph $G=(V,E,w)$, $w$ being a function assigning non-negative weights to nodes, and a set of query nodes $Q$, find a connected subgraph $H$ of $G$, spanning all nodes in $Q$, and also maximizing the following function
$$f(H) = \sum_{v\in V_H}w(v) + \min_{v \in V_H}\deg_H(v)$$
with $V_H \subseteq V$ being the set of nodes in $H$.
Keep in mind I need an easy to implement approximation algorithm to use as a baseline for comparison, so any LP or SDP ways to solve it, won't be of much help.