1. We show an approximation preserving reduction from set cover. Let be an instance of the set cover problem and define the node-weighted Steiner tree instance where , and for each set to be 1 and 0 otherwise (both edges and vertices).
One can verify that any solution to the Steiner tree instance can be transformed to a solution to the set cover instance of the same cost and vice versa.
2. We describe an iterative algorithm that in each iteration contract an induced subgraph of that contains at least two terminals. The induced subgraph to contract is a spider that minimizes the cost of the subgraph per terminal it contains. A spider is a tree whose leafs are terminals and all vertices have degree two beside the leafs and, possibly, another vertex. We can assume w.l.o.g that the weights of the terminals are zero, and observe that any Steiner tree can be decomposed into spiders, thus, in any iteration, the cost we pay per terminal is at most , where is the number of terminals left at the beginning of the th iteration. The total cost, then, is no more than .