Algorithm to find minimum spanning tree of chosen vertices

Question

One can use Prim's algorithm or Kruskal's algorithm to find the minimum spanning tree/graph of a collection of vertices/nodes and edges/links. What I want though, is an algorithm that finds the minimum spanning graph of this collection, but the resulting graph needs to include only arbitrarily chosen nodes, instead of all nodes. It's okay if the resulting graph includes more nodes than just those needed.

Does such an algorithm exist? Perhaps one could just use Prim's (or Kruskal's) algorithm after modifying the graph to include only the needed nodes? But, I'm not sure how to modify the graph to do so while maintaining its connectedness.

For example, say we have a diamond shaped starting graph (with costs of links in brackets):

    A
(2)/ \(1)
  B   C
(2)\ /(5)
    D

Now, we arbitrarily decide that only nodes A and D are needed. If we started at A, we'd still want it to take the left path, because ((2 + 2) < (1 + 5)).

Say we modify the graph slightly:

    A
(2)/ \(1) (2)
  B   C ------E
(2)\ /(5)
    D

If we decide that only nodes A, D, and E are needed, we realize that the path with the minimum cost is not necessarily the one with the fewest links. Taking A--B--D and A--C--E costs 7, but A--C--D and C--E costs 8.

Answer 1

What you want to find is a discrete Steiner tree. When not all vertices in the graph are mandatory but the tree is allowed to split at the optional vertices, the problem is NP-hard.

Wikipedia says (linked above) of this problem: it is believed that arbitrarily good approximation ratios cannot in general be achieved in polynomial time. There is a polynomial-time algorithm that finds a factor 1.39 approximation of a minimum Steiner tree.