Minimum Steiner Arborescenes approximation: intuitions?

Question

Given a (positively) weighted directed graph G, a set of query nodes T and a root r, finding the minimum steiner arborescence containing the query nodes and rooted at r is an NP-hard problem

However, I'm having troubles finding a clearly explained approximation algorithm for this problem. This is quite unlike the undirected version of finding a minimum steiner tree, where the (general) 2-approximation algorithm is quite clear .

Does anybody know how to tackle the directed version? What is the state of the art in terms of approximations? Can't find much about it on the web.

Thanks!

Answer 1

The problem is approximable within approximation ratio $O(|S|^\epsilon)$ for every $\epsilon > 0$. For every fixed $\epsilon > 0$ cannot be approximated within ratio $\log^{2-\epsilon} n$, unless $NP \subseteq ZTIME(n^{polylog(n)})$.

See problem 1.2 and bibliography here: http://theory.cs.uni-bonn.de/info5/steinerkompendium/netcompendium.pdf