On directed Steiner trees

The directed Steiner tree problem is the following: given a directed graph G = (V, E) with weights on the edges, a set of terminals S C V, and a root vertex r, find a minimum weight out-branching T rooted at r, such that all vertices in S are included in T. This problem is known to be NP-hard. Recently, non-trivial polynomial time approximation algorithms have been developed for this problem with worst case approximation guarantees of O(kϵ) for any fixed ϵ > 0. We consider a natural LP relaxation of this problem. Using a dual formulation we construct a simple deterministic (D + l)-approximation algorithm for a special case when the subgraph induced by V \ S is a tree with depth D (for example, this can be shown to include the group Steiner tree problem as a special case, by the loss of poly-log factors in the approximation guarantee). We also show that this LP has an integrality gap of φ(y√k) for the general problem.