A Logarithmic Integrality Gap Bound for Directed Steiner Tree in   Quasi-bipartite Graphs
  We demonstrate that the integrality gap of the natural cut-based LP relaxation for the directed Steiner tree problem is $O(\log k)$ in quasi-bipartite graphs with $k$ terminals. Such instances can be seen to generalize set cover, so the integrality gap analysis is tight up to a constant factor. A novel aspect of our approach is that we use the primal-dual method; a technique that is rarely used in designing approximation algorithms for network design problems in directed graphs. 
Zachary Friggstad, Jochen Koenemann, Mohammad Shadravan
cs.DS