  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.
