A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs

@article{Friggstad2016ALI,
title={A Logarithmic Integrality Gap Bound for Directed Steiner Tree in Quasi-bipartite Graphs},
author={Zachary Friggstad and J. K{\"o}nemann and M. Shadravan},
journal={ArXiv},
year={2016},
volume={abs/1604.08132}
}

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.