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

TL;DR

This paper proves that the integrality gap for the directed Steiner tree problem in quasi-bipartite graphs is logarithmic in the number of terminals, using a primal-dual approach, which is novel for directed network design.

Contribution

It establishes a tight logarithmic bound on the integrality gap for the problem in quasi-bipartite graphs using a primal-dual method, a novel approach in this context.

Findings

Integrality gap is $O(\log k)$ for quasi-bipartite graphs.

Primal-dual method effectively bounds the integrality gap.

Results generalize set cover, showing tight bounds up to a constant.

Abstract

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.