A Constant-Factor Approximation for Quasi-bipartite Directed Steiner Tree on Minor-Free Graphs

TL;DR

This paper presents the first constant-factor approximation algorithm for quasi-bipartite Directed Steiner Tree problems on minor-free graphs, using a refined primal-dual approach.

Contribution

It introduces a novel primal-dual algorithm for quasi-bipartite instances on minor-free graphs with specific approximation guarantees.

Findings

Approximation guarantee of O(r*sqrt(log r)) for K_r-minor-free graphs

Approximation guarantee of 20 for planar graphs

Upper bounds on integrality gaps for the LP relaxation

Abstract

We give the first constant-factor approximation algorithm for quasi-bipartite instances of Directed Steiner Tree on graphs that exclude fixed minors. In particular, for $K_r$-minor-free graphs our approximation guarantee is $O(r\cdot\sqrt{\log r})$ and, further, for planar graphs our approximation guarantee is 20. Our algorithm uses the primal-dual scheme. We employ a more involved method of determining when to buy an edge while raising dual variables since, as we show, the natural primal-dual scheme fails to raise enough dual value to pay for the purchased solution. As a consequence, we also demonstrate integrality gap upper bounds on the standard cut-based linear programming relaxation for the Directed Steiner Tree instances we consider.