An $O(\log k)$-Approximation for Directed Steiner Tree in Planar Graphs

TL;DR

This paper introduces an efficient approximation algorithm with logarithmic ratio for the Directed Steiner Tree problem in planar graphs, improving solution quality for network design tasks involving multiple terminals.

Contribution

It provides the first $O( ext{log }k)$-approximation for directed Steiner tree in planar graphs and extends the approach to the multi-rooted case with additional approximation guarantees.

Findings

Achieves $O( ext{log }k)$-approximation for directed Steiner tree in planar graphs.

Extends the approximation to multi-rooted Steiner problems with $O(R + ext{log }k)$ ratio.

Demonstrates the effectiveness of the approach in planar graph network design scenarios.

Abstract

We present an $O(\log k)$-approximation for both the edge-weighted and node-weighted versions of \DST in planar graphs where $k$ is the number of terminals. We extend our approach to \MDST (in general graphs \MDST and \DST are easily seen to be equivalent but in planar graphs this is not the case necessarily) in which we get an $O(R+\log k)$-approximation for planar graphs for where $R$ is the number of roots.