From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs
Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, Weihao, Zhu

TL;DR
This paper advances approximation algorithms for the Directed Steiner Tree problem and related problems in planar graphs, achieving poly-logarithmic approximations and analyzing LP relaxations, building on recent separator theorem techniques.
Contribution
It introduces a tree embedding technique for planar digraphs, leading to new poly-logarithmic approximations for DST and related problems, and analyzes the LP relaxation's integrality gap in planar graphs.
Findings
Poly-logarithmic approximations for DST, Group Steiner Tree, Covering Steiner Tree, and Polymatroid Steiner Tree in planar digraphs.
The LP relaxation for DST in planar graphs has an $O( ext{log}^2 k)$ integrality gap.
Improved approximation ratios for multi-rooted problems in planar digraphs.
Abstract
In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph , a root vertex and a set of terminals. The goal is to find a min-cost subgraph that connects to each of the terminals. DST admits an -approximation in quasi-polynomial time, and an -approximation for any fixed in polynomial-time. Resolving the existence of a polynomial-time poly-logarithmic approximation is a major open problem in approximation algorithms. In a recent work, Friggstad and Mousavi [ICALP 2023] obtained a simple and elegant polynomial-time -approximation for DST in planar digraphs via Thorup's shortest path separator theorem. We build on their work and obtain several new results on DST and related problems. - We develop a tree embedding technique for rooted problems in planar…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
