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From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs

Authors Chandra Chekuri, Rhea Jain, Shubhang Kulkarni , Da Wei Zheng , Weihao Zhu

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Author Details

Chandra Chekuri
  • University of Illinois, Urbana-Champaign, IL, USA
Rhea Jain
  • University of Illinois, Urbana-Champaign, IL, USA
Shubhang Kulkarni
  • University of Illinois, Urbana-Champaign, IL, USA
Da Wei Zheng
  • University of Illinois, Urbana-Champaign, IL, USA
Weihao Zhu
  • University of Illinois, Urbana-Champaign, IL, USA

Funding

Chandra Chekuri and Rhea Jain are partially supported by NSF grants CCF-1907937 and CCF-2402667. Weihao Zhu acknowledges funding support from a graduate fellowship of the CS department.

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Chandra Chekuri, Rhea Jain, Shubhang Kulkarni, Da Wei Zheng, and Weihao Zhu. From Directed Steiner Tree to Directed Polymatroid Steiner Tree in Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 42:1-42:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.42

Abstract

In the Directed Steiner Tree (DST) problem the input is a directed edge-weighted graph G = (V,E), a root vertex r and a set S ⊆ V of k terminals. The goal is to find a min-cost subgraph that connects r to each of the terminals. DST admits an O(log² k/log log k)-approximation in quasi-polynomial time [Grandoni et al., 2022; Rohan Ghuge and Viswanath Nagarajan, 2022], and an O(k^{ε})-approximation for any fixed ε > 0 in polynomial-time [Alexander Zelikovsky, 1997; Moses Charikar et al., 1999]. 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 [Zachary Friggstad and Ramin Mousavi, 2023] obtained a simple and elegant polynomial-time O(log k)-approximation for DST in planar digraphs via Thorup’s shortest path separator theorem [Thorup, 2004]. 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 digraphs via an interpretation of the recursion in [Zachary Friggstad and Ramin Mousavi, 2023]. Using this we obtain polynomial-time poly-logarithmic approximations for Group Steiner Tree [Naveen Garg et al., 2000], Covering Steiner Tree [Goran Konjevod et al., 2002] and the Polymatroid Steiner Tree [Gruia Călinescu and Alexander Zelikovsky, 2005] problems in planar digraphs. All these problems are hard to approximate to within a factor of Ω(log² n/log log n) even in trees [Eran Halperin and Robert Krauthgamer, 2003; Grandoni et al., 2022]. 
- We prove that the natural cut-based LP relaxation for DST has an integrality gap of O(log² k) in planar digraphs. This is in contrast to general graphs where the integrality gap of this LP is known to be Ω(√k) [Leonid Zosin and Samir Khuller, 2002] and Ω(n^{δ}) for some fixed δ > 0 [Shi Li and Bundit Laekhanukit, 2022]. 
- We combine the preceding results with density based arguments to obtain poly-logarithmic approximations for the multi-rooted versions of the problems in planar digraphs. For DST our result improves the O(R + log k) approximation of [Zachary Friggstad and Ramin Mousavi, 2023] when R = ω(log² k).

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Routing and network design problems
Keywords
  • Directed Planar Graphs
  • Submodular Functions
  • Steiner Tree
  • Network Design

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