License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2023.13
URN: urn:nbn:de:0030-drops-188389
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18838/
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Friggstad, Zachary ; Mousavi, Ramin

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

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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⋅√(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.

BibTeX - Entry

@InProceedings{friggstad_et_al:LIPIcs.APPROX/RANDOM.2023.13,
  author =	{Friggstad, Zachary and Mousavi, Ramin},
  title =	{{A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)},
  pages =	{13:1--13:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-296-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{275},
  editor =	{Megow, Nicole and Smith, Adam},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18838},
  URN =		{urn:nbn:de:0030-drops-188389},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2023.13},
  annote =	{Keywords: Directed Steiner tree, Combinatorial optimization, approximation algorithms}
}

Keywords: Directed Steiner tree, Combinatorial optimization, approximation algorithms
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023)
Issue Date: 2023
Date of publication: 04.09.2023

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