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

Authors Zachary Friggstad, Ramin Mousavi

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Zachary Friggstad
  • Department of Computing Science, University of Alberta, Canada
Ramin Mousavi
  • Department of Computing Science, University of Alberta, Canada

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Zachary Friggstad and Ramin Mousavi. A Constant-Factor Approximation for Quasi-Bipartite Directed Steiner Tree on Minor-Free Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 13:1-13:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.13

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
Keywords
  • Directed Steiner tree
  • Combinatorial optimization
  • approximation algorithms

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