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Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem

Authors Josefine Foos, Stephan Held , Yannik Kyle Dustin Spitzley

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Josefine Foos
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany
Stephan Held
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany
Yannik Kyle Dustin Spitzley
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany

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Josefine Foos, Stephan Held, and Yannik Kyle Dustin Spitzley. Tighter Approximation for the Uniform Cost-Distance Steiner Tree Problem. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2023.19

Abstract

Uniform cost-distance Steiner trees minimize the sum of the total length and weighted path lengths from a dedicated root to the other terminals. They are applied when the tree is intended for signal transmission, e.g. in chip design or telecommunication networks. They are a special case of general cost-distance Steiner trees, where different distance functions are used for total length and path lengths.
We improve the best published approximation factor for the uniform cost-distance Steiner tree problem from 2.39 [Khazraei and Held, 2021] to 2.05. If we can approximate the minimum-length Steiner tree problem arbitrarily well, our algorithm achieves an approximation factor arbitrarily close to 1+1/√2. This bound is tight in the following sense. We also prove the gap 1+1/√2 between optimum solutions and the lower bound which we and all previous approximation algorithms for this problem use.
Similarly to previous approaches, we start with an approximate minimum-length Steiner tree and split it into subtrees that are later re-connected. To improve the approximation factor, we split it into components more carefully, taking the cost structure into account, and we significantly enhance the analysis.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • cost-distance Steiner tree
  • approximation algorithm
  • uniform

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