Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree

Authors Yury Makarychev , Max Ovsiankin , Erasmo Tani



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Author Details

Yury Makarychev
  • Toyota Technological Institute at Chicago, IL, USA
Max Ovsiankin
  • Toyota Technological Institute at Chicago, IL, USA
Erasmo Tani
  • University of Chicago, IL, USA

Acknowledgements

We would like to thank the anonymous reviewers for providing references to relevant prior work and for comments that improved the presentation of this paper.

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Yury Makarychev, Max Ovsiankin, and Erasmo Tani. Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 111:1-111:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.111

Abstract

We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a vector cost c_e ∈ ℝ_{≥0}^𝓁. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the 𝓁_p-norm of the obtained cost vector (we assume that p ≥ 1 is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results.

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Shortest paths
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Routing and network design problems
Keywords
  • Shortest Path
  • Asymmetric Group Steiner Tree
  • Sum-of-Squares

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