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A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP

Authors Susanne Armbruster , Matthias Mnich , Martin Nägele

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Author Details

Susanne Armbruster
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany
Matthias Mnich
  • Hamburg University of Technology, Institute for Algorithms and Complexity, Hamburg, Germany
Martin Nägele
  • Department of Mathematics, ETH Zurich, Zurich, Switzerland

Funding

  • Mnich, Matthias: Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), project MN 59/4-1.
  • Nägele, Martin: Partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXZ-2047/1 - 390685813, and the Swiss National Science Foundation (grant no. P500PT_206742).

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Susanne Armbruster, Matthias Mnich, and Martin Nägele. A (3/2 + 1/e)-Approximation Algorithm for Ordered TSP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 1:1-1:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.1

Abstract

We present a new (3/2 + 1/e)-approximation algorithm for the Ordered Traveling Salesperson Problem (Ordered TSP). Ordered TSP is a variant of the classic metric Traveling Salesperson Problem (TSP) where a specified subset of vertices needs to appear on the output Hamiltonian cycle in a given order, and the task is to compute a cheapest such cycle. Our approximation guarantee of approximately 1.868 holds with respect to the value of a natural new linear programming (LP) relaxation for Ordered TSP. Our result significantly improves upon the previously best known guarantee of 5/2 for this problem and thereby considerably reduces the gap between approximability of Ordered TSP and metric TSP. Our algorithm is based on a decomposition of the LP solution into weighted trees that serve as building blocks in our tour construction.

Subject Classification

ACM Subject Classification
  • Theory of computation → Discrete optimization
  • Theory of computation → Routing and network design problems
  • Theory of computation → Rounding techniques
Keywords
  • Travelling Salesperson Problem
  • precedence constraints
  • linear programming
  • approximation algorithms

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