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Counting Locally Optimal Tours in the TSP

Authors Bodo Manthey , Jesse van Rhijn

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Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Discrete optimization
Keywords
  • Travelling salesman problem
  • probabilistic analysis
  • local search
  • heuristics
  • 2-opt

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Abstract

We show that the problem of counting 2-optimal tours in instances of the Travelling Salesperson Problem (TSP) on complete graphs is #P-complete. In addition, we show that the expected number of 2-optimal tours in random instances of the TSP on complete graphs is O(1.2098ⁿ √{n!}). Based on numerical experiments, we conjecture that the true bound is at most O(√{n!}), which is approximately the square root of the total number of tours.

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Bodo Manthey and Jesse van Rhijn. Counting Locally Optimal Tours in the TSP. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 73:1-73:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.MFCS.2025.73

Author Details

Bodo Manthey
  • Faculty of Electrical Engineering, Mathematics, and Computer Science, University of Twente, The Netherlands
Jesse van Rhijn
  • Faculty of Electrical Engineering, Mathematics, and Computer Science, University of Twente, The Netherlands

Funding

  • van Rhijn, Jesse: Supported by NWO grant OCENW.KLEIN.176.

Acknowledgements

We thank Alexander E. Black for helpful discussions that led to the geometric perspective outlined in Section 4.4.

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