The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5

Authors Sophia Heimann , Hung P. Hoang , Stefan Hougardy



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Author Details

Sophia Heimann
  • Research Institute for Discrete Mathematics, University of Bonn, Germany
Hung P. Hoang
  • Algorithms and Complexity Group, Faculty of Informatics, TU Wien, Austria
Stefan Hougardy
  • Research Institute for Discrete Mathematics and Hausdorff Center for Mathematics, University of Bonn, Germany

Acknowledgements

This work was initiated at the "Discrete Optimization" trimester program of the Hausdorff Institute of Mathematics, University of Bonn, Germany in 2021. We would like to thank the organizers for providing excellent working conditions and inspiring atmosphere.

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Sophia Heimann, Hung P. Hoang, and Stefan Hougardy. The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 84:1-84:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.84

Abstract

The k-Opt algorithm is a local search algorithm for the Traveling Salesman Problem. Starting with an initial tour, it iteratively replaces at most k edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the Traveling Salesman Problem with the k-Opt neighborhood is complete for the class PLS (polynomial time local search) and that the k-Opt algorithm can have exponential running time for any pivot rule. However, his proof requires k ≫ 1000 and has a substantial gap. We show the two properties above for a much smaller value of k, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). In particular, we prove the PLS-completeness for k ≥ 17 and the exponential running time for k ≥ 5.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Traveling Salesman Problem
  • k-Opt algorithm
  • PLS-completeness

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References

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