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# Improved Smoothed Analysis of 2-Opt for the Euclidean TSP

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LIPIcs.ISAAC.2023.52.pdf
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## Acknowledgements

We thank Ashkan Safari and Tjark Vredeveld for many useful discussions.

## Cite As

Bodo Manthey and Jesse van Rhijn. Improved Smoothed Analysis of 2-Opt for the Euclidean TSP. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.52

## Abstract

The 2-opt heuristic is a simple local search heuristic for the Travelling Salesperson Problem (TSP). Although it usually performs well in practice, its worst-case running time is poor. Attempts to reconcile this difference have used smoothed analysis, in which adversarial instances are perturbed probabilistically. We are interested in the classical model of smoothed analysis for the Euclidean TSP, in which the perturbations are Gaussian. This model was previously used by Manthey & Veenstra, who obtained smoothed complexity bounds polynomial in n, the dimension d, and the perturbation strength σ^{-1}. However, their analysis only works for d ≥ 4. The only previous analysis for d ≤ 3 was performed by Englert, Röglin & Vöcking, who used a different perturbation model which can be translated to Gaussian perturbations. Their model yields bounds polynomial in n and σ^{-d}, and super-exponential in d. As the fact that no direct analysis exists for Gaussian perturbations that yields polynomial bounds for all d is somewhat unsatisfactory, we perform this missing analysis. Along the way, we improve all existing smoothed complexity bounds for Euclidean 2-opt with Gaussian perturbations.

## 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
• smoothed analysis
• probabilistic analysis
• local search
• heuristics
• 2-opt

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## References

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