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# On the Approximation Ratio of the k-Opt and Lin-Kernighan Algorithm for Metric and Graph TSP

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## Acknowledgements

I want to thank Fabian Henneke, Stefan Hougardy, Yvonne Omlor, Heiko Röglin and Fabian Zaiser for reading this paper and making helpful remarks.

## Cite As

Xianghui Zhong. On the Approximation Ratio of the k-Opt and Lin-Kernighan Algorithm for Metric and Graph TSP. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 83:1-83:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ESA.2020.83

## Abstract

The k-Opt and Lin-Kernighan algorithm are two of the most important local search approaches for the Metric TSP. Both start with an arbitrary tour and make local improvements in each step to get a shorter tour. We show that for any fixed k ≥ 3 the approximation ratio of the k-Opt algorithm for Metric TSP is O(√[k]{n}). Assuming the Erdős girth conjecture, we prove a matching lower bound of Ω(√[k]{n}). Unconditionally, we obtain matching bounds for k = 3,4,6 and a lower bound of Ω(n^{2/(3k-3)}). Our most general bounds depend on the values of a function from extremal graph theory and are tight up to a factor logarithmic in the number of vertices unconditionally. Moreover, all the upper bounds also apply to a parameterized version of the Lin-Kernighan algorithm with appropriate parameter. We also show that the approximation ratio of k-Opt for Graph TSP is Ω(log(n)/(log log(n))) and O({log(n)/(log log(n))}^{log₂(9)+ε}) for all ε > 0.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Approximation algorithms analysis
##### Keywords
• traveling salesman problem
• metric TSP
• graph TSP
• k-Opt algorithm
• Lin-Kernighan algorithm
• approximation algorithm
• approximation ratio.

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