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DOI: 10.4230/LIPIcs.ESA.2020.83

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

Authors: Xianghui Zhong

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

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.

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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)

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@InProceedings{zhong:LIPIcs.ESA.2020.83,
  author =	{Zhong, Xianghui},
  title =	{{On the Approximation Ratio of the k-Opt and Lin-Kernighan Algorithm for Metric and Graph TSP}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{83:1--83:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.83},
  URN =		{urn:nbn:de:0030-drops-129497},
  doi =		{10.4230/LIPIcs.ESA.2020.83},
  annote =	{Keywords: traveling salesman problem, metric TSP, graph TSP, k-Opt algorithm, Lin-Kernighan algorithm, approximation algorithm, approximation ratio.}
}

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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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