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

Author Xianghui Zhong

Thumbnail PDF


  • Filesize: 0.51 MB
  • 13 pages

Document Identifiers

Author Details

Xianghui Zhong
  • University of Bonn, Germany


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

Cite AsGet BibTex

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)


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
  • traveling salesman problem
  • metric TSP
  • graph TSP
  • k-Opt algorithm
  • Lin-Kernighan algorithm
  • approximation algorithm
  • approximation ratio.


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Noga Alon, Shlomo Hoory, and Nathan Linial. The Moore bound for irregular graphs. Graphs and Combinatorics, 18(1):53-57, March 2002. URL: https://doi.org/10.1007/s003730200002.
  2. Clark T. Benson. Minimal regular graphs of girths eight and twelve. Canadian Journal of Mathematics, 18:1091–1094, 1966. URL: https://doi.org/10.4153/CJM-1966-109-8.
  3. Jon J. Bentley. Fast algorithms for geometric traveling salesman problems. ORSA Journal on computing, 4(4):387-411, 1992. URL: https://doi.org/10.1287/ijoc.4.4.387.
  4. William G. Brown. On graphs that do not contain a thomsen graph. Canadian Mathematical Bulletin, 9(3):281-285, 1966. Google Scholar
  5. Barun Chandra, Howard Karloff, and Craig Tovey. New results on the old k-opt algorithm for the traveling salesman problem. SIAM Journal on Computing, 28(6):1998-2029, 1999. URL: https://doi.org/10.1137/S0097539793251244.
  6. Nicos Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Technical report, Carnegie-Mellon Univ Pittsburgh Pa Management Sciences Research Group, 1976. Google Scholar
  7. Matthias Englert, Heiko Röglin, and Berthold Vöcking. Worst case and probabilistic analysis of the 2-opt algorithm for the TSP. Algorithmica, 68(1):190-264, January 2014. URL: https://doi.org/10.1007/s00453-013-9801-4.
  8. Paul Erdős and Alfréd Rényi. On a problem of graph theory. Magyar Tudományos Akadémia Math. Kuató Int. Közl., 7:623-641, 1962. Google Scholar
  9. Paul Erdős, Alfréd Rényi, and Vera T. Sós. On a problem of graph theory. Studia Scientiarum Mathematicarum Hungarica, 1:215-235, 1966. Google Scholar
  10. Paul Erdős. Extremal problems in graph theory. In Proc. Symp. Theory of Graphs and its Applications, pages 29-36, 1963. Google Scholar
  11. Jacob Fox and Benny Sudakov. Decompositions into subgraphs of small diameter. Combinatorics, Probability and Computing, 19(5-6):753-774, 2010. URL: https://doi.org/10.1017/S0963548310000040.
  12. Michael R Garey and David S Johnson. Computers and intractability, volume 174. freeman San Francisco, 1979. Google Scholar
  13. Stefan Hougardy, Fabian Zaiser, and Xianghui Zhong. The approximation ratio of the 2-opt heuristic for the metric traveling salesman problem. Operations Research Letters, 48(4):401-404, 2020. URL: https://doi.org/10.1016/j.orl.2020.05.007.
  14. David S. Johnson. Local optimization and the traveling salesman problem. In International colloquium on automata, languages, and programming, pages 446-461. Springer, 1990. URL: https://doi.org/10.1007/BFb0032050.
  15. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations, pages 85-103. Springer, 1972. URL: https://doi.org/10.1007/978-3-540-68279-0_8.
  16. Bernhard Korte and Jens Vygen. Combinatorial Optimization: Theory and Algorithms. Springer Publishing Company, Incorporated, 4th edition, 2007. URL: https://doi.org/10.1007/978-3-642-24488-9.
  17. Marvin Künnemann and Bodo Manthey. Towards understanding the smoothed approximation ratio of the 2-opt heuristic. In International Colloquium on Automata, Languages, and Programming, pages 859-871. Springer, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_70.
  18. Felix Lazebnik, Vasiliy A. Ustimenko, and Andrew J. Woldar. A new series of dense graphs of high girth. Bulletin of the American mathematical society, 32(1):73-79, 1995. URL: https://doi.org/10.1090/S0273-0979-1995-00569-0.
  19. Asaf Levin and Uri Yovel. Nonoblivious 2-opt heuristics for the traveling salesman problem. Networks, 62(3):201-219, 2013. URL: https://doi.org/10.1002/net.21512.
  20. Shen Lin and Brian W. Kernighan. An effective heuristic algorithm for the traveling-salesman problem. Operations research, 21(2):498-516, 1973. URL: https://doi.org/10.1287/opre.21.2.498.
  21. Ján Plesník. Bad examples of the metric traveling salesman problem for the 2-change heuristic. Acta Mathematica Universitatis Comenianae, 55:203-207, 1986. Google Scholar
  22. Gerhard Reinelt. The traveling salesman: computational solutions for TSP applications. Springer-Verlag, 1994. URL: https://doi.org/10.1007/3-540-48661-5.
  23. Daniel J Rosenkrantz, Richard E Stearns, and Philip M Lewis, II. An analysis of several heuristics for the traveling salesman problem. SIAM journal on computing, 6(3):563-581, 1977. URL: https://doi.org/10.1007/978-1-4020-9688-4_3.
  24. A. I. Serdjukov. Some extremal bypasses in graphs [in Russian]. Upravlyaemye Sistemy, 17:76-79, 1978. Google Scholar
  25. Robert Singleton. On minimal graphs of maximum even girth. Journal of Combinatorial Theory, 1(3):306-332, 1966. URL: https://doi.org/10.1016/S0021-9800(66)80054-6.
  26. Rephael Wenger. Extremal graphs with no C^4’s, C^6’s, or C^10’s. J. Comb. Theory, Ser. B, 52(1):113-116, 1991. URL: https://doi.org/10.1016/0095-8956(91)90097-4.
  27. David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, New York, NY, USA, 1st edition, 2011. Google Scholar