The Approximation Ratio of the 2-Opt Heuristic for the Euclidean Traveling Salesman Problem

Authors Ulrich A. Brodowsky, Stefan Hougardy

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Ulrich A. Brodowsky
  • Pontsheide 20, 52076 Aachen, Germany
Stefan Hougardy
  • Research Institute for Discrete Mathematics, Universität Bonn, Germany


We thank the anonymous referees for several useful comments.

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Ulrich A. Brodowsky and Stefan Hougardy. The Approximation Ratio of the 2-Opt Heuristic for the Euclidean Traveling Salesman Problem. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 18:1-18:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


The 2-Opt heuristic is a simple improvement heuristic for the Traveling Salesman Problem. It starts with an arbitrary tour and then repeatedly replaces two edges of the tour by two other edges, as long as this yields a shorter tour. We will prove that for Euclidean Traveling Salesman Problems with n cities the approximation ratio of the 2-Opt heuristic is Θ(log n / log log n). This improves the upper bound of O(log n) given by Chandra, Karloff, and Tovey [Barun Chandra et al., 1999] in 1999.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • traveling salesman problem
  • metric TSP
  • Euclidean TSP
  • 2-Opt
  • approximation algorithm


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