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Towards Understanding the Smoothed Approximation Ratio of the 2-Opt Heuristic

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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Abstract

The 2-Opt heuristic is a very simple, easy-to-implement local search heuristic for the traveling salesman problem. While it usually provides good approximations to the optimal tour in experiments, its worst-case performance is poor.

In an attempt to explain the approximation performance of 2-Opt, we analyze the smoothed approximation ratio of 2-Opt. We obtain a bound of \(O(\log (1/\sigma ))\) for the smoothed approximation ratio of 2-Opt. As a lower bound, we prove that the worst-case lower bound of \(\Omega (\frac{\log n}{\log \log n})\) for the approximation ratio holds for \(\sigma = O(1/\sqrt{n})\).

Our main technical novelty is that, different from existing smoothed analyses, we do not separately analyze objective values of the global and the local optimum on all inputs, but simultaneously bound them on the same input.

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Author information

Authors and Affiliations

  1. Saarbrücken Graduate School of Computer Science, Max Planck Institute for Informatics, Saarbrücken, Germany

    Marvin Künnemann

  2. University of Twente, Enschede, The Netherlands

    Bodo Manthey

Authors
  1. Marvin Künnemann
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  2. Bodo Manthey
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Corresponding author

Correspondence to Marvin Künnemann .

Editor information

Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Künnemann, M., Manthey, B. (2015). Towards Understanding the Smoothed Approximation Ratio of the 2-Opt Heuristic. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_70

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_70

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-47671-0

  • Online ISBN: 978-3-662-47672-7

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