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

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)

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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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Saarbrücken Graduate School of Computer ScienceMax Planck Institute for InformaticsSaarbrückenGermany
  2. 2.University of TwenteEnschedeThe Netherlands

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