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Documents authored by Hougardy, Stefan

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Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2024.84
The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5

Authors: Sophia Heimann, Hung P. Hoang, and Stefan Hougardy

Published in: LIPIcs, Volume 297, 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)

Abstract
The k-Opt algorithm is a local search algorithm for the Traveling Salesman Problem. Starting with an initial tour, it iteratively replaces at most k edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the Traveling Salesman Problem with the k-Opt neighborhood is complete for the class PLS (polynomial time local search) and that the k-Opt algorithm can have exponential running time for any pivot rule. However, his proof requires k ≫ 1000 and has a substantial gap. We show the two properties above for a much smaller value of k, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). In particular, we prove the PLS-completeness for k ≥ 17 and the exponential running time for k ≥ 5.
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Sophia Heimann, Hung P. Hoang, and Stefan Hougardy. The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 84:1-84:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{heimann_et_al:LIPIcs.ICALP.2024.84,
  author =	{Heimann, Sophia and Hoang, Hung P. and Hougardy, Stefan},
  title =	{{The k-Opt Algorithm for the Traveling Salesman Problem Has Exponential Running Time for k ≥ 5}},
  booktitle =	{51st International Colloquium on Automata, Languages, and Programming (ICALP 2024)},
  pages =	{84:1--84:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-322-5},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{297},
  editor =	{Bringmann, Karl and Grohe, Martin and Puppis, Gabriele and Svensson, Ola},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2024.84},
  URN =		{urn:nbn:de:0030-drops-202270},
  doi =		{10.4230/LIPIcs.ICALP.2024.84},
  annote =	{Keywords: Traveling Salesman Problem, k-Opt algorithm, PLS-completeness}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.18
The Approximation Ratio of the 2-Opt Heuristic for the Euclidean Traveling Salesman Problem

Authors: Ulrich A. Brodowsky and Stefan Hougardy

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
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.
Cite as

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)

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@InProceedings{brodowsky_et_al:LIPIcs.STACS.2021.18,
  author =	{Brodowsky, Ulrich A. and Hougardy, Stefan},
  title =	{{The Approximation Ratio of the 2-Opt Heuristic for the Euclidean Traveling Salesman Problem}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{18:1--18:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.18},
  URN =		{urn:nbn:de:0030-drops-136634},
  doi =		{10.4230/LIPIcs.STACS.2021.18},
  annote =	{Keywords: traveling salesman problem, metric TSP, Euclidean TSP, 2-Opt, approximation algorithm}
}
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