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Documents authored by Hougardy, Stefan

Document
DOI: 10.4230/LIPIcs.STACS.2026.54
A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm

Authors: Stefan Hougardy and Bart Zondervan

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
In the Strip Packing problem, we are given a vertical strip of fixed width and unbounded height, along with a set of axis‑parallel rectangles. The task is to place all rectangles within the strip, without overlaps, while minimizing the height of the packing. This problem is known to be NP-hard. The Bottom-Left Algorithm is a simple and widely used heuristic for Strip Packing. Given a fixed order of the rectangles, it places them one by one, always choosing the lowest feasible position in the strip and, in case of ties, the leftmost one. Baker, Coffman, and Rivest proved in 1980 that the Bottom-Left Algorithm has approximation ratio 3 if the rectangles are sorted by decreasing width [Brenda S. Baker et al., 1980]. For the past 45 years, no alternative ordering has been found that improves this bound. We introduce a new rectangle ordering and show that with this ordering the Bottom-Left Algorithm achieves a 13/6 approximation for the Strip Packing problem.
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Stefan Hougardy and Bart Zondervan. A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 54:1-54:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{hougardy_et_al:LIPIcs.STACS.2026.54,
  author =	{Hougardy, Stefan and Zondervan, Bart},
  title =	{{A 13/6-Approximation for Strip Packing via the Bottom-Left Algorithm}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{54:1--54:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.54},
  URN =		{urn:nbn:de:0030-drops-255432},
  doi =		{10.4230/LIPIcs.STACS.2026.54},
  annote =	{Keywords: Approximation Algorithm, Strip Packing, Bottom-Left Algorithm, Rectangle Packing}
}
Document
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.
Cite as

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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