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Euclidean TSP in Narrow Strips

Authors Henk Alkema, Mark de Berg, Sándor Kisfaludi-Bak

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ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Computational geometry
  • Euclidean TSP
  • bitonic TSP
  • fixed-parameter tractable algorithms

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Abstract

We investigate how the complexity of {Euclidean TSP} for point sets P inside the strip (-∞,+∞)×[0,δ] depends on the strip width δ. We obtain two main results.  
- For the case where the points have distinct integer x-coordinates, we prove that a shortest bitonic tour (which can be computed in O(n log²n) time using an existing algorithm) is guaranteed to be a shortest tour overall when δ ⩽ 2√2, a bound which is best possible. 
- We present an algorithm that is fixed-parameter tractable with respect to δ. More precisely, our algorithm has running time 2^{O(√δ)} n² for sparse point sets, where each 1×δ rectangle inside the strip contains O(1) points. For random point sets, where the points are chosen uniformly at random from the rectangle [0,n]× [0,δ], it has an expected running time of 2^{O(√δ)} n² + O(n³).

Cite As Get BibTex

Henk Alkema, Mark de Berg, and Sándor Kisfaludi-Bak. Euclidean TSP in Narrow Strips. In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SoCG.2020.4

Author Details

Henk Alkema
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Mark de Berg
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Sándor Kisfaludi-Bak
  • Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

Funding

The work in this paper is supported by the Netherlands Organisation for Scientific Research (NWO) through Gravitation-grant NETWORKS-024.002.003.

Acknowledgements

We thank Remco van der Hofstad for discussions about the probabilistic analysis of an earlier version of the algorithm.

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