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).
@InProceedings{alkema_et_al:LIPIcs.SoCG.2020.4, author = {Alkema, Henk and de Berg, Mark and Kisfaludi-Bak, S\'{a}ndor}, title = {{Euclidean TSP in Narrow Strips}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {4:1--4:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-143-6}, ISSN = {1868-8969}, year = {2020}, volume = {164}, editor = {Cabello, Sergio and Chen, Danny Z.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.4}, URN = {urn:nbn:de:0030-drops-121628}, doi = {10.4230/LIPIcs.SoCG.2020.4}, annote = {Keywords: Computational geometry, Euclidean TSP, bitonic TSP, fixed-parameter tractable algorithms} }
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