We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature -1. Let α denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an n^{O(log² n)max(1,1/α)} algorithm for Hyperbolic TSP. This is quasi-polynomial time if α is at least some absolute constant, and it grows to n^O(√n) as α decreases to log² n/√n. (For even smaller values of α, we can use a planarity-based algorithm of Hwang et al. (1993), which gives a running time of n^O(√n).)
@InProceedings{kisfaludibak:LIPIcs.SoCG.2020.55, author = {Kisfaludi-Bak, S\'{a}ndor}, title = {{A Quasi-Polynomial Algorithm for Well-Spaced Hyperbolic TSP}}, booktitle = {36th International Symposium on Computational Geometry (SoCG 2020)}, pages = {55:1--55:15}, 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.2020.55}, URN = {urn:nbn:de:0030-drops-122135}, doi = {10.4230/LIPIcs.SoCG.2020.55}, annote = {Keywords: Computational geometry, Hyperbolic geometry, Traveling salesman} }
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