A quasi-polynomial algorithm for well-spaced hyperbolic TSP: https://arxiv.org/abs/2002.05414
This new preprint by Sándor Kisfaludi-Bak (accepted to SoCG) just came out and caught my attention. TSP is NP-hard for Euclidean points or close-together hyperbolic points. This paper shows that it's much easier when the points are widely spaced in the hyperbolic plane. The idea is to separate the input by a short line segment that the solution crosses few times and apply dynamic programming.
This is the second time I read about a problem becoming easier in hyperbolic space. I forgot the other one, I'll have to hunt for it! I can see what's happening with TSP here - there must be more problems like this! Thanks for inspiring!
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