A quasipolynomial algorithm for wellspaced hyperbolic TSP
Abstract
We study the traveling salesman problem in the hyperbolic plane of Gaussian curvature $1$. Let $\alpha$ denote the minimum distance between any two input points. Using a new separator theorem and a new rerouting argument, we give an $n^{O(\log^2 n)\max(1,1/\alpha)}$ algorithm for Hyperbolic TSP. This is quasipolynomial time if $\alpha$ is at least some absolute constant, and it grows to $n^{O(\sqrt{n})}$ as $\alpha$ decreases to $\log^2 n/\sqrt{n}$. (For even smaller values of $\alpha$, we can use a planaritybased algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 arXiv:
 arXiv:2002.05414
 Bibcode:
 2020arXiv200205414K
 Keywords:

 Computer Science  Computational Geometry;
 Computer Science  Data Structures and Algorithms
 EPrint:
 SoCG 2020