A quasi-polynomial algorithm for well-spaced 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 quasi-polynomial 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 planarity-based algorithm of Hwang et al. (1993), which gives a running time of $n^{O(\sqrt{n})}$.)