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 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(log 2 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 log 2 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) .