A quasi-polynomial algorithm for well-spaced hyperbolic TSP

02/13/2020

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by   Sándor Kisfaludi-Bak, et al.

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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)).)