# Euclidean TSP, Motorcycle Graphs, and Other New Applications of Nearest-Neighbor Chains

## TL;DR

This paper demonstrates the versatility of the nearest-neighbor chain algorithm by applying it to various geometric problems, achieving improved time complexities and novel solutions in computational geometry and matching models.

## Contribution

It introduces new applications of the nearest-neighbor chain algorithm to geometric problems, including Euclidean TSP, motorcycle graphs, and stable matching variants, with improved algorithms and complexities.

## Key findings

- Euclidean TSP greedy tour in O(n log n) time
- Motorcycle graphs computed in O(n^{4/3+ε}) time
- Linear-time 2-approximation for 1D geometric set cover

## Abstract

We show new applications of the nearest-neighbor chain algorithm, a technique that originated in agglomerative hierarchical clustering. We apply it to a diverse class of geometric problems: we construct the greedy multi-fragment tour for Euclidean TSP in $O(n\log n)$ time in any fixed dimension and for Steiner TSP in planar graphs in $O(n\sqrt{n}\log n)$ time; we compute motorcycle graphs (which are a central part in straight skeleton algorithms) in $O(n^{4/3+\varepsilon})$ time for any $\varepsilon>0$; we introduce a narcissistic variant of the $k$-attribute stable matching model, and solve it in $O(n^{2-4/(k(1+\varepsilon)+2)})$ time; we give a linear-time $2$-approximation for a 1D geometric set cover problem with applications to radio station placement.