Geometric Planar Networks on Bichromatic Points

TL;DR

This paper investigates classical geometric graph problems on bichromatic collinear points, demonstrating that most can be solved efficiently in linear time, contrasting with their NP-hardness in general Euclidean settings.

Contribution

The paper provides the first linear-time algorithms for Hamiltonian path, minimum spanning tree, and minimum perfect matching on bichromatic collinear points.

Findings

Most problems are solvable in linear time for collinear bichromatic points.

Contrasts NP-hardness in general Euclidean plane with linear-time solutions in this special case.

Extends understanding of geometric graph problems on restricted point configurations.

Abstract

We study three classical graph problems - Hamiltonian path, minimum spanning tree, and minimum perfect matching on geometric graphs induced by bichromatic (red and blue) points. These problems have been widely studied for points in the Euclidean plane, and many of them are NP-hard. In this work, we consider these problems for collinear points. We show that almost all of these problems can be solved in linear time in this setting.