On the MST-ratio: Theoretical Bounds and Complexity of Finding the Maximum

TL;DR

This paper investigates the MST-ratio for red and blue point sets, establishing its computational complexity, providing approximation algorithms, and analyzing average behavior in random point distributions.

Contribution

It proves NP-hardness of finding the maximum MST-ratio, offers a quadratic-time 3-approximation algorithm, and analyzes the average MST-ratio in random point sets.

Findings

Maximum MST-ratio is NP-hard to compute.

A quadratic-time 3-approximation algorithm exists.

Average MST-ratio over all colorings is at least (n-2)/(n-1).

Abstract

Given a finite set of red and blue points in $\Rspace^d$, the MST-ratio is defined as the total length of the Euclidean minimum spanning trees of the red points and the blue points, divided by the length of the Euclidean minimum spanning tree of their union. The MST-ratio has recently gained attention due to its direct interpretation in topological models for studying point sets with applications in spatial biology. The maximum MST-ratio of a point set is the maximum MST-ratio over all proper colorings of its points by red and blue. We prove that finding the maximum MST-ratio of a given point set is NP-hard when the dimension is part of the input. Moreover, we present a quadratic-time $3$-approximation algorithm for this problem. As part of the proof, we show that, in any metric space, the maximum MST-ratio is smaller than $3$. Additionally, we study the average MST-ratio over all colorings of a set of $n$ points. We show that this average is always at least $\frac{n-2}{n-1}$, and for $n$ random points uniformly distributed in a $d$-dimensional unit cube, the average tends to $\sqrt[d]{2}$ in expectation as $n$ approaches infinity.