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

Afrouz Jabal Ameli, Faezeh Motiei, Morteza Saghafian

TL;DR

This work analyzes how the total lengths of Euclidean minimum spanning trees (EMSTs) of color classes compare to the EMST of the whole point set, introducing the MAX-MST-ratio problem for both abstract graphs and geometric point sets. It establishes strong computational barriers: MAX-MST-ratio is NP-hard and hard to approximate in the abstract graph setting, and MAX-EMST-ratio is NP-hard in the geometric setting. The authors provide a universal upper bound of $3-\frac{4}{n-1}$ in metric spaces, a simple $O(n^2)$-time $3$-approximation for MAX-EMST-ratio, and show that bipartite colorings can yield near-optimal EMST-ratios in practice. They also analyze average-case behavior, proving a lower bound of $\frac{n-2}{n-1}$ for the average EMST-ratio and showing that for random points in $[0,1]^d$ the average tends to $\sqrt[d]{2}$ as $n\to\infty$, supported by experimental results.

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.

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

TL;DR

This work analyzes how the total lengths of Euclidean minimum spanning trees (EMSTs) of color classes compare to the EMST of the whole point set, introducing the MAX-MST-ratio problem for both abstract graphs and geometric point sets. It establishes strong computational barriers: MAX-MST-ratio is NP-hard and hard to approximate in the abstract graph setting, and MAX-EMST-ratio is NP-hard in the geometric setting. The authors provide a universal upper bound of in metric spaces, a simple -time -approximation for MAX-EMST-ratio, and show that bipartite colorings can yield near-optimal EMST-ratios in practice. They also analyze average-case behavior, proving a lower bound of for the average EMST-ratio and showing that for random points in the average tends to as , supported by experimental results.

Abstract

Given a finite set of red and blue points in , 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 -approximation algorithm for this problem. As part of the proof, we show that, in any metric space, the maximum MST-ratio is smaller than . Additionally, we study the average MST-ratio over all colorings of a set of points. We show that this average is always at least , and for random points uniformly distributed in a -dimensional unit cube, the average tends to in expectation as approaches infinity.
Paper Structure (5 sections, 4 theorems, 3 equations, 1 figure)

This paper contains 5 sections, 4 theorems, 3 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Work
  3. Outline
  4. Maximum MST-ratio of Abstract Graphs
  5. Maximum MST-ratio of Geometric Point Sets

Key Result

theorem thmcountertheorem

For every $0<\varepsilon\le 1$, there is no polynomial-time $O(n^{1-\varepsilon})$-approximation with weights restricted to $1$ and $n$ for MAX-MST-ratio problem, unless P=NP.

Figures (1)

  • Figure 1: The point set $P$ shown by square nodes is $3$ vertices of an equilateral triangle of side length $1$. The black edges form $\text{EMST}(P)$ with length $2$. Adding an extra vertex in the center, the green edges form the minimum Steiner tree of $P$ with length $\sqrt{3}$.

Theorems & Definitions (7)

  • theorem thmcountertheorem
  • proof
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • theorem thmcountertheorem
  • proof