Throttling for metric dimension and its variants

TL;DR

This work extends metric-dimension theory by introducing throttling variants that minimize the sum of a resolving-dimension parameter and a radius (cost) parameter. It defines distance-$r$ throttling for standard, edge, and mixed metric dimensions, establishes NP-hardness for computing these throttling numbers, and provides a suite of tight asymptotic bounds and exact values for numerous graph families. Key contributions include a universal lower bound on throttling in terms of the size of the target family, precise results for complete graphs, paths, cycles, trees, circulants, grids, and hypercubes, and a unifying DP/IP framework to compute throttling numbers. The findings illuminate fundamental limits of compact embeddings with low-dimensional, bounded-magnitude coordinates and offer practical guidance for embedding and navigation tasks, while also leaving open questions about coefficient tightness and connected-graph hardness proofs.

Abstract

Metric dimension is a graph parameter that has been applied to robot navigation and finding low-dimensional vector embeddings. Throttling entails minimizing the sum of two available resources when solving certain graph problems. In this paper, we introduce throttling for metric dimension, edge metric dimension, and mixed metric dimension. In the context of vector embeddings, metric dimension throttling finds a low-dimensional, low-magnitude embedding with integer coordinates. We show that computing the throttling number is NP-hard for all three variants. We give formulas for the throttling numbers of special families of graphs, and characterize graphs with extremal throttling numbers. We also prove that the minimum possible throttling number of a graph of order $n$ is $Θ\left(\frac{\log{n}}{\log{\log{n}}}\right)$, while the minimum possible throttling number of a tree of order $n$ is $Θ(n^{1/3})$ or $Θ(n^{1/2})$ depending on the variant of metric dimension.

Figures (6)

• Figure 5.1: A $O(n^{1/3})$ metric dimension throttling configuration for a tree of order $n$.
• Figure 5.2: Metric dimension throttling numbers for $C_n$ and $P_n$, $1\leq n\leq 50$.
• Figure 5.3: A metric dimension throttling configuration for $\operatorname{Circ}_{n}\!\left( S \right)$ with $n=17$, $\ell=2$, $S=\{1,2\}$, and we place $6$ landmarks in $3$ groups of $2$ adjacent landmarks.
• Figure 5.4: Metric dimension throttling numbers for $Q_d$, $1\leq d\leq 5$.
• Figure 6.1: An optimal edge metric dimension throttling configuration for $C_n$ with $n=25$, $k=3$, $r=3$, and the landmarks are at distance $3$ and $6$ alternating, and an edge $e$ and $e'$ from the second part of the proof are marked.

Theorems & Definitions (86)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3