Throttling for metric dimension and its variants

2025-10-01

AI Summary

This paper initiates the first systematic study of throttling for metric dimension, edge metric dimension, and mixed metric dimension—aiming to jointly minimize locating set size and embedding dimension/amplitude for low-dimensional, low-amplitude integer vector embeddings. Using techniques from graph theory, combinatorial optimization, and computational complexity, we prove that computing each of the three throttling numbers is NP-hard. We derive exact throttling formulas for fundamental graph classes—including paths, cycles, complete graphs, and trees—and characterize extremal graph structures achieving these bounds. Moreover, we establish asymptotically tight lower bounds on the minimum throttling number: Θ(log n / log log n) for general graphs and Θ(n¹ᐟ³) for trees. The central contribution is the introduction and foundational development of *metric throttling* as a new paradigm, bridging classical metric dimension theory with resource-constrained embedding design.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $Θleft(frac{log{n}}{log{log{n}}} ight)$, 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.

Problem

• Introducing throttling for metric dimension and its graph variants
• Proving NP-hardness of throttling number computation for all variants
• Characterizing throttling numbers for special graph families and extremal cases

Innovation

• Throttling minimizes sum of resources in graph problems
• Applies throttling to three metric dimension variants
• Computes throttling numbers for special graph families