Realizing Metric Spaces with Convex Obstacles

2025-09-18

AI Summary

This paper investigates how geometric properties of obstacles—specifically convexity, disjointness, and fatness—affect the geodesic metric induced in three- and higher-dimensional Euclidean spaces, asking whether arbitrary finite metric spaces can be $(1+varepsilon)$-isometrically embedded into such obstacle environments. Method: We combine geodesic distance analysis, doubling-space TSP approximation techniques, and structural characterization of obstacle-induced metrics. Contribution/Results: We prove that a set of congruent convex triangles—without requiring fatness or strict separation—suffices to approximate any finite metric space arbitrarily closely, showing fatness is unnecessary for metric realizability. Furthermore, when obstacles are simultaneously convex, fat, and pairwise disjoint, Euclidean TSP admits a PTAS; however, dropping any one of these properties renders the problem APX-hard. Our work establishes fundamental trade-offs between geometric constraints, metric realizability, and computational tractability.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic 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

The presence of obstacles has a major impact on distance computation, motion planning, and visibility. While these problems are well studied in the plane, our understanding in three and higher dimensions is still limited. We investigate how different obstacle properties affect the induced geodesic metric in three-dimensional space. A finite metric space is said to be approximately realizable by a collection of obstacles if, for any $varepsilon>0$, it can be embedded into the free space around the obstacles with geodesic distance and worst-case distortion $1+varepsilon$. We focus on three key properties-convexity, disjointness, and fatness-and analyze how omitting each of them influences realizability. Our main result shows that if fatness is dropped, then every finite metric space can be realized with distortion $1+varepsilon$ using convex, pairwise disjoint obstacles in $mathbb{R}^3$, even if all obstacles are congruent equilateral triangles. Moreover, if we enforce fatness but drop convexity or disjointness, the same realizability still holds. Our results have important implications on the approximability of TSP with Obstacles, a natural variant of TSP introduced recently by Alkema et al. (ESA 2022). Specifically, we use the recent results of Banerjee et al. on TSP in doubling spaces (FOCS 2024) and of Chew et al. on distances among obstacles (Information Processing Letters 2002) to show that TSP with Obstacles admits a PTAS if the obstacles are convex, fat, and pairwise disjoint. If any of these three properties is dropped, then our results, combined with the APX-hardness of Metric TSP, demonstrate that TSP with Obstacles is APX-hard.

Problem

• Investigating obstacle properties' impact on geodesic metrics
• Analyzing realizability of finite metric spaces with obstacles
• Determining conditions for TSP with Obstacles approximability

Innovation

• Convex disjoint triangle obstacles realize metrics
• Non-fat convex obstacles enable metric embedding
• Fat non-convex obstacles allow geodesic realization