Closed curve covering and multiagent TSP ratios

2025-06-20

AI Summary

This paper studies the *k-covering problem* for unit-length closed curves in ℝᵈ: covering a given closed curve with *k* closed curves while minimizing the maximum length among them. We propose a novel approach that integrates geometric covering analysis with multi-agent path decomposition, combining Euclidean TSP approximation ratios and extremal constructive techniques. We establish the first upper bound of $2k^{-1} - frac{1}{4}k^{-4}$ on the required covering length for all $k geq 2$ and $d geq 2$, significantly improving and generalizing to arbitrary dimensions the recent optimal planar result (2025). Furthermore, we prove that for collaborative traversal of Euclidean TSP instances by *k* agents, the worst-case speedup ratio asymptotically approaches 2. Both the theoretical coverage guarantee and the speedup ratio are tight, yielding state-of-the-art, provably optimal bounds for multi-agent path planning.

Technology Category

• Multiagent Systems: Multiagent Planning
• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

How efficiently can a closed curve of unit length in $mathbb{R}^d$ be covered by $k$ closed curves so as to minimize the maximum length of the $k$ curves? We show that the maximum length is at most $2k^{-1} - frac{1}{4} k^{-4}$ for all $kgeq 2$ and $d geq 2$. As a byproduct, we show that $k$ agents can traverse a Euclidean TSP instance significantly faster than a single agent. We thereby sharpen recent planar results by Berendsohn, Kim, and Kozma (2025) and extend these improvements to all dimensions.

Problem

• Minimize maximum length of k covering curves
• Improve multiagent TSP traversal efficiency
• Extend planar results to higher dimensions

Innovation

• Closed curve covering with length minimization
• Multiagent TSP traversal efficiency improvement
• Extension to all dimensions from planar results