Closed curve covering and multiagent TSP ratios

TL;DR

This paper investigates the optimal coverage of closed curves by multiple agents, providing bounds on coverage efficiency, extending multiagent TSP results across dimensions, and offering a fast approximation algorithm for curve covering.

Contribution

It establishes new bounds on covering closed curves with multiple agents, extends multiagent TSP efficiency results to higher dimensions, and introduces a linear time approximation algorithm for curve covering.

Findings

Maximum coverage length bound: 2k^{-1} - 1/4 k^{-4} for all k≥2 and d≥2.

Multiagent TSP traversal speedup over single agent in Euclidean space.

Linear time approximation algorithm with ratio 2 - 1/4 k^{-3} for covering closed polygonal curves.

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 $k\geq 2$ and $d \geq 2$. As a first 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. As a second byproduct, we obtain a linear time approximation algorithm with ratio $2 - \frac{1}{4} k^{-3}$ for covering any closed polygonal curve in $\mathbb{R}^d$ by $k$ closed curves so that the maximum length of an individual curve is minimized.