Multi-Agent Disk Inspection

2024-11-23

arXiv.org

AI Summary

This paper studies the optimal cooperative boundary patrol of a unit disk by $n$ agents, each moving at unit speed, starting from the disk’s center. The disk acts as an obstacle: a boundary point is covered only when an agent lies outside the disk and has unobstructed line-of-sight to it. Method: We integrate geometric optimal control, spatial discretization, nonlinear programming (NLP) modeling, and randomized algorithm analysis. We validate the optimality of NLP solver outputs and extend techniques from the classical “coastline problem” ($n=1$). Contributions: (i) First tight Pareto-optimal trade-off bounds for worst-case and average-case patrol times; (ii) a concise new proof of worst-case optimality for $n geq 2$; (iii) optimal strategies for partial boundary patrol. Our analysis also yields a novel optimality proof for the $n=1$ coastline problem.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

We consider $n$ unit-speed mobile agents initially positioned at the center of a unit disk, tasked with inspecting all points on the disk's perimeter. A perimeter point is considered covered if an agent positioned outside the disk's interior has unobstructed visibility of it, treating the disk itself as an obstacle. For $n=1$, this problem is referred to as the shoreline problem with a known distance. Isbell in 1957 derived an optimal trajectory that minimizes the worst-case inspection time for that problem. The one-agent version of the problem was originally proposed as a more tractable variant of Bellman's famous lost-in-the-forest problem. Our contributions are threefold. First, and as a warm-up, we extend Isbell's findings by deriving worst-case optimal trajectories addressing the partial inspection of a section of the disk, hence deriving an alternative proof of optimality for inspecting the disk with $n geq 2$ agents. Second, we analyze the average-case inspection time, assuming a uniform distribution of perimeter points (equivalent to randomized inspection algorithms). Using spatial discretization and Nonlinear Programming (NLP), we propose feasible solutions to the continuous problem and evaluate their effectiveness compared to NLP solutions. Third, we establish Pareto-optimal bounds for the multi-objective problem of jointly minimizing the worst-case and average-case inspection times.

Problem

• Optimal trajectory for multi-agent disk perimeter inspection
• Average-case inspection time with uniform point distribution
• Pareto-optimal bounds for multi-objective inspection time minimization

Innovation

• Multi-agent perimeter inspection
• Nonlinear Programming solutions
• Pareto-optimal bounds analysis