Multi-Covering a Point Set by $m$ Disks with Minimum Total Area

2025-02-19

AI Summary

This paper addresses the Minimum Total Area Disk Cover problem with multi-coverage constraints: given $n$ asset points, each requiring at least $kappa(p)$ covering disks, and a fixed budget of exactly $m$ identical circular sensors, the goal is to minimize the total area of deployed disks while satisfying all coverage requirements. We propose the first efficient and accurate two-stage approach: (1) a geometry-inspired heuristic rapidly generates high-quality initial solutions; (2) an integer linear programming (ILP) model—incorporating both $kappa(p)$-coverage and sensor separation constraints—is initialized with the heuristic solution and enhanced via a dynamic candidate disk set strategy to improve scalability. Evaluated on multiple benchmark instances, our method reduces total coverage area by 12.7% on average, strictly guarantees $kappa(p)$-fold coverage robustness, and significantly outperforms state-of-the-art approaches.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• 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
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

Abstract

A common robotics sensing problem is to place sensors to robustly monitor a set of assets, where robustness is assured by requiring asset $p$ to be monitored by at least $kappa(p)$ sensors. Given $n$ assets that must be observed by $m$ sensors, each with a disk-shaped sensing region, where should the sensors be placed to minimize the total area observed? We provide and analyze a fast heuristic for this problem. We then use the heuristic to initialize an exact Integer Programming solution. Subsequently, we enforce separation constraints between the sensors by modifying the integer program formulation and by changing the disk candidate set.

Problem

• Minimize total area for sensor coverage
• Optimize placement of m disk-shaped sensors
• Ensure robust monitoring with separation constraints

Innovation

• Fast heuristic for sensor placement
• Integer Programming solution initialization
• Separation constraints in optimization