Optimally Guarding Perimeters and Regions with Mobile Range Sensors

TL;DR

This paper studies the optimal deployment of mobile sensors with range capabilities to guard arbitrary perimeters and regions, providing complexity results and approximation algorithms, including an FPTAS for certain cases.

Contribution

It introduces new approximation algorithms and complexity results for guarding problems with mobile sensors, including an FPTAS for perimeter guarding with single-segment sensors.

Findings

NP-hardness of computing 1.152-approximate solutions

Development of a (2+ε)-approximation algorithm

Effective ILP-based near-optimal solutions

Abstract

We investigate the problem of using mobile robots equipped with 2D range sensors to optimally guard perimeters or regions, i.e., 1D or 2D sets. Given such a set of arbitrary shape to be guarded, and $k$ mobile sensors where the $i$-th sensor can guard a circular region with a variable radius $r_i$, we seek the optimal strategy to deploy the $k$ sensors to fully cover the set such that $\max r_i$ is minimized. On the side of computational complexity, we show that computing a $1.152$-optimal solution for guarding a perimeter or a region is NP-hard, i.e., the problem is hard to approximate. The hardness result on perimeter guarding holds when each sensor may guard at most two disjoint perimeter segments. On the side of computational methods, for the guarding perimeters, we develop a fully polynomial time approximation scheme (FPTAS) for the special setting where each sensor may only guard a single continuous perimeter segment, suggesting that the aforementioned hard-to-approximate result on the two-disjoint-segment sensing model is tight. For the general problem, we first describe a polynomial-time (2+$\epsilon)$-approximation algorithm as an upper bound, applicable to both perimeter guarding and region guarding. This is followed by a high-performance integer linear programming (ILP) based method that computes near-optimal solutions. Thorough computational benchmarks as well as evaluation on potential application scenarios demonstrate the effectiveness of these algorithmic solutions.