Tracking Mobile Intruders in an Art Gallery: Guard Deployment Strategies, Fundamental Limitations, and Performance Guarantees

TL;DR

This paper investigates guard deployment strategies for tracking mobile intruders in polygonal environments, formulates the problem as a multi-robot task allocation challenge, and proposes algorithms with performance guarantees, addressing both simple and complex polygons.

Contribution

It introduces a novel approach to guard deployment and tracking in polygonal environments, including NP-hardness analysis, partition-based algorithms, and extensions to multiple intruders and non-simple polygons.

Findings

NP-hardness of minimum guard speed problem

Partition-based guard allocation algorithm

Effective strategies for multiple intruders

Abstract

This paper addresses the problem of tracking mobile intruders in a polygonal environment. We assume that a team of diagonal guards is deployed inside the polygon to provide mobile coverage. First, we formulate the problem of tracking a mobile intruder inside a polygonal environment as a multi-robot task allocation (MRTA) problem. Leveraging on guard deployment strategies in art gallery problems for mobile coverage, we show that the problem of finding the minimum speed of guards to persistently track a single mobile intruder is NP-hard. Next, for a given maximum speed of the intruder and the guards, we propose a technique to partition a polygon, and compute a feasible allocation of guards to the partitions. We prove the correctness of the proposed algorithm, and show its completeness for a specific class of inputs. We classify the guards based on the structural properties of the partitions allocated to them. Based on the classification, we propose motion strategy for the guards to track the mobile intruder when it is located in the partition allocated to the guard. Finally, we extend the proposed technique to address guard deployment and allocation strategies for non-simple polygons and multiple intruders.