Guarding a Polygon Without Losing Touch

TL;DR

This paper addresses the Art Gallery Problem by developing optimal and near-optimal algorithms for mobile agents to guard an unknown polygon while maintaining visibility connectivity, with applications in distributed multi-agent systems.

Contribution

It introduces a centralized $O(n)$ algorithm and two distributed algorithms for guarding polygons with visibility connectivity constraints, despite agents having limited initial knowledge.

Findings

Centralized algorithm runs in optimal $O(n)$ time.

Distributed algorithms are near-optimal despite limited perception.

Guarding ensures a connected network that fully covers the polygon.

Abstract

We study the classical Art Gallery Problem first proposed by Klee in 1973 from a mobile multi-agents perspective. Specifically, we require an optimally small number of agents (also called guards) to navigate and position themselves in the interior of an unknown simple polygon with $n$ vertices such that the collective view of all the agents covers the polygon. We consider the visibly connected setting wherein agents must remain connected through line of sight links -- a requirement particularly relevant to multi-agent systems. We first provide a centralized algorithm for the visibly connected setting that runs in time $O(n)$, which is of course optimal. We then provide algorithms for two different distributed settings. In the first setting, agents can only perceive relative proximity (i.e., can tell which of a pair of objects is closer) whereas they can perceive exact distances in the second setting. Our distributed algorithms work despite agents having no prior knowledge of the polygon. Furthermore, we provide lower bounds to show that our distributed algorithms are near-optimal. Our visibly connected guarding ensures that (i) the guards form a connected network and (ii) the polygon is fully guarded. Consequently, this guarding provides the distributed infrastructure to execute any geometric algorithm for this polygon.