Guarding Polygons with Holes

TL;DR

This paper addresses the problem of deploying multiple mobile agents to guard polygons with holes, providing efficient algorithms under different computational models to ensure complete visibility and connectivity.

Contribution

It introduces deterministic algorithms for cooperative guarding of polygons with holes, optimizing for rounds, agent count, and memory under varying computational capabilities.

Findings

Deterministic $O(n)$ round algorithm in the stronger model.

Algorithm uses at most $(n+h)/2$ agents with $O( ext{log} n)$ memory.

Weaker model algorithm runs in $O(n^4)$ rounds with at most $(n+2h)/2$ agents.

Abstract

We study the Cooperative Guarding problem for polygons with holes in a mobile multi-agents setting. Given a set of agents, initially deployed at a point in a polygon with $n$ vertices and $h$ holes, we require the agents to collaboratively explore and position themselves in such a way that every point in the polygon is visible to at least one agent and that the set of agents are visibly connected. We study the problem under two models of computation, one in which the agents can compute exact distances and angles between two points in its visibility, and one in which agents can only compare distances and angles. In the stronger model, we provide a deterministic $O(n)$ round algorithm to compute such a cooperative guard set while not requiring more than $\frac{n + h}{2}$ agents and $O(\log n)$ bits of persistent memory per agent. In the weaker model, we provide an $O(n^4)$ round algorithm, that does not require more than $\frac{n+2h}{2}$ agents.