Dispersive Vertex Guarding for Simple and Non-Simple Polygons

TL;DR

This paper investigates the computational complexity of dispersive vertex guarding in polygons, proving NP-completeness for polygons with holes at a certain dispersion distance and providing an optimal guarding algorithm for simple polygons.

Contribution

It establishes NP-completeness for dispersive guarding in polygons with holes and presents an optimal guard placement algorithm for simple polygons.

Findings

NP-complete to decide dispersive guarding in polygons with holes at distance 2

Polynomial algorithm for placing guards in simple polygons with dispersion at least 2

Tight bound showing simple polygons cannot have dispersion greater than 2 with vertex guards

Abstract

We study the Dispersive Art Gallery Problem with vertex guards: Given a polygon $\mathcal{P}$, with pairwise geodesic Euclidean vertex distance of at least $1$, and a rational number $\ell$; decide whether there is a set of vertex guards such that $\mathcal{P}$ is guarded, and the minimum geodesic Euclidean distance between any two guards (the so-called dispersion distance) is at least $\ell$. We show that it is NP-complete to decide whether a polygon with holes has a set of vertex guards with dispersion distance $2$. On the other hand, we provide an algorithm that places vertex guards in simple polygons at dispersion distance at least $2$. This result is tight, as there are simple polygons in which any vertex guard set has a dispersion distance of at most $2$.