A Note on Edge Guards in Art Galleries

TL;DR

This paper introduces a heuristic method for placing edge guards on polygon walls to cover inward sides, using at most N/4 guards, offering insights into the broader Art Gallery Problem.

Contribution

It proposes a heuristic algorithm for a weaker version of the Art Gallery Problem focusing on inward wall coverage with at most N/4 guards.

Findings

Heuristic algorithm effectively arranges edge guards on polygon walls.

Supports the idea that solving the weaker problem aids in understanding the full problem.

Provides bounds on the number of guards needed for inward coverage.

Abstract

We examine the Art Gallery Problem with Edge Guards. We present a heuristic algorithm to arrange edge guards to guard only the inward side of the walls of any N-vertex simple polygonal gallery using at most roof (N/4) edge guards - a weakened version of Toussaint's conjecture on the number of edge guards that can guard an entire simple polygon. Our study indicates that solving this weaker problem could give a handle on the full problem.