An Approximation Algorithm for the Art Gallery Problem

TL;DR

This paper introduces the first logarithmic approximation algorithm for the point guard art gallery problem in simple polygons, improving previous approaches and identifying an error in related work.

Contribution

It presents a novel $O( ext{log OPT})$-approximation algorithm for the problem, combining existing ideas and correcting prior inaccuracies.

Findings

First $O( ext{log OPT})$-approximation algorithm for simple polygons.

Identifies and corrects a mistake in previous related work.

Provides theoretical bounds for guard placement in polygons.

Abstract

Given a simple polygon $\mathcal{P}$ on $n$ vertices, two points $x,y$ in $\mathcal{P}$ are said to be visible to each other if the line segment between $x$ and $y$ is contained in $\mathcal{P}$. The Point Guard Art Gallery problem asks for a minimum set $S$ such that every point in $\mathcal{P}$ is visible from a point in $S$. The set $S$ is referred to as guards. Assuming integer coordinates and a specific general position assumption, we present the first $O(\log \text{OPT})$-approximation algorithm for the point guard problem for simple polygons. This algorithm combines ideas of a paper of Efrat and Har-Peled [Inf. Process. Lett. 2006] and Deshpande et. al. [WADS 2007]. We also point out a mistake in the latter.