Art Gallery Plus Single Specular-reflection

TL;DR

This paper introduces an approximation algorithm for a variant of the Art Gallery problem where guards can see through single reflections on polygon edges, potentially reducing the number of guards needed.

Contribution

It presents an $ ext{O}( ext{log} n)$-approximation algorithm for guard placement in polygons with single specular reflections, extending classical Art Gallery problem solutions.

Findings

Reflection can reduce the number of guards needed in practice.

The algorithm converts the problem into a Set Cover problem.

Reflection allows guards to see through larger portions of the polygon.

Abstract

Given a simple polygon $\cal P$, in the Art Gallery problem, the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which the boundaries of $\cal P$ are replaced by single specular-reflection edges, allowing the view rays to reflect once per collision with an edge. This property allows the guards to see through the reflections, thereby viewing a larger portion of the polygon. For this problem, the position of the guards in $\cal P$ can be determined with our proposed $\mathcal{O}(\log n)$-approximation algorithm. Besides presenting an algorithm with the mentioned approximation factor, we will see that reflection can decrease the number of guards in practice. The proposed algorithm converts the generalized problem to the Set Cover problem.