Sometimes Two Irrational Guards are Needed

TL;DR

This paper demonstrates that in the art gallery problem, even two guards may need to be placed at irrational coordinates to optimally cover a polygon, closing a gap in understanding when irrational guards are necessary.

Contribution

It proves that optimal solutions with two guards can require irrational coordinates, extending previous results about the necessity of irrational guards.

Findings

Two guards may need irrational coordinates for optimal coverage.

It completes the characterization of when irrational guards are necessary.

The result applies to polygons with rational coordinates.

Abstract

In the art gallery problem, we are given a closed polygon $P$, with rational coordinates and an integer $k$. We are asked whether it is possible to find a set (of guards) $G$ of size $k$ such that any point $p\in P$ is seen by a point in $G$. We say two points $p$, $q$ see each other if the line segment $pq$ is contained inside $P$. It was shown by Abrahamsen, Adamaszek, and Miltzow that there is a polygon that can be guarded with three guards, but requires four guards if the guards are required to have rational coordinates. In other words, an optimal solution of size three might need to be irrational. We show that an optimal solution of size two might need to be irrational. Note that it is well-known that any polygon that can be guarded with one guard has an optimal guard placement with rational coordinates. Hence, our work closes the gap on when irrational guards are possible to occur.