On $r$-Guarding Thin Orthogonal Polygons

TL;DR

This paper studies the problem of guarding thin orthogonal polygons with a new approach that reduces the computational complexity from super-polynomial to linear time in the case of tree polygons, and extends to polygons with holes or thickness.

Contribution

It introduces a more efficient linear-time algorithm for r-guarding thin orthogonal polygons that are tree polygons, and generalizes the approach to polygons with holes or thickness, making it fixed-parameter tractable.

Findings

Linear-time algorithm for r-guarding thin tree polygons

NP-hardness of guarding polygons with holes even if thin

Extension to polygons with holes or thickness, fixed-parameter tractable

Abstract

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.