The Parameterized Hardness of Art Gallery Problems

TL;DR

This paper establishes tight lower bounds on the computational complexity of the Art Gallery problems, showing they cannot be solved faster than certain exponential bounds unless a major complexity hypothesis fails.

Contribution

It proves parameterized hardness results for both Point and Vertex Guard Art Gallery problems, nearly matching existing algorithms' upper bounds.

Findings

No $f(k)n^{o(k / \log k)}$ algorithms unless ETH fails

Lower bounds closely match known $n^{O(k)}$ algorithms

Results apply to both point and vertex guard variants

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 Vertex Guard Art Gallery problem asks for such a set $S$ subset of the vertices of $\mathcal{P}$. A point in the set $S$ is referred to as a guard. For both variants, we rule out any $f(k)n^{o(k / \log k)}$ algorithm, where $k := |S|$ is the number of guards, for any computable function $f$, unless the Exponential Time Hypothesis fails. These lower bounds almost match the $n^{O(k)}$ algorithms that exist for both problems.