The Point-Boundary Art Gallery Problem is $\exists\mathbb{R}$-hard

TL;DR

This paper proves that the point-boundary art gallery problem is computationally as hard as solving systems of polynomial equations, establishing its $orall ext{R}$-completeness and simplifying previous complexity proofs.

Contribution

It establishes the $orall ext{R}$-completeness of the point-boundary art gallery problem and introduces simplified, hand-constructible gadgets for the proof.

Findings

The problem is $orall ext{R}$-complete.

Simplified gadgets can be constructed by hand.

Proof techniques can be applied to related problems.

Abstract

We resolve the complexity of the point-boundary variant of the art gallery problem, showing that it is $\exists\mathbb{R}$-complete, meaning that it is equivalent under polynomial time reductions to deciding whether a system of polynomial equations has a real solution. The art gallery problem asks whether there is a configuration of {\it guards} that together can see every point inside of an {\it art gallery} modeled by a simple polygon. The original version of this problem (which we call the point-point variant) was shown to be $\exists\mathbb{R}$-hard [Abrahamsen, Adamaszek, and Miltzow, JACM 2021], but the complexity of the variant where guards only need to guard the walls of the art gallery was left as an open problem. We show that this variant is also $\exists\mathbb{R}$-hard. Our techniques can also be used to greatly simplify the proof of $\exists\mathbb{R}$-hardness of the point-point art gallery problem. The gadgets in previous work could only be constructed by using a computer to find complicated rational coordinates with specific algebraic properties. All of our gadgets can be constructed by hand and can be verified with simple geometric arguments.