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

TL;DR

The paper proves that the point-boundary variant of the art gallery problem is $ exists\mathbb{R}$-complete by constructing a geometric reduction from the problem ETR-INV-REV. It introduces a suite of geometric gadgets—guard regions, constraint nooks, copy nooks, and constraint gadgets (including inversion and addition variants)—that encode the real-valued constraints of the source problem into guard placements in a polygon, with guard positions mapped to variables in $[\tfrac{1}{2},2]$. The authors provide a hands-on, compass-and-straightedge-style construction that avoids irrational coordinates, and they show how the gadget interactions yield the necessary monotone constraints to simulate the target inequalities. The reduction runs in polynomial time and yields $ exists\mathbb{R}$-hardness for the point-boundary variant; the approach also simplifies earlier $ exists\mathbb{R}$-hardness proofs for the point-point variant and suggests broader applicability to related AGP variants. Overall, the work establishes tight complexity for natural visibility problems and strengthens connections between computational geometry and the existential theory of the reals.

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.

Figures (30)

• Figure 1: Left: A wedge ($W$) and its visibility region ($V$). Right: The intersection of the visibility regions of the three wedges shown forms a guard segment. This method of using $3$ wedges to form a guard segment is due to Bertschinger, El Maalouly, Miltzow, Schnider and Weber HomotopyUniversality and is a slight simplification of the method for forming guard segments from ExistsRHardness.
• Figure 2: Left: A nook with nook segment $AB$, opening points $E$ restricting $A$ and $F$ restricting $B$, and partial visibility region $V$. The outer parts of $V$ (lighter) can only see part of the nook segment. Right: This art gallery can be guarded by two guards, but there is no such configuration where the nook segment on the left is guarded entirely by a single guard. This creates a continuous constraint between the positions of the two guards.
• Figure 3: A schematic of the art gallery that we construct (not to scale). The variables are encoded by positions of guards on the variable segments (red, middle). The constraint gadgets on the bottom right enforce constraints like $xy\ge 1$ or $x+y\le z$ on the variables encoded by the input segments (red, bottom right). The copy nooks copy the values of the appropriate variables onto the input segments. The wedges creating the segment-shaped guard regions can be seen along the top and bottom walls of the art gallery.
• Figure 4: Four wedges create a point-shaped guard region.
• Figure 5: A copy nook, shown in the situation where the guard positions don't fully guard the nook.

Theorems & Definitions (38)

• Definition 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Theorem 2.4
• proof
• Lemma 3.1
• proof
• Lemma 4.1