The Art Gallery Problem is ∃ℝ-complete

2022 ◽  
Vol 69 (1) ◽  
pp. 1-70
Author(s):  
Mikkel Abrahamsen ◽  
Anna Adamaszek ◽  
Tillmann Miltzow

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL

A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.


1999 ◽  
Vol 09 (01) ◽  
pp. 63-79
Author(s):  
XUEHOU TAN

We study the art gallery problem restricted to edge guards and straight walkable polygons. An edge guard is the guard that patrols individual edges of the polygon. A simple polygon P is called straight walkable if there are two vertices s and t in P and we can move two points montonically on two polygonal chains of P from s to t, one clockwise and the other counterclockwise, such that two points are always mutually visible. For instance, monotone polygons and spiral polygons are straight walkable. We show that ⌊(n+2)/5⌋ edge guards are always sufficient to watch and n-vertex gallery of this type. Furthermore, we also show that if the given polygon is straight walkable and rectilinear, then ⌊(n+3)/6⌋ edge guards are sufficient. Both of our upper bounds match the known lower bounds.


Author(s):  
J. Czyzowicz ◽  
E. Rivera-Campo ◽  
N. Santoro ◽  
J. Urrutia ◽  
J. Zaks

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