The Art Gallery Problem

2006 ◽  
pp. 8-10
Author(s):  
Alan Tucker
Keyword(s):  
Art Gallery ◽  
Art Gallery Problem

Tight bounds for the rectangular art gallery problem

1992 ◽  
pp. 105-112 ◽  
Author(s):  
J. Czyzowicz ◽  
E. Rivera-Campo ◽  
N. Santoro ◽  
J. Urrutia ◽  
J. Zaks
Keyword(s):  
Art Gallery ◽  
Art Gallery Problem

POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL
Keyword(s):  
Polynomial Time ◽  
Art Gallery ◽  
Minimum Cardinality ◽  
Decomposition Problem ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition

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.

The Art Gallery Problem is ∃ℝ-complete

2022 ◽  
Vol 69 (1) ◽  
pp. 1-70
Author(s):  
Mikkel Abrahamsen ◽  
Anna Adamaszek ◽  
Tillmann Miltzow
Keyword(s):  
Simple Polygon ◽  
Art Gallery ◽  
Arbitrary Degree ◽  
Minimum Cardinality ◽  
Art Gallery Problem ◽  
Algebraic Set ◽  
Classic Problem ◽  
Finite Set ◽  
Roots Of Polynomials ◽  
System Of Polynomial Equations

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.

scholarly journals Proactive Deployment of Aerial Drones for Coverage over Very Uneven Terrains: A Version of the 3D Art Gallery Problem

Sensors ◽  
2019 ◽  
Vol 19 (6) ◽  
pp. 1438 ◽  
Author(s):  
Andrey Savkin ◽  
Hailong Huang
Keyword(s):  
Computer Simulations ◽  
Simple Algorithm ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Uneven Terrain ◽  
Upper Estimate

The paper focuses on surveillance and monitoring using aerial drones. The aim is to estimate the minimal number of drones necessary to monitor a given area of a very uneven terrain. The proposed problem may be viewed as a drone version of the 3D Art Gallery Problem. A computationally simple algorithm to calculate an upper estimate of the minimal number of drones together with their locations is developed. Computer simulations are conducted to demonstrate the effectiveness of the proposed method.

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