Computational Complexity of the Chromatic Art Gallery Problem for Orthogonal Polygons

2020 ◽  
pp. 146-157
Author(s):  
Chuzo Iwamoto ◽  
Tatsuaki Ibusuki
Keyword(s):  
Computational Complexity ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Orthogonal Polygons

LINEAR-TIME 3-APPROXIMATION ALGORITHM FOR THE r-STAR COVERING PROBLEM

2012 ◽  
Vol 22 (02) ◽  
pp. 103-141 ◽  
Author(s):  
ANDRZEJ LINGAS ◽  
AGNIESZKA WASYLEWICZ ◽  
PAWEŁ ŻYLIŃSKI
Keyword(s):  
Approximation Algorithm ◽  
Linear Time ◽  
Covering Problem ◽  
The Novel ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition ◽  
Orthogonal Polygons ◽  
Complexity Status

The complexity status of the minimum r-star cover problem for orthogonal polygons had been open for many years, until 2004 when Ch. Worman and J. M. Keil proved it to be polynomially tractable (Polygon decomposition and the orthogonal art gallery problem, IJCGA 17(2) (2007), 105-138). However, since their algorithm has Õ(n17)-time complexity, where Õ(·) hides a polylogarithmic factor, and thus it is not practical, in this paper we present a linear-time 3-approximation algorithm. Our approach is based upon the novel partition of an orthogonal polygon into so-called o-star-shaped orthogonal polygons.

An Exact Algorithm for the Discrete Chromatic Art Gallery Problem

2014 ◽  
pp. 59-73
Author(s):  
Maurício J. O. Zambon ◽  
Pedro J. de Rezende ◽  
Cid C. de Souza
Keyword(s):  
Exact Algorithm ◽  
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

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

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.

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