scholarly journals Art Gallery Problem with Rook and Queen Vision

2021 ◽  
Vol 37 (2) ◽  
pp. 621-642
Author(s):  
Hannah Alpert ◽  
Érika Roldán
Keyword(s):  
Np Hard ◽  
Art Gallery ◽  
Bipartite Matching ◽  
Art Gallery Problem ◽  
Square Grid ◽  
Minimum Number

AbstractHow many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that $$\lfloor {\frac{n}{2}} \rfloor $$ ⌊ n 2 ⌋ rooks or $$\lfloor {\frac{n}{3}} \rfloor $$ ⌊ n 3 ⌋ queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard a polyomino is NP-hard. These results also apply to d-dimensional rooks and queens on d-dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.

OPTIMUM GUARD COVERS AND m-WATCHMEN ROUTES FOR RESTRICTED POLYGONS

1993 ◽  
Vol 03 (01) ◽  
pp. 85-105 ◽  
Author(s):  
SVANTE CARLSSON ◽  
BENGT J. NILSSON ◽  
SIMEON NTAFOS
Keyword(s):  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Np Hard ◽  
Art Gallery ◽  
Art Galleries ◽  
Art Gallery Problem ◽  
Natural Connection ◽  
Route Problem ◽  
Minimum Number

A watchman, in the terminology of art galleries, is a mobile guard. We consider several watchman and guard problems for different classes of polygons. We introduce the notion of vision spans along a path or route which provide a natural connection between the art gallery problem, the m-watchmen routes problem and the watchman route problem. We prove that finding the minimum number of vision points, i.e., static guards, along a shortest watchman route is NP-hard. We provide a linear time algorithm to compute the best set of static guards in a histogram polygon. The m-watchmen routes problem, minimize total length of routes for m watchmen, is NP-hard for simple polygons. We give a Θ(n3+n2m2)-time algorithm to compute the best set of m watchmen in a histogram.

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

scholarly journals Optimal Art Gallery Localization is NP-hard

2020 ◽  
Vol 88 ◽  
pp. 101607 ◽  
Author(s):  
Prosenjit Bose ◽  
Jean-Lou De Carufel ◽  
Alina Shaikhet ◽  
Michiel Smid
Keyword(s):  
Np Hard ◽  
Art Gallery

OPTIMAL BINARY SPACE PARTITIONS FOR SEGMENTS IN THE PLANE

2012 ◽  
Vol 22 (03) ◽  
pp. 187-205 ◽  
Author(s):  
MARK DE BERG ◽  
AMIRALI KHOSRAVI
Keyword(s):  
Recursive Partitioning ◽  
Np Hard ◽  
Line Segments ◽  
Minimum Number ◽  
Disjoint Line

An optimal BSP for a set S of disjoint line segments in the plane is a BSP for S that produces the minimum number of cuts. We study optimal BSPs for three classes of BSPs, which differ in the splitting lines that can be used when partitioning a set of fragments in the recursive partitioning process: free BSPs can use any splitting line, restricted BSPs can only use splitting lines through pairs of fragment endpoints, and auto-partitions can only use splitting lines containing a fragment. We obtain the following two results: • It is NP-hard to decide whether a given set of segments admits an auto-partition that does not make any cuts. • An optimal restricted BSP makes at most 2 times as many cuts as an optimal free BSP for the same set of segments.

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