Parameterized Analysis of Art Gallery and Terrain Guarding

Agrawal, Akanksha; Zehavi, Meirav

doi:10.1007/978-3-030-50026-9_2

Cited by 6 publications

(8 citation statements)

References 47 publications

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“…Note that the guards can still be placed anywhere in the polygon. Following the language of Agrawal and Zehavi [7], we call this problem version the Point-Vertex Art Gallery. This problem is contained in NP, for a short proof see Appendix B.…”

Section: Resultsmentioning

confidence: 99%

“…Surprisingly, the art gallery problem is probably not contained in NP, as it is ∃R-complete [3]. In recent years, the art gallery problem has been studied also from various other theoretical perspectives: approximation [12,19], smoothed analysis [18,22], and parameterized complexity [6][7][8]13]. From a practical perspective, various research groups developed, implemented, and tested algorithms that are exact, reliable, and relatively fast, but theoretically may run forever [14,24,27,44].…”

Section: Art Gallery Problemmentioning

confidence: 99%

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Topological Art in Simple Galleries

Bertschinger¹,

El²,

Miltzow³

et al. 2022

Symposium on Simplicity in Algorithms (SOSA)

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Let P be a simple polygon, then the art gallery problem is looking for a minimum set of points (guards) that can see every point in P. We say two points a, b ∈ P can see each other if the line segment seg(a, b) is contained in P. We denote by V (P) the family of all minimum guard placements. The Hausdorff distance makes V (P) a metric space and thus a topological space. We show homotopy-universality, that is, for every semi-algebraic set S there is a polygon P such that V (P) is homotopy equivalent to S. Furthermore, for various concrete topological spaces T , we describe instances I of the art gallery problem such that V (I ) is homeomorphic to T .

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Section: Resultsmentioning

confidence: 99%

Section: Art Gallery Problemmentioning

confidence: 99%

Topological Art in Simple Galleries

Bertschinger¹,

El²,

Miltzow³

et al. 2022

Symposium on Simplicity in Algorithms (SOSA)

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show abstract

“…In fact, visual sensors’ automatic placement is a particular case of solving the well-known art gallery problem (AGP) in three-dimensional space with a given set of constraints [ 1 ]. It can be stated as follows.…”

Section: Introductionmentioning

confidence: 99%

Automatic Placement of Visual Sensors in a Smart Space to Ensure Required PPM Level in Specified Regions of Interest

Khutornoi

Kobyzhev

Vatamaniuk

2022

Sensors

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This work is devoted to a cost-effective method for the automatic placement of visual sensors within a smart room to ensure the requirements for its design. Various unique conditions make the process of manually placing sensors time consuming and can also lead to a decrease in system efficiency. To automate the design process, we solve a multi-objective optimization problem known as the art gallery problem in 3D, modified as follows. For the specified regions of interest within a smart room, the required pixels per meter level (PPM) should be ensured. The optimization criteria are visibility of the room and the cost of equipment. To meet these criteria, we describe a room model with doors, windows, and obstacles represented in such a way as to consider their impact on the field of view of the sensors. To model sensor placement, a genetic algorithm is used. The optimal solution is selected from the Pareto front by means of the technique for order of preference by similarity to ideal solution (TOPSIS). The developed method’s effectiveness has been tested on modeling real premises of various types. The method is flexible because of the assignment of weights to certain aspects when placing sensors. Further, it can be scalable to other types of sensors.

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“…A third example is the number of reflex vertices of a polygon. This parameter gave raise to some FPT algorithms for the art gallery problem [1].…”

Section: Introductionmentioning

confidence: 99%

Local Complexity of Polygons

Klute,

Reddy,

Miltzow

2021

Preprint

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Many problems in Discrete and Computational Geometry deal with simple polygons or polygonal regions. Many algorithms and data-structures perform considerably faster, if the underlying polygonal region has low local complexity. One obstacle to make this intuition rigorous, is the lack of a formal definition of local complexity. Here, we give two possible definitions and show how they are related in a combinatorial sense. We say that a polygon P has point visibility width w = pvw , if there is no point q ∈ P that sees more than w reflex vertices. We say that a polygon P has chord visibility width w = cvw , if there is no chord c = seg(a, b) ⊂ P that sees more than w reflex vertices. We show that cvw ≤ pvw O( pvw ) , for any simple polygon. Furthermore, we show that there exists a simple polygon with cvw ≥ 2 Ω( pvw ) . * Supported by the Netherlands Organisation for Scientific Research (NWO) under project no. 612.001.651.† Supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681.‡ The second author's full last name consists of two words and is Mallik Reddy. However, she consistently refers to herself with the first word of her last name being abbreviated.§ Supported by the NWO Veni project EAGER.

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Parameterized Analysis of Art Gallery and Terrain Guarding

Cited by 6 publications

References 47 publications

Topological Art in Simple Galleries

Topological Art in Simple Galleries

Automatic Placement of Visual Sensors in a Smart Space to Ensure Required PPM Level in Specified Regions of Interest

Local Complexity of Polygons

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