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Irrational Guards are Sometimes Needed

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In this paper we study the art gallery problem, which is one of the fundamental problems in computational geometry. The objective is to place a minimum number of guards inside a simple polygon so that the guards together can see the whole polygon. We say that a guard at position x sees a point y if the line segment xy is contained in the polygon. Despite an extensive study of the art gallery problem, it remained an open question whether there are polygons given by integer coordinates that require guard positions with irrational coordinates in any optimal solution. We give a positive answer to this question by constructing a monotone polygon with integer coordinates that can be guarded by three guards only when we allow to place the guards at points with irrational coordinates. Otherwise, four guards are needed. By extending this example, we show that for every n, there is a polygon which can be guarded by 3n guards with irrational coordinates but needs 4n guards if the coordinates have to be rational. Subsequently, we show that there are rectilinear polygons given by integer coordinates that require guards with irrational coordinates in any optimal solution.
OriginalsprogEngelsk
Titel33rd International Symposium on Computational Geometry (SoCG 2017)
RedaktørerBoris Aronov, Matthew J. Katz
ForlagSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Publikationsdato2017
Sider1-15
Artikelnummer3
ISBN (Trykt)978-3-95977-038-5}
DOI
StatusUdgivet - 2017
Begivenhed33rd International Symposium on Computational Geometry - Brisbane, Australien
Varighed: 4 jul. 20177 jul. 2017
Konferencens nummer: 33

Konference

Konference33rd International Symposium on Computational Geometry
Nummer33
LandAustralien
ByBrisbane
Periode04/07/201707/07/2017
NavnLeibniz International Proceedings in Informatics (LIPIcs)
Vol/bind77

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