Algorithms for art gallery illumination

Maximilian Ernestus (), Stephan Friedrichs (), Michael Hemmer (), Jan Kokemüller (), Alexander Kröller (), Mahdi Moeini () and Christiane Schmidt ()
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Maximilian Ernestus: TU Braunschweig, IBR, Algorithms Group
Stephan Friedrichs: Max Planck Institute for Informatics
Michael Hemmer: TU Braunschweig, IBR, Algorithms Group
Jan Kokemüller: TU Braunschweig, IBR, Algorithms Group
Alexander Kröller: TU Braunschweig, IBR, Algorithms Group
Mahdi Moeini: Technical University of Kaiserslautern
Christiane Schmidt: Linköping University

Journal of Global Optimization, 2017, vol. 68, issue 1, No 2, 23-45

Abstract: Abstract The art gallery problem (AGP) is one of the classical problems in computational geometry. It asks for the minimum number of guards required to achieve visibility coverage of a given polygon. The AGP is well-known to be NP-hard even in restricted cases. In this paper, we consider the AGP with fading (AGPF): A polygonal region is to be illuminated with light sources such that every point is illuminated with at least a global threshold, light intensity decreases over distance, and we seek to minimize the total energy consumption. Choosing fading exponents of zero, one, and two are equivalent to the AGP, laser scanner applications, and natural light, respectively. We present complexity results as well as a negative solvability result. Still, we propose two practical algorithms for AGPF with fixed light positions (e.g. vertex guards) independent of the fading exponent, which we demonstrate to work well in practice. One is based on a discrete approximation, the other on non-linear programming by means of simplex-partitioning strategies. The former approach yields a fully polynomial-time approximation scheme for the AGPF with fixed light positions. The latter approach obtains better results in our experimental evaluation.

Keywords: Art gallery problem; Fading; Computational geometry; Linear program; Non-linear program; Lipschitz function; Algorithm engineering; 65D18; 68W25; 68W40; 68U05; 90C05; 90C30; 90C90 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10898-016-0452-2

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