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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2021.44
URN: urn:nbn:de:0030-drops-138433
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/13843/
Hengeveld, Simon B. ;
Miltzow, Tillmann
A Practical Algorithm with Performance Guarantees for the Art Gallery Problem
Abstract
Given a closed simple polygon P, we say two points p,q see each other if the segment seg(p,q) is fully contained in P. The art gallery problem seeks a minimum size set G ⊂ P of guards that sees P completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods. As the art gallery problem is ∃ ℝ-complete, it seems unlikely to avoid algebraic methods, for any exact algorithm, without additional assumptions.
In this paper, we introduce the notion of vision-stability. In order to describe vision-stability consider an enhanced guard that can see "around the corner" by an angle of δ or a diminished guard whose vision is by an angle of δ "blocked" by reflex vertices. A polygon P has vision-stability δ if the optimal number of enhanced guards to guard P is the same as the optimal number of diminished guards to guard P. We will argue that most relevant polygons are vision-stable. We describe a one-shot vision-stable algorithm that computes an optimal guard set for vision-stable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision-stable polygon. We implemented an iterative vision-stable algorithm and show its practical performance is slower, but comparable with other state-of-the-art algorithms. The practical implementation can be found at: https://github.com/simonheng/AGPIterative. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord c of a polygon, we denote by n(c) the number of vertices visible from c. The chord-visibility width (cw(P)) of a polygon is the maximum n(c) over all possible chords c. The set of vision-stable polygons admit an FPT algorithm when parameterized by the chord-visibility width. Furthermore, the one-shot algorithm runs in FPT time when parameterized by the number of reflex vertices.
BibTeX - Entry
@InProceedings{hengeveld_et_al:LIPIcs.SoCG.2021.44,
author = {Hengeveld, Simon B. and Miltzow, Tillmann},
title = {{A Practical Algorithm with Performance Guarantees for the Art Gallery Problem}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {44:1--44:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-184-9},
ISSN = {1868-8969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13843},
URN = {urn:nbn:de:0030-drops-138433},
doi = {10.4230/LIPIcs.SoCG.2021.44},
annote = {Keywords: Art Gallery, Parametrized complexity, Integer Programming, Visibility}
}
