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A Practical Algorithm with Performance Guarantees for the Art Gallery Problem

Authors Simon B. Hengeveld , Tillmann Miltzow



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Simon B. Hengeveld
  • Université Rennes 1, France
Tillmann Miltzow
  • Utrecht University, The Netherlands

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Simon B. Hengeveld and Tillmann Miltzow. A Practical Algorithm with Performance Guarantees for the Art Gallery Problem. In 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 189, pp. 44:1-44:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.SoCG.2021.44

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Art Gallery
  • Parametrized complexity
  • Integer Programming
  • Visibility

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