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A Practical Algorithm with Performance Guarantees for the Art~Gallery Problem. (arXiv:2007.06920v1 [cs.CG])

[Submitted on 14 Jul 2020] Download PDF Abstract: Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $Gsubset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve…

[Submitted on 14 Jul 2020]

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Abstract: Given a closed simple polygon $P$, we say two points $p,q$ see each other if
the segment $pq$ is fully contained in $P$. The art gallery problem seeks a
minimum size set $Gsubset P$ of guards that sees $P$ completely. The only
currently correct algorithm to solve the art gallery problem exactly uses
algebraic methods and is attributed to Sharir. As the art gallery problem is
ER-complete, it seems unlikely to avoid algebraic methods, 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 $delta$ or a diminished guard whose
vision is by an angle of $delta$ “blocked” by reflex vertices. A polygon $P$
has vision stability $delta$ 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
visionstable 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 visionstable algorithm and show its practical
performance is slower, but comparable with other state of the art algorithms.
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-width of a polygon is the maximum $n(c)$ over all possible
chords $c$. The set of vision stable polygons admits an FPT algorithm when
parametrized by the chord-width. Furthermore, the one-shot algorithm runs in
FPT time, when parameterized by the number of reflex vertices.

Submission history

From: Tillmann Miltzow [view email]

[v1]
Tue, 14 Jul 2020 09: 09: 22 UTC (1,627 KB)

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