The Art Gallery Problem is $\exists \mathbb{R}$-complete
We prove that the art gallery problem is equivalent under polynomial time reductions to deciding whether a system of polynomial equations over the real numbers has a solution. The art gallery problem is a classical problem in computational geometry. Given a simple polygon $P$ and an integer $k$, the...
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Main Authors | , , |
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Format | Journal Article |
Language | English |
Published |
23.04.2017
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Subjects | |
Online Access | Get full text |
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Summary: | We prove that the art gallery problem is equivalent under polynomial time
reductions to deciding whether a system of polynomial equations over the real
numbers has a solution. The art gallery problem is a classical problem in
computational geometry. Given a simple polygon $P$ and an integer $k$, the goal
is to decide if there exists a set $G$ of $k$ guards within $P$ such that every
point $p\in P$ is seen by at least one guard $g\in G$. Each guard corresponds
to a point in the polygon $P$, and we say that a guard $g$ sees a point $p$ if
the line segment $pg$ is contained in $P$.
The art gallery problem has stimulated extensive research in geometry and in
algorithms. However, the complexity status of the art gallery problem has not
been resolved. It has long been known that the problem is $\text{NP}$-hard, but
no one has been able to show that it lies in $\text{NP}$. Recently, the
computational geometry community became more aware of the complexity class
$\exists \mathbb{R}$. The class $\exists \mathbb{R}$ consists of problems that
can be reduced in polynomial time to the problem of deciding whether a system
of polynomial equations with integer coefficients and any number of real
variables has a solution. It can be easily seen that $\text{NP}\subseteq
\exists \mathbb{R}$. We prove that the art gallery problem is $\exists
\mathbb{R}$-complete, implying that (1) any system of polynomial equations over
the real numbers can be encoded as an instance of the art gallery problem, and
(2) the art gallery problem is not in the complexity class $\text{NP}$ unless
$\text{NP}=\exists \mathbb{R}$. As a corollary of our construction, we prove
that for any real algebraic number $\alpha$ there is an instance of the art
gallery problem where one of the coordinates of the guards equals $\alpha$ in
any guard set of minimum cardinality. That rules out many geometric approaches
to the problem. |
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DOI: | 10.48550/arxiv.1704.06969 |