  We resolve the complexity of the point-boundary variant of the art gallery
problem, showing that it is $\exists\mathbb{R}$-complete, meaning that it is
equivalent under polynomial time reductions to deciding whether a system of
polynomial equations has a real solution.
  Introduced by Victor Klee in 1973, the art gallery problem concerns finding
configurations of \emph{guards} which together can see every point inside of an
\emph{art gallery} shaped like a polygon. The original version of this problem
has previously been shown to $\exists\mathbb{R}$-hard, but until now the
complexity of the variant where guards only need to guard the walls of the art
gallery was an open problem.
  Our results can also be used to provide a simpler proof of the
$\exists\mathbb{R}$-hardness of the point-point art gallery problem. In
particular, we show how the algebraic constraints describing a polynomial
system of equations can occur somewhat naturally in an art gallery setting.
