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.

翻译：我们解决了艺术画廊问题的点界变体的复杂性, 显示它是 $\ exptions\ mathb{R}$- complete, 意思是, 在多元时间缩减下, 它等同于决定一个多面方程式系统是否有一个真正的解决方案。 1973 年由 Victor Klee 引入, 艺术画廊问题涉及寻找\ emph{ guards} 的配置, 这些配置可以一起看到像多边形的 \ emph{ art 画廊中的每一个点。 这个问题的原始版本以前被显示为$\ exptions\ mathb{R} $- hard, 但直到现在, 变体的复杂性, 即仅需要守卫艺术画廊墙的警卫是一个开放的问题。 我们的结果也可以用来提供更简单的证据, 证明点画廊问题的 $\ dephenctions\ mathb{R} $- hard 。 特别是, 我们展示了描述多面方程式系的代数形限制是如何在艺术画廊设置上发生的。