Given a simple polygon $\cal P$, in the Art Gallery problem the goal is to find the minimum number of guards needed to cover the entire $\cal P$, where a guard is a point and can see another point $q$ when $\overline{pq}$ does not cross the edges of $\cal P$. This paper studies a variant of the Art Gallery problem in which guards are restricted to lie on a dense grid inside $\cal P$. In the general problem, guards can be anywhere inside or on the boundary of $\cal P$. The general problem is called the \emph{point} guarding problem. It was proved that the point guarding problem is APX-complete, meaning that we cannot do better than a constant-factor approximation algorithm unless $P = NP$. A huge amount of research is committed to the studies of combinatorial and algorithmic aspects of this problem, and as of this time, we could not find a constant factor approximation for simple polygons. The last best-known approximation factor for point guarding a simple polygon was $\mathcal{O}(\log (|OPT|))$ introduced by E. Bonnet and T. Miltzow in 2020, where $|OPT|$ is the size of the optimal solution. Here, we propose an algorithm with a constant approximation factor for the point guarding problem where the location of guards is restricted to a grid. The running time of the proposed algorithm depends on the number of cells of the grid. The approximation factor is constant regardless of the grid we use, the running time could be super-polynomial if the grid size becomes exponential.
翻译:鉴于一个简单的多边形$\ cal P$, 在艺术美术馆的问题中, 目标是找到覆盖整个$\ cal P$所需的最起码的警卫人数。 在这种情况下, 一名警卫是一个点, 当$\ overline{ pq} 美元没有跨过美元P$的边缘时, 可以看到另一个点 $ q$ 。 本文研究艺术美术问题的变种, 在这种变种中, 警卫只能停留在$\ cal P$ 之内的密集网格上。 在一般的问题中, 警卫可以出现在任何内部或边界 $\ calP$ 的边界上。 通常的问题就是 \ emph{ ppoint} 看守问题。 事实证明, 防守点是 APX 问题完全的 APX, 这意味着除非$( overline{ pququal ) corrupical ral ral ral ral ral_ ral_ ral_ ral_ ral_ ral ral ral______ ral_ ral___ ral___________________b_____________________________________________________________________________________________________________________________________________________________________________________________________________________________________