Art Gallery Localization (AGL) is the problem of placing a set T of broadcast towers in a simple polygon P in order for a point to locate itself in the interior. For any point $p\in P$: for each tower $t\in T\cap V(p)$ (where $V(p)$ denotes the visibility polygon of p) the point p receives the coordinates of t and the Euclidean distance between t and p. From this information p can determine its coordinates. We study the computational complexity of AGL problem. We show that the problem of determining the minimum number of broadcast towers that can localize a point anywhere in a simple polygon P is NP-hard. We show a reduction from Boolean Three Satisfiability problem to our problem and give a proof that the reduction takes polynomial time.

画廊本地化（AGL）是将广播塔的集合T放置在一个简单的多边形P中的问题，目的是使一个点将其自身定位在内部。对于任何一点$p\in P$：对于每个塔 $Ť\in Ť\cap V（p）$ （哪里 $V（p）$表示能见度多边形的p）的点p接收的坐标吨和之间的欧几里德距离吨和p。根据该信息，p可以确定其坐标。我们研究了AGL问题的计算复杂性。我们表明，确定可以将点定位在简单多边形P中任何位置的广播塔的最小数量的问题是NP-难的。我们展示了从布尔三可满足性问题到我们的问题的减少，并证明了减少需要多项式时间。