Given a closed simple polygon $P$, we say two points $p,q$ see each other if the segment $pq$ is fully contained in $P$. The art gallery problem seeks a minimum size set $G\subset P$ of guards that sees $P$ completely. The only currently correct algorithm to solve the art gallery problem exactly uses algebraic methods and is attributed to Sharir. As the art gallery problem is ER-complete, it seems unlikely to avoid algebraic methods, without additional assumptions. In this paper, we introduce the notion of vision stability. In order to describe vision stability consider an enhanced guard that can see "around the corner" by an angle of $\delta$ or a diminished guard whose vision is by an angle of $\delta$ "blocked" by reflex vertices. A polygon $P$ has vision stability $\delta$ if the optimal number of enhanced guards to guard $P$ is the same as the optimal number of diminished guards to guard $P$. We will argue that most relevant polygons are vision stable. We describe a one-shot vision stable algorithm that computes an optimal guard set for visionstable polygons using polynomial time and solving one integer program. It guarantees to find the optimal solution for every vision stable polygon. We implemented an iterative visionstable algorithm and show its practical performance is slower, but comparable with other state of the art algorithms. Our iterative algorithm is inspired and follows closely the one-shot algorithm. It delays several steps and only computes them when deemed necessary. Given a chord $c$ of a polygon, we denote by $n(c)$ the number of vertices visible from $c$. The chord-width of a polygon is the maximum $n(c)$ over all possible chords $c$. The set of vision stable polygons admits an FPT algorithm when parametrized by the chord-width. Furthermore, the one-shot algorithm runs in FPT time, when parameterized by the number of reflex vertices.

翻译：给定一个封闭简单多边形 $P$，若线段 $pq$ 完全包含于 $P$ 中，则称两点 $p,q$ 彼此可见。艺术画廊问题寻求一个最小规模的集合 $G\subset P$（即守卫），使其能够完全监视 $P$。当前唯一能够精确求解艺术画廊问题的正确算法采用代数方法，归功于 Sharir。由于艺术画廊问题是 ER-完全的，在没有额外假设的情况下，很难避免使用代数方法。本文引入了视觉稳定性的概念。为描述视觉稳定性，考虑一种增强型守卫，其可以“绕过拐角”以 $\delta$ 角度观察；或一种削弱型守卫，其视野被反射顶点以 $\delta$ 角度“阻挡”。若多边形 $P$ 所需的最优增强型守卫数量与最优削弱型守卫数量相同，则称 $P$ 具有视觉稳定性 $\delta$。我们将论证大部分相关多边形均具有视觉稳定性。我们描述了一种一次性视觉稳定算法，该算法在多项式时间内通过求解一个整数规划，为视觉稳定多边形计算最优守卫集，并保证为每个视觉稳定多边形找到最优解。我们实现了一种迭代视觉稳定算法，并展示其实际性能虽较慢，但仍与其他前沿算法相当。该迭代算法受一次性算法启发并紧密遵循其思路，将多个步骤延迟执行，仅在必要时才计算。给定多边形的一条弦 $c$，记 $n(c)$ 为从 $c$ 可见的顶点数量。多边形的弦宽为所有可能弦 $c$ 对应的 $n(c)$ 最大值。视觉稳定多边形集合在以弦宽为参数时存在固定参数可解（FPT）算法。此外，一次性算法在以反射顶点数量为参数时亦可在 FPT 时间内运行。