一种针对美术馆问题的具有性能保证的实用算法
A Practical Algorithm with Performance Guarantees for the Art Gallery Problem
AI总结：
本文针对美术馆问题提出视觉稳定性概念，并设计了一次性和迭代式算法，在视觉稳定多边形上通过整数规划保证求得最优守卫集合，同时给出以弦宽和反射顶点数为参数的FPT运行时间保证。
摘要：
给定一个封闭的简单多边形 $P$，如果线段 $pq$ 完全包含在 $P$ 中，则称两个点 $p,q$ 可以互相看见。美术馆问题寻求一个最小规模的守卫集合 $G\subset P$，使得 $G$ 能够完全看见 $P$。目前唯一正确的精确求解美术馆问题的算法使用代数方法，并被归功于 Sharir。由于美术馆问题是 ER-完全的，在没有额外假设的情况下，似乎不太可能避免使用代数方法。在本文中，我们引入了视觉稳定性（vision stability）的概念。为了描述视觉稳定性，考虑一个增强型守卫，它可以“绕过拐角”以 $δ$ 的角度看见，或者考虑一个减弱型守卫，其视觉被反射顶点以 $δ$ 的角度“阻挡”。如果守卫 $P$ 所需的最优增强型守卫数量与守卫 $P$ 所需的最优减弱型守卫数量相同，则称多边形 $P$ 具有视觉稳定性 $δ$。我们将论证大多数相关多边形都是视觉稳定的。我们描述了一种一次性的视觉稳定算法，该算法使用多项式时间并求解一个整数规划来计算视觉稳定多边形的最优守卫集合。它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式视觉稳定算法，并展示了其实际性能较慢，但与其他最先进算法相当。我们的迭代算法受到一次性算法的启发并紧密遵循它。它延迟了若干步骤，并且仅在认为必要时才计算这些步骤。给定多边形的一条弦 $c$，我们用 $n(c)$ 表示从 $c$ 可见的顶点数量。多边形的弦宽（chord-width）是所有可能弦 $c$ 上 $n(c)$ 的最大值。视觉稳定多边形集合在以弦宽为参数时存在一个 FPT 算法。此外，一次性算法在以反射顶点数量为参数时以 FPT 时间运行。
英文摘要：
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 $δ$ or a diminished guard whose vision is by an angle of $δ$ "blocked" by reflex vertices. A polygon $P$ has vision stability $δ$ 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.
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