Art Gallery is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of Art Gallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of Art Gallery are all W[1]-hard with respect to k [Bonnet and Miltzow, ESA'16], the classic variant has an O(log k)-approximation algorithm [Bonnet and Miltzow, SoCG'17], and it may require irrational guards [Abrahamsen et al., SoCG'17]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be \exists R-complete~[Abrahamsen et al., STOC'18]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Center Workshop, 2016]: Is Art Gallery FPT with respect to r, the number of reflex vertices? We focus on the variant where G and C are all the vertices of P, called Vertex-Vertex Art Gallery. We show that Vertex-Vertex Art Gallery is solvable in time r^{O(r^2)}n^{O(1)}. Our approach also extends to assert that Vertex-Boundary Art Gallery and Boundary-Vertex Art Gallery are both FPT. We utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-Vertex Art Gallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions.

中文翻译：

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保护近凸多边形的参数化复杂度

Art Gallery 是计算几何中的一个基本可见性问题。输入包含一个简单的多边形 P，（可能是无限的）P 内点的集合 G 和 C，以及一个整数 k；任务是决定是否最多可以在 G 中的点上放置 k 个守卫，以便 C 中的每个点对至少一个守卫可见。在 Art Gallery 的经典公式中，G 和 C 由 P 内的所有点组成。其他众所周知的变体将 G 和 C 限制为由 P 边界上的所有点或 P 的所有顶点组成。最近，取得了三个新的重要发现：上述 Art Gallery 变体对于 k [Bonnet and Miltzow, ESA'16] 都是 W[1]-hard，经典变体具有 O(log k)-近似算法 [ Bonnet 和 Miltzow，SoCG'17]，它可能需要非理性的保护 [Abrahamsen 等人，SOCG'17]。基于第三个结果，经典变体和 G 仅由 P 边界上的所有点组成的情况都被证明是\exists R-complete~[Abrahamsen et al., STOC'18]。即使 G 和 C 都只包含 P 边界上的所有点，也不知道问题出在 NP 中。鉴于第一个发现，Giannopoulos [Lorentz Center Workshop, 2016] 提出了以下问题：Art Gallery FPT 是否关于 r，反射顶点的数量？我们关注 G 和 C 都是 P 的所有顶点的变体，称为 Vertex-Vertex Art Gallery。我们证明 Vertex-Vertex Art Gallery 在时间 r^{O(r^2)}n^{O(1)} 是可解的。我们的方法还扩展到断言 Vertex-Boundary Art Gallery 和 Boundary-Vertex Art Gallery 都是 FPT。我们利用“