发表机构
Indian Statistical Institute（印度统计研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文首次给出弱可见多边形中见证集问题的精确多项式时间算法，包括离散和连续两种设置，并证明离散情形的最优性。
AI 中文摘要
在经典的艺术画廊问题（AGP）中，守卫被放置在多边形中，使得它们共同看到每个点。见证集问题（WSP）由Amit、Mitchell和Packer提出，是AGP的自然对偶问题。在本文中，我们研究了弱可见多边形（WVP）中的WSP，即每个点都能从某条固定边上的某个点看到的简单多边形。见证集是一组点，其可见区域两两不相交，因此没有单个守卫能看到其中两个点。因此，最大见证集给出了守卫数量的下界。已知的WSP精确多项式时间算法仅适用于单调山形多边形，这是WVP的一个真子类。我们首次给出了WVP中WSP的精确多项式时间算法，适用于两种设置。在离散见证集问题（DiscWSP）中，见证来自给定的$m$个点，我们在具有$n$个顶点的多边形上以$O(n + m \log(n+m))$时间找到最大见证子集。该算法基于一个结构事实：WVP的可见性交集图（其中两个点相邻如果它们的可见区域相交）是一个梯形图，即两条平行线之间梯形的交集图。此外，这类图类真包含区间图和置换图，这可能在图论中具有独立意义。我们还证明了在代数决策树模型中，对于$m = \Theta(n)$的实例，存在$\Omega(n \log n)$的下界，因此我们的DiscWSP算法是最优的。在连续见证集问题（ContWSP）中，见证可以是多边形的任意点，我们给出了一个精确算法，运行时间为$O(n \log n + \rho^{2}(n + \rho^{2}))$，其中$\rho$是反射顶点的数量。
英文摘要
In the classical Art Gallery Problem (AGP), guards are placed in a polygon so that together they see every point. The Witness Set Problem (WSP), introduced by Amit, Mitchell, and Packer, is a natural dual to the AGP. In this paper, we study the WSP in weak visibility polygons (WVPs), the simple polygons in which every point is seen from some point of one fixed edge. A witness set is a set of points whose visibility regions are pairwise disjoint, so that no single guard sees two of them. A maximum witness set, therefore, lower-bounds the guard number. Exact polynomial-time algorithms for the WSP are known only for monotone mountains, a proper subclass of WVPs. We give the first exact polynomial-time algorithms for the WSP in WVPs, in two settings. In the Discrete Witness Set Problem (DiscWSP), the witnesses come from a given set of $m$ points, and we find a maximum witness subset in $O(n + m \log(n+m))$ time on an $n$-vertex polygon. The algorithm rests on a structural fact: the visibility intersection graph of a WVP, in which two points are adjacent if their visibility regions intersect, is a trapezoid graph, that is, an intersection graph of trapezoids between two parallel lines. Moreover, the class of these graphs properly contains the interval graphs and the permutation graphs, which may be of independent interest in graph theory. We also prove an $Ω(n \log n)$ lower bound in the algebraic decision-tree model for instances with $m = Θ(n)$, so our algorithm for DiscWSP is optimum. In the Continuous Witness Set Problem (ContWSP), a witness may be any point of the polygon, and we give an exact algorithm running in $O(n \log n + ρ^{2}(n + ρ^{2}))$ time, where $ρ$ is the number of reflex vertices.
