旋转半平面搜索多面体
Searching Polyhedra by Rotating Half-Planes
摘要：
探照灯调度问题首先在2D多边形中进行了研究，其目标是让固定位置的控球后卫旋转探照灯以抓住逃避的入侵者。在这里，问题扩展到3D多面体，现在的警卫边界段旋转半平面的照明。在仔细地详细描述3D模型之后，建立了几个结果。第一个是一个几乎直接扩展的平面单向扫描策略，我们称之为穷举警卫，一个泛化，成功，尽管没有明确定义的概念，在3D的平面“顺时针旋转”。接下来有两个结果：每个具有r>0反射边的多面体最多可以通过r^2个适当放置的保护来搜索，而如果多面体是正交的，则仅r个保护就足够了。（最小化搜索给定多面体的警卫数量很容易被视为NP困难。最后，我们表明，决定一组给定的警卫是否有一个成功的搜索时间表是强NP-难的，并决定一个给定的目标区域是否可搜索的是强PSPACE-难的，即使是正交多面体。一些周边的结果证明途中，这些中心定理，和几个开放的问题仍然为今后的工作。
英文摘要：
The Searchlight Scheduling Problem was first studied in 2D polygons, where the goal is for point guards in fixed positions to rotate searchlights to catch an evasive intruder. Here the problem is extended to 3D polyhedra, with the guards now boundary segments who rotate half-planes of illumination. After carefully detailing the 3D model, several results are established. The first is a nearly direct extension of the planar one-way sweep strategy using what we call exhaustive guards, a generalization that succeeds despite there being no well-defined notion in 3D of planar "clockwise rotation". Next follow two results: every polyhedron with r>0 reflex edges can be searched by at most r^2 suitably placed guards, whereas just r guards suffice if the polyhedron is orthogonal. (Minimizing the number of guards to search a given polyhedron is easily seen to be NP-hard.) Finally we show that deciding whether a given set of guards has a successful search schedule is strongly NP-hard, and that deciding if a given target area is searchable at all is strongly PSPACE-hard, even for orthogonal polyhedra. A number of peripheral results are proved en route to these central theorems, and several open problems remain for future work.
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