简单廊道周界防护的改进近似法
Improved Approximation for Guarding Simple Galleries from the Perimeter
摘要：
我们提供了一个O（log log OPT）-近似算法的问题，守卫一个简单的多边形与警卫的周长。首先，我们设计了一个多项式时间算法，用于为与我们的守护问题相关的命中集的实例构建大小为O（1/1/2 log log 1/2）的ε网。然后，我们应用Bronnimann和Goodrich的技术，建立一个近似算法，从这个ε-网络发现者。随着一个简单的多边形P，我们的算法作为输入的一个有限的一组潜在的保护位置，必须包括多边形的顶点。如果一个有限的一组潜在的警卫位置没有指定，例如，当警卫可以放置在任何地方的周边，我们使用一个已知的离散化技术的成本，使算法的运行时间可能线性的最长和最短的距离之间的比率顶点。我们的算法是第一个改进O（log OPT）-近似算法，使用通用的网络查找有限VC维集系统。
英文摘要：
We provide an O(log log OPT)-approximation algorithm for the problem of guarding a simple polygon with guards on the perimeter. We first design a polynomial-time algorithm for building epsilon-nets of size O(1/epsilon log log 1/epsilon) for the instances of Hitting Set associated with our guarding problem. We then apply the technique of Bronnimann and Goodrich to build an approximation algorithm from this epsilon-net finder. Along with a simple polygon P, our algorithm takes as input a finite set of potential guard locations that must include the polygon's vertices. If a finite set of potential guard locations is not specified, e.g. when guards may be placed anywhere on the perimeter, we use a known discretization technique at the cost of making the algorithm's running time potentially linear in the ratio between the longest and shortest distances between vertices. Our algorithm is the first to improve upon O(log OPT)-approximation algorithms that use generic net finders for set systems of finite VC-dimension.
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