最大曝光问题
The Maximum Exposure Problem
摘要：
给定平面中的一组点$P$和轴对齐的矩形$\mathcal{R}$，如果点$p \in P$位于$\mathcal{R}$中的所有矩形之外，则该点称为\emph{exposed}。在\emph{max-exposure problem}中，给定整数参数$k$，我们希望从$\mathcal{R}$中删除$k$矩形，以便最大化暴露点的数量。我们表明，这个问题是NP-难的，假设合理的复杂性结构也很难近似，即使在$\mathcal{R}$矩形是两个固定的矩形的翻译。然而，如果$\mathcal{R}$只包括一个单一的矩形的平移，我们提出了一个多项式时间近似计划。对于由一般矩形定义的值域空间，我们提出了一个简单的$O(k)$双准则近似算法，即通过删除$O(k^2)$矩形，我们可以暴露至少$Ω(1/k)$的最优点数.
英文摘要：
Given a set of points $P$ and axis-aligned rectangles $\mathcal{R}$ in the plane, a point $p \in P$ is called \emph{exposed} if it lies outside all rectangles in $\mathcal{R}$. In the \emph{max-exposure problem}, given an integer parameter $k$, we want to delete $k$ rectangles from $\mathcal{R}$ so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in $\mathcal{R}$ are translates of two fixed rectangles. However, if $\mathcal{R}$ only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For range space defined by general rectangles, we present a simple $O(k)$ bicriteria approximation algorithm; that is by deleting $O(k^2)$ rectangles, we can expose at least $Ω(1/k)$ of the optimal number of points.
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