The Maximum Exposure Problem

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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