The maximum exposure problem

Abstract

Given a set of points P and axis-aligned rectangles R in the plane, a point p∈P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from 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 R are translates of two fixed rectangles. However, if 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(k2) rectangles, we can expose at least Ω(1/k) of the optimal number of points.