The maximum exposure problem

Given a set of points P and axis-aligned rectangles R in the plane, point p∈P is called exposed if it lies outside all R. In max-exposure problem, given an integer parameter k, we want to delete k from so as maximize number points. We show that problem NP-hard assuming plausible complexity conjectures also hard approximate even when are translates two fixed rectangles. However, only consists single rectangle, present polynomial-time approximation scheme. For range space defined by general rectangles, simple O(k) bicriteria algorithm; deleting O(k2) can expose at least Ω(1/k) optimal