Maximizing the Number of Visible Labels on a Rotating Map

For a map that can be rotated, we consider the following problem. There are a number of feature points on the map, each having a geometric object as a label. The goal is to find the largest subset of these labels such that when the map is rotated and the labels remain vertical, no two labels in the subset intersect. We show that, even if the labels are vertical bars of zero width, this problem remains NP-hard, and present a polynomial approximation scheme for solving it. We also introduce a new variant of the problem for vertical labels of zero width, in which any label that does not appear in the output must be coalesced with a label that does. Coalescing a subset of the labels means to choose a representative among them and set its label height to the sum of the individual label heights.