We show that the following variant of labeling rotating maps is NP-hard, and present a polynomial approximation scheme for solving it. The input is a set of feature points on a map, to each of which a vertical bar of zero width is assigned. The goal is to choose the largest subsets of the bars such that when the map is rotated and the labels remain vertical, none of the bars intersect. We extend this algorithm to the general case where labels are arbitrary objects.