Motivated by the problem of labeling maps, we investigate the problem of computing a large non-intersecting subset in a set of n rectangles in the plane. Our results are as follows. In O(n log n) time, we can nd an O(log n)-factor approximation of the maximum subset in a set of n arbitrary axis-parallel rectangles in the plane. If all rectangles have unit height, we can nd a 2-approximation in O(n log n) time. Extending this result, we obtain a (1 + 1 k )-approximation in time O(n log n + n 2k?1 ) time, for any integer k 1.
In two dimensions, Agarwal et al. REF presented a static algorithm which computes O(log n) approximation of the maximum IS of n arbitrary axis-parallel rectangles in time O(n log n).
Label Placement by Maximum Independent Set in Rectangles
Computational Geometry: Theory and Applications
Mathematics Computer Science