Polynomial-time approximation schemes for packing and piercing fat objects

We consider two problems: given a collection of n fat objects in a fixed dimension, (1) (packing) find the maximum subcollection of pairwise disjoint objects, and (2) (piercing) find the minimum point set that intersects every object. Recently, Erlebach, Jansen, and Seidel gave a polynomial-time approximation scheme (PTAS) for the packing problem, based on a shifted hierarchical subdivision method. Using shifted quadtrees, we describe a similar algorithm for packing but with a smaller time bound. Erlebach et al.'s algorithm requires polynomial space. We describe a different algorithm, based on geometric separators, that requires only linear space. This algorithm can also be applied to piercing, yielding the first PTAS for that problem.