Polynomial-Time Approximation Schemes for Circle Packing Problems

We consider the problem of packing a set of circles into a minimum number of unit square bins. We give an asymptotic approximation scheme (APTAS) when we have resource augmentation in one dimension, that is, we may use bins of height 1+gamma, for some small gamma > 0. As a corollary, we also obtain an APTAS for the circle strip packing problem, whose objective is to pack a set of circles into a strip of unit width and minimum height. These are the first approximation schemes for these problems. Our algorithm is based on novel ideas of iteratively separating small and large items, and may be extended to more general packing problems. For example, we also obtain APTAS's for the corresponding problems of packing d-dimensional spheres under the L-p-norm.