An Algorithm for Packing Circles of Two Types in a Fixed Size Container with Non-Euclidean Metric

The paper deals with the problem of optimal packing of two sets of circles (2-D spheres) into a simply connected container. The number of circles is given. The radii of these circles are equal within each set, but, generally speaking, they differ between sets. There are two different statements of a such problem. The simplest one is when the circles of a larger radius are located first, and then smaller circles are packed into the gaps. Solving of a such problem, in fact, reduces to a two-fold solution of the equal circles packing problem. However, the procedure is complicated by the fact that in the second solution the container will be a multiply connected set. We consider a more complex formulation: it is required to maximize the radii of the circles when their ratio is fixed. The circle packing problem is usually studied in the case when the distance between points is Euclidean and even then belongs to the class of NP-hard problems. We assume that the distance is determined by means of some special metric, which, generally speaking, is not Euclidean. The special numerical algorithm is suggested and implemented. It based on optical-geometric approach, which is developed by the authors in recent years and previously used only for packing circles of equal radius. The results of computational experiment are presented and discussed.