Beam search and non-linear programming tools for the circular packing problem

This article studies the circular packing problem (CPP) which consists of packing n circles of known radii into the smallest containing circle C. The objective is to determine the coordinates of the centres of the circles as well as the radius of C. CPP, which is a variant of the 2D open-dimension problem, embeds two extremely difficult problems: a pure continuous optimisation problem and a combinatorial one. This article provides experimental proof that the continuous and combinatorial aspects of the problem should not be tackled independently, and need to be considered simultaneously. Specifically, it proposes two beam search algorithms. The first algorithm is a two-stage approach. Stage one uses a filtered beam search to identify the best ordering of the circles whereas stage two uses a recovering beam search along with non-linear optimisation to find the best position of each circle given the ordering identified during stage one. The second algorithm embeds both searches into a single search tree where each level has two sublevels: sublevel one applies a filtered beam search to identify the circle to be packed whereas sublevel two applies a recovering beam search along with non-linear optimisation to identify its position. The computational results further show that the second algorithm obtains very good results.