Adaptive beam search lookahead algorithms for the circular packing problem

This paper addresses the circular packing problem (CPP), which consists in packing n circles C_i, each of known radius r_i, i∈N=(1, …, n), into the smallest containing circle C. The objective is to determine the radius r of C as well as the coordinates (x_i, y_i) of the center of C_i, i∈N. CPP is solved using two adaptive algorithms that adopt a binary search to determine r, and a beam search to check the feasibility of packing n circles into C when the radius is fixed at r. A node of level ℓ, ℓ=1, …, n, of the beam search tree corresponds to a partial packing of ℓ circles of N into C. The potential of each node of the tree is assessed using a lookahead strategy that, starting with the partial packing of the current node, assigns each unpacked circle to its maximum hole degree position. The beam search stops either when the lookahead strategy identifies a feasible packing or when it has fathomed all nodes. The computational tests on a set of benchmark instances show the effectiveness of the proposed adaptive algorithms.