Designing Algorithms with Multi-sphere Scheme

In this paper, we deal with a packing problem that asks to place a given set of objects such as non-convex poly- topes or objects with curved surfaces in a container in R² or R³ so that the resulting layout of the objects has no penetration of the objects and no protrusion of the objects from the container. Imamichi and Nagamochi proposed a new algorithm scheme, called multi-sphere scheme, which first approximates each object by a set of spheres and finds a layout with no overlap as a local optimal solution of an unconstraint nonlinear program for the sets of shperes. We focus on the case that all objects are rigid, and describe how to design algorithms with the scheme when motions of objects are specified. We handle three variations of motions of objects: "translation," "translation with a fixed direction," and "translation and rotation by an arbitrary angle." We then develop an iterated local search algorithm combining a local search algorithm based on a nonlinear program and perturbation that swaps two objects for the case where "translation and rotation by an arbitrary angle" is allowed. Computational experiments disclose our algorithm can find a layout with nearly no penetration and protrusion quickly.