Approximating Smallest Containers for Packing Three-Dimensional Convex Objects

We investigate the problem of computing a minimum volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NPhard so that we cannot expect to find exact polynomial time algorithms. We give constant ratio approximation algorithms for packing axis-parallel (rectangular) cuboids under translation into an axis-parallel (rectangular) cuboid as container, for cuboids under rigid motions into an axis-parallel cuboid or into an arbitrary convex container, and for packing convex polyhedra under rigid motions into an axis-parallel cuboid or arbitrary convex container. This work gives the first approximability results for the computation of minimum volume containers for the objects described.