Approximating Smallest Containers for Packing Three-dimensional Convex Objects

TL;DR

This paper develops constant ratio approximation algorithms for packing 3D convex objects into minimal-volume containers, addressing NP-hard variants and providing the first known approximability results for these packing problems.

Contribution

It introduces the first approximation algorithms with constant ratios for packing convex objects into minimal containers, covering various object and container configurations.

Findings

Provides the first constant ratio approximation algorithms for 3D convex packing.

Addresses NP-hard variants of the container minimization problem.

Extends results to different object and container types, including convex polyhedra.

Abstract

We investigate the problem of computing a minimal-volume container for the non-overlapping packing of a given set of three-dimensional convex objects. Already the simplest versions of the problem are NP-hard 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 minimal volume containers for the objects described.