Packing 2D Disks into a 3D Container

Abstract

In this article, we consider the problem of finding in three dimensions a minimum-volume axisparallel box into which a given set of unit-radius disks can be packed under translations. The problem is neither known to be NP-hard nor to be in NP. We give a constant-factor approximation algorithm based on reduction to finding a shortest Hamiltonian path in a weighted graph. As a byproduct, we can show that there is no finite size container into which all unit disks can be packed simultaneously.

Online Sorting and Translational Packing of Convex Polygons

“…As two cubes with sidelengths 1/2 + ε, for any ε > 0, cannot be packed in the unit cube, this shows that the critical density of packing cubes into a unit cube is 1/2 d−1 for any d ≥ 1. Alt, Cheong, Park, and Scharf [7] showed that there exist n 2D unit disks embedded in 3D (with different normal vectors) such that whenever they are placed in a non-overlapping way, their bounding box has volume Ω( √ n). It follows that when rotations are not allowed, the critical density of packing convex bodies of bounded diameter into a cube is 0, or, in other words, that one kilogram of potatoes cannot always be put into a finite sack by translation.…”

“…In simplified terms, it is then proved that for some constant δ = δ(d) > 0, any sequence of axis-parallel boxes of diameter and total area at most δ can be packed online in the d-dimensional unit hypercube. Determining whether the critical density of translational and online packing convex 2D polygons is positive remained an interesting question: On one hand, this packing problem is harder than the 2D offline version which has positive critical density (Theorem 6 (d)), and on the other hand, it is easier than the 3D online version which has 0 critical density (since also the 3D offline version has 0 critical density [7]). In this paper, we prove that the 2D online version also has critical density 0 (Theorem 2 (d)).…”