The Pencil Packing Problem ∗

We consider the following three-dimensional packing problem that arises in multi-body motion planning in an environment with obstacles: Given an n×n×n regular grid of voxels (cubes), with each voxel labeled as “empty” or “occupied”, determine the maximum number of “pencils” that can be packed within the empty voxels, where a pencil is a union of n voxels that form an axis-parallel strip (i.e., a 1× 1× n box). No pencil is allowed to contain an occupied voxel, and no two pencils can have a shared voxel (since they form a packing). We consider also the dual (covering) problem in which we want to find the minimum number of empty “covering” voxels such that every pencil is intersected by at least one covering voxel. We show that both problems are NP-Hard and we give some approximation algorithms. We have evaluated our approximation algorithms experimentally and found that they perform very well in practice.