Online Packing of Orthogonal Polygons

Abstract

While rectangular and box-shaped objects dominate the classic discourse of theoretic investigations, a fascinating frontier lies in packing more complex shapes. Given recent insights that convex polygons do not allow for constant competitive online algorithms for diverse variants under translation, we study orthogonal polygons, in particular of small complexity. For translational packings of orthogonal 6-gons, we show that the competitive ratio of any online algorithm that aims to pack the items into a minimal number of unit bins is in $\Omega(n / \log n)$, where $n$ denotes the number of objects. In contrast, we show that constant competitive algorithms exist when the orthogonal 6-gons are symmetric or small. For (orthogonally convex) orthogonal 8-gons, we show that the trivial $n$-competitive algorithm, which places each item in its own bin, is best-possible, i.e., every online algorithm has an asymptotic competitive ratio of at least $n$. This implies that for general orthogonal polygons, the trivial algorithm is best possible. Interestingly, for packing degenerate orthogonal polygons (with thickness $0$), called skeletons, the change in complexity is even more drastic. While constant competitive algorithms for 6-skeletons exist, no online algorithm for 8-skeletons achieves a competitive ratio better than $n$. For other packing variants of orthogonal 6-gons under translation, our insights imply the following consequences. The asymptotic competitive ratio of any online algorithm is in $\Omega(n / \log n)$ for strip packing, and there exist online algorithms with competitive ratios in $O(1)$ for perimeter packing, or in $O(\sqrt{n})$ for minimizing the area of the bounding box. Moreover, the critical packing density is positive (if every object individually fits into the interior of a unit bin).