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 $Ω(n / \log n)$, 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 $Ω(n / \log n)$ 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).

翻译：虽然矩形和盒状物体主导着理论研究的经典讨论，但一个引人入胜的前沿领域在于更复杂形状的打包问题。鉴于最近的研究表明，在多类平移变体问题中，凸多边形不存在常数竞争比的在线算法，我们转而研究正交多边形，特别是复杂度较低的正交多边形。对于正交六边形的平移打包，我们证明了，当目标是将物品打包到最少数量单位箱子中时，任何在线算法的竞争比均为 $Ω(n / \log n)$，其中 $n$ 表示物体数量。相比之下，我们证明了当正交六边形为对称或小尺寸时，存在常数竞争比的算法。对于（正交凸）正交八边形，我们证明平凡 $n$ 竞争比算法（即将每个物品放入其独立箱子）是最优的，即每个在线算法的渐近竞争比至少为 $n$。这意味着对于一般正交多边形，平凡算法是最优的。有趣的是，对于打包厚度为 $0$ 的退化正交多边形（称为骨架），其复杂度变化更为剧烈：尽管六边形骨架存在常数竞争比算法，但八边形骨架的任何在线算法都无法达到优于 $n$ 的竞争比。对于正交六边形平移打包的其他变体，我们的研究结论具有以下推论。条带打包中任何在线算法的渐近竞争比均为 $Ω(n / \log n)$；周长打包存在竞争比为 $O(1)$ 的在线算法；而最小化边界框面积时存在竞争比为 $O(\sqrt{n})$ 的在线算法。此外，若每个物体都能单独放入单位箱子内部，则临界打包密度为正数。