We investigate several online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container, while the aim is to minimize the used space. Among other variants, we consider strip packing and bin packing, where the container is the infinite horizontal strip $[0,\infty)\times [0,1]$ or a collection of $1 \times 1$ bins, respectively. We draw interesting connections to the following online sorting problem OnlineSorting$[\gamma,n]$: We receive a stream of real numbers $s_1,\ldots,s_n$, $s_i\in[0,1]$, one by one. Each real must be placed in an array $A$ with $\gamma n$ initially empty cells without knowing the subsequent reals. The goal is to minimize the sum of differences of consecutive reals in $A$. The offline optimum is to place the reals in sorted order so the cost is at most $1$. We show that for any $\Delta$-competitive online algorithm of OnlineSorting$[\gamma,n]$, it holds that $\gamma \Delta \in\Omega(\log n/\log \log n)$. We use this lower bound to prove the non-existence of competitive algorithms for various online translational packing problems of convex polygons, among them strip packing, bin packing and perimeter packing. This also implies that there exists no online algorithm that can pack all streams of pieces of diameter and total area at most $\delta$ into the unit square. These results are in contrast to the case when the pieces are restricted to rectangles, for which competitive algorithms are known. Likewise, the offline versions of packing convex polygons have constant factor approximation algorithms. As a complement, we also include algorithms for both online sorting and translation-only online strip packing with non-trivial competitive ratios. Our algorithm for strip packing relies on a new technique for recursively subdividing the strip into parallelograms of varying height, thickness and slope.

翻译：我们研究了几类在线装箱问题，其中凸多边形逐个到达并需不可逆地放入容器，目标是最小化使用的空间。在众多变体中，我们特别考虑了条形装箱和箱式装箱，其中容器分别为无限水平条带$[0,\infty)\times [0,1]$或一组$1 \times 1$的箱子。我们发现了与以下在线排序问题OnlineSorting$[\gamma,n]$的有趣关联：我们依次接收实数流$s_1,\ldots,s_n$，$s_i\in[0,1]$。每个实数必须放入具有$\gamma n$个初始空单元格的数组$A$中，且无法预知后续实数。目标是最小化$A$中相邻实数差值之和。离线最优解是将实数按排序顺序放置，使得代价不超过$1$。我们证明：对于OnlineSorting$[\gamma,n]$的任何$\Delta$-竞争在线算法，必然有$\gamma \Delta \in\Omega(\log n/\log \log n)$。利用这一下界，我们证明了凸多边形多种在线平移装箱问题（包括条形装箱、箱式装箱和周长装箱）不存在竞争算法。这也意味着不存在任何在线算法能将直径和总面积均不超过$\delta$的所有片段流装入单位正方形。这些结果与已知存在竞争算法的矩形片段情形形成鲜明对比。同样地，凸多边形的离线装箱问题具有常数因子近似算法。作为补充，我们还给出了在线排序和仅平移在线条形装箱的非平凡竞争比算法。我们的条形装箱算法依赖于一种新方法：将条带递归细分为高度、厚度和斜率各异的平行四边形。