Optimal packing of objects in containers is a critical problem in various real-life and industrial applications. This paper investigates the two-dimensional packing of convex polygons without rotations, where only translations are allowed. We study different settings depending on the type of containers used, including minimizing the number of containers or the size of the container based on an objective function. Building on prior research in the field, we develop polynomial-time algorithms with improved approximation guarantees upon the best-known results by Alt, de Berg and Knauer, as well as Aamand, Abrahamsen, Beretta and Kleist, for problems such as Polygon Area Minimization, Polygon Perimeter Minimization, Polygon Strip Packing, and Polygon Bin Packing. Our approach utilizes a sequence of object transformations that allows sorting by height and orientation, thus enhancing the effectiveness of shelf packing algorithms for polygon packing problems. In addition, we present efficient approximation algorithms for special cases of the Polygon Bin Packing problem, progressing toward solving an open question concerning an O(1)-approximation algorithm for arbitrary polygons.

翻译：物体在容器中的最优装填是各类现实与工业应用中的关键问题。本文研究了仅允许平移（不允许旋转）的二维凸多边形装填问题。我们根据所用容器类型的不同，探讨了多种场景，包括基于目标函数最小化容器数量或容器尺寸。在既有研究基础上，我们针对多边形面积最小化、多边形周长最小化、多边形条带装填以及多边形箱式装填等问题，开发了多项式时间算法，其近似保证优于Alt、de Berg与Knauer以及Aamand、Abrahamsen、Beretta与Kleist等学者目前已知的最佳结果。我们的方法采用一系列对象变换，使得能够按高度和方向进行排序，从而增强了货架装填算法对多边形装填问题的有效性。此外，我们还针对多边形箱式装填问题的特殊情形提出了高效近似算法，为解决任意多边形O(1)近似算法的开放性问题取得了进展。