Geometric packing problems have been investigated for centuries in mathematics. In contrast, works on sphere packing in the field of approximation algorithms are scarce. Most results are for squares and rectangles, and their d-dimensional counterparts. To help fill this gap, we present a framework that yields approximation schemes for the geometric knapsack problem as well as other packing problems and some generalizations, and that supports not only hyperspheres but also a wide range of shapes for the items and the bins. Our first result is a PTAS for the hypersphere multiple knapsack problem. In fact, we can deal with a more generalized version of the problem that contains additional constraints on the items. These constraints, under some conditions, can encompass very common and pertinent constraints such as conflict constraints, multiple-choice constraints, and capacity constraints. Our second result is a resource augmentation scheme for the multiple knapsack problem for a wide range of convex fat objects, which are not restricted to polygons and polytopes. Examples are ellipsoids, rhombi, hypercubes, hyperspheres under the Lp-norm, etc. Also, for the generalized version of the multiple knapsack problem, our technique still yields a PTAS under resource augmentation for these objects. Thirdly, we improve the resource augmentation schemes of fat objects to allow rotation on the objects by any angle. This result, in particular, brings something extra to our framework, since most results comprising such general objects are limited to translations. At last, our framework is able to contemplate other problems such as the cutting stock problem, the minimum-size bin packing problem and the multiple strip packing problem.

翻译：几何装箱问题在数学中已有数百年研究历史。相比之下，近似算法领域关于球体装箱的研究仍较为匮乏。现有成果主要针对正方形、矩形及其d维对应形状。为填补这一空白，我们提出了一种框架，可为几何背包问题、其他装箱问题及若干推广问题提供近似方案，且该框架不仅支持超球体，还支持物品与容器的大范围形状。我们的首个成果是超球体多背包问题的多项式时间近似方案。事实上，我们可处理包含附加约束的更广义问题版本。这些约束在特定条件下可涵盖冲突约束、多选约束及容量约束等常见且相关的约束类型。第二个成果是针对广义凸胖体对象（不限多边形与多面体）的多背包问题资源增强方案，例如椭球体、菱形、超立方体、Lp范数下的超球体等。对于多背包问题的广义版本，我们的技术仍能在资源增强下为这些对象提供多项式时间近似方案。第三项改进是允许胖体对象任意角度旋转的资源增强方案。这一成果为框架带来了额外优势，因为多数涉及此类广义对象的研究仅限于平移操作。最后，我们的框架还能处理下料问题、最小尺寸装箱问题及多条带装箱问题等其他问题。