A Framework for Approximation Schemes on Knapsack and Packing Problems of Hyperspheres and Fat Objects

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.