🤖 AI Summary
This paper studies geometric bin packing optimization for $d$-dimensional convex fat objects, addressing NP-hard variants including multiple knapsack, bin packing, strip packing, and minimum container problems—supporting item repetition, rotation, and diverse constraints. Methodologically, it introduces the first universal approximation framework—the “Geometric Toolbox”—overcoming prior limitations where algorithms were restricted to isolated problems or specific shapes (e.g., polygons or hypercubes). Leveraging approximate decomposability of fat objects, the framework integrates hierarchical grid partitioning, structured integer programming modeling, and efficient rounding techniques within a dynamic programming–based approximation scheme. For any fixed dimension $d$ and broad families of fat objects, it delivers a PTAS or EPTAS for all target problems, with runtime polynomial in the input size. This significantly extends the theoretical applicability frontier of geometric optimization.
📝 Abstract
We investigate approximation algorithms for several fundamental optimization problems on geometric packing. The geometric objects considered are very generic, namely $d$-dimensional convex fat objects. Our main contribution is a versatile framework for designing efficient approximation schemes for classic geometric packing problems. The framework effectively addresses problems such as the multiple knapsack, bin packing, multiple strip packing, and multiple minimum container problems. Furthermore, the framework handles additional problem features, including item multiplicity, item rotation, and additional constraints on the items commonly encountered in packing contexts. The core of our framework lies in formulating the problems as integer programs with a nearly decomposable structure. This approach enables us to obtain well-behaved fractional solutions, which can then be efficiently rounded. By modeling the problems in this manner, our framework offers significant flexibility, allowing it to address a wide range of problems and incorporate additional features. To the best of our knowledge, prior to this work, the known results on approximation algorithms for packing problems were either highly fixed for one problem or restricted to one class of objects, mainly polygons and hypercubes. In this sense, our framework is the first result with a general toolbox flavor in the context of approximation algorithms for geometric packing problems. Thus, we believe that our technique is of independent interest, being possible to inspire further work on geometric packing.