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.

翻译：本文研究了若干几何装箱基础优化问题的近似算法。所考虑的几何对象具有高度一般性，即$d$维凸胖对象。我们的主要贡献是提出了一种通用框架，可用于为经典几何装箱问题设计高效近似方案。该框架能有效处理多重背包、装箱、多重条带装箱及多重最小容器等问题，同时支持项目多重性、项目旋转以及装箱场景中常见的额外项目约束等特征。框架的核心在于将问题建模为具有近似可分解结构的整数规划，从而获得性质良好的分数解，并实现高效舍入。通过这种建模方式，本框架展现出显著的灵活性，能够处理广泛的问题类别并整合额外特征。据我们所知，在本研究之前，装箱问题近似算法的已知结果要么高度局限于单一问题，要么受限于单一对象类别（主要是多边形和超立方体）。在此意义上，本框架是几何装箱问题近似算法领域中首个具有通用工具箱性质的研究成果。因此，我们相信该技术具有独立价值，有望推动几何装箱问题的后续研究。