The minimal convex hulls of disks problem is to find such arrangements of circular disks in the plane that minimize the length of the convex hull boundary. The mixed-integer non-linear programming model, named MinPerim [17], works only for small to moderate-sized problems. Here we propose a polylithic framework of the problem for big problem instances by combining the following algorithms and models: (i) A fast disk-packing algorithm VOROPACK-D based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model minDPCH for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm QuickhullDisk to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model MinPerim taking the output of QuickhullDisk as its input. We present complete analytic solutions for small problems up to four disks and a semi-analytic mixed-integer linear programming model which yields exact solutions for strip packing problems with up to one thousand congruent disks. It turns out that the proposed polylithic approach works fine for large problem instances containing up to 1,000 disks. Monolithic and polylithic solutions using minDPCH usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

圆盘的最小凸包问题是要在平面上找到这样的圆形圆盘布置，以使凸包边界的长度最小化。名为MinPerim [17]的混合整数非线性规划模型仅适用于中小型问题。在这里，我们通过结合以下算法和模型，针对大问题实例提出了一个多方面的问题框架：（i）一种基于Voronoi图的快速磁盘打包算法VOROPACK-D，用于打包磁盘的非线性编程（NLP）模型， NLP模型minDPCH用于最小化凸包的离散周长；（ii）快速凸包算法QuickhullDisk计算磁盘布置的凸包及其周长；（ⅲ）混合整数NLP模型MinPerim服用的输出QuickhullDisk作为其输入。我们提出了针对最多四个磁盘的小问题的完整解析解决方案，以及一个半解析混合整数线性规划模型，该模型可为多达一千个同等磁盘的带材包装问题提供精确的解决方案。事实证明，所提出的多块方法对于包含多达1,000个磁盘的大型问题实例效果很好。使用minDPCH的单片和多片解决方案通常胜过其他方法。与[17]中的结果相比，多块方法产生了更好的解决方案，并为进一步研究提供了基准套件。