Near optimal minimal convex hulls of disks

Abstract

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.