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 [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 based on Voronoi diagrams, non-linear programming (NLP) models for packing disks, and an NLP model for minimizing the discretized perimeter of convex hull; (ii) A fast convex-hull algorithm to compute the convex hulls of disk arrangements and their perimeter lengths; (iii) A mixed-integer NLP model taking the output of 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 usually outperform other approaches. The polylithic approach yields better solutions than the results in [17] and provides a benchmark suite for further research.

The ropelength of complex knots

“…When N is greater than 2, the roplength is constrained by a theorem of Cantarella et al [37] stating that the length of a component in an ideal link cannot be smaller than that of the shape which minimizes the convex hull around identical disks. For small N the minimal hull length can be derived or computed, and has a known asymptotic limit for large N, but is nontrivial for intermediate N [38]. A minimal configuration for the four-fold Hopf link, in which four rounded squares bound four tightly packed stadia, is shown in figure 2(b).…”

The Ropelength of Complex Knots

“…When N is greater than 2, a theorem of Cantarella et al [34] stating that the length of a component in an ideal link cannot be smaller than that of the shape which minimizes the convex hull around identical disks. For small N the minimal hull length can be derived or computed, and has a known asymptotic limit for large N, but is nontrivial for intermediate N [35]. A minimal configuration for the 4-fold Hopf link, in which four rounded squares bound four tightly packed stadia, is shown in Fig.…”