This is the online supplement for the paper
T. Berthold, D. Kamp, G. Mexi, S. Pokutta, I. Polik. Out-of-the-Box Global Optimization for Packing Problems: New Models and Improved Solutions. Submitted to the Journal of Global Optimization (JoGO), 2026.
Pack n circles with variable radii into the unit square, maximizing the sum of radii.
- Instances: n in {1, ..., 60}
- Directory:
n{n}/ - Files:
circlepacking_n{n}.txt,circlepacking_n{n}.pdf - Solution format: Line 1:
n, Line 2:sum_radii, Lines 3+:x_i y_i r_iper circle
Pack n circles with variable radii into a rectangle of perimeter 4, maximizing the sum of radii.
- Instances: n in {1, ..., 60}
- Directory:
n{n}/ - Files:
rectangle_n{n}.txt,rectangle_n{n}.pdf - Solution format: Line 1:
n, Line 2:sum_radii, Lines 3+:x_i y_i r_iper circle (rectangle width inferred as 1 + max(x_i + r_i) - max(y_i + r_i) clamped to [0, 1])
Pack n regular m-gons with unit circumradius into a regular l-gon, minimizing the circumradius R of the outer polygon.
- Instances: l, m in {3, ..., 7}, n in {1, ..., 30}
- Directory:
l{l}/m{m}/n{n}/ - Files:
polygon_n{n}_m{m}_l{l}.txt,polygon_n{n}_m{m}_l{l}.pdf - Solution format: Line 1:
l m n, Line 2:R, Lines 3+:x_i y_i theta_iper inner polygon
Pack n Platonic solids of type m with unit circumradius into a Platonic solid of type l, minimizing the circumradius R of the outer solid.
Solid types: 1 = Tetrahedron, 2 = Octahedron, 3 = Cube, 4 = Icosahedron, 5 = Dodecahedron.
- Instances: l, m in {1, ..., 5}, n in {1, ..., 30}
- Directory:
l{l}/m{m}/n{n}/ - Files:
platonic_n{n}_m{m}_l{l}.txt,platonic_n{n}_m{m}_l{l}.pdf,platonic_n{n}_m{m}_l{l}.html - Solution format: Line 1:
l m n, Line 2:R, Lines 3+:x_i y_i z_i theta_i iota_i kappa_iper inner solid (ZYX rotation angles)
Each problem directory contains a results.dat file listing all instances with rounded objective values:
- Square and Rectangle: sum of radii floored to 5 decimal places (maximization)
- Polygon and Platonic: circumradius ceiled to 5 decimal places (minimization)