Packing of Circles on Square Flat Torus as Global Optimization of Mixed Integer Nonlinear problem

TL;DR

This paper presents a general approach to solving discrete geometry packing problems by formulating them as mixed-integer nonlinear optimization problems and applying branch-and-bound solvers, successfully confirming an optimal packing conjecture for N=9 circles.

Contribution

It introduces a novel application of global optimization techniques to circle packing on a square flat torus, confirming a longstanding conjecture using advanced computational methods.

Findings

Confirmed the optimal packing for N=9 circles on a square flat torus.

Demonstrated the effectiveness of branch-and-bound solvers in discrete geometry problems.

Provided a general framework for solving similar packing problems as mixed-integer nonlinear programs.

Abstract

The article demonstrates rather general approach to problems of discrete geometry: treat them as global optimization problems to be solved by one of general purpose solver implementing branch-and-bound algorithm (B&B). This approach may be used for various types of problems, i.e. Tammes problems, Thomson problems, search of minimal potential energy of micro-clusters, etc. Here we consider a problem of densest packing of equal circles in special geometrical object, so called square flat torus $\mathbb{R}^2/\mathbb{Z}^2$ with the induced metric. It is formulated as Mixed-Integer Nonlinear Problem with linear and non-convex quadratic constraints. The open-source B&B-solver SCIP, http://scip.zib.de, and its parallel implementation ParaSCIP, http://ug.zib.de, had been used in computing experiments to find "very good" approximations of optimal arrangements. The main result is a confirmation of the conjecture on optimal packing for N=9 that was published in 2012 by O. Musin and A. Nikitenko. To do that, ParaSCIP took about 2000 CPU*hours (16 hours x 128 CPUs) of cluster HPC4/HPC5, National Research Centre "Kurchatov Institute", http://ckp.nrcki.ru