Improved interval methods for solving circle packing problems in the unit square

In this work computer-assisted optimality proofs are given for the problems of finding the densest packings of 31, 32, and 33 non-overlapping equal circles in a square. In a study of 2005, a fully interval arithmetic based global optimization method was introduced for the problem class, solving the cases 28, 29, 30. Until now, these were the largest problem instances solved on a computer. Using the techniques of that paper, the estimated solution time for the next three cases would have been 3–6 CPU months. In the present paper this former method is improved in both its local and global search phases. We discuss a new interval-based polygon representation of the core local method for eliminating suboptimal regions, which has a simpler implementation, easier proof of correctness, and faster behaviour than the former one. Furthermore, a modified strategy is presented for the global phase of the search, including improved symmetry filtering and tile pattern matching. With the new method the cases n=31,32,33\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=31,32,33$$\end{document} have been solved in 26, 61, and 13 CPU hours, giving high precision enclosures for all global optimizers and the optimum value. After eliminating the hardware and compiler improvements since the former study, the new proof technique became roughly about 40–100 times faster than the previous one. In addition, the new implementation is suitable for solving the next few circle packing instances with similar computational effort.