Packing equal circles in a square: a deterministic global optimization approach

Abstract

In this paper the problem of packing n equal circles into the unit square will be considered. Starting from a general rectangular branch-and-bound algorithm, many tools, which exploit the special structure of the problem and properties fulfilled by some of its solutions, will be introduced and discussed. Computational results will be presented and, in particular, the optimality within a given tolerance of best known solutions in the literature for n=10–35, n=38,39 will be proved, with the exception of the case n=32 for which a new solution has been detected and proved to be optimal within the given tolerance. Moreover, a new solution for n=37 has been detected, but not yet proved to be optimal within the given tolerance.