Static Symmetry Breaking in Circle Packing

We present new Static Symmetry-Breaking Inequalities (SSBI) [11,6] for the problem of packing equal circles in a square [9]. The new SSBIs provide a marked computational improvement with respect to past work [1], though not yet at the level where a purely Mathematical Programming (MP) based spatial Branch-and-Bound (sBB) can be competitive with a Branch-and-Bound (BB) “boosted” by combinatorial and geometrical devices such as [9]. We consider the following formulation of Circle Packing in a Square (CPS) problem: given N ∈ N and L ∈ N, can N non-overlapping circles of unit radius be arranged in a square of side 2L? This is equivalent to the more usual formulation where one maximizes the number of non-overlapping circles of unit radius in a square of side 2L with L ∈ Q+: it suffices to consider the usual correspondence (via bisection) of optimization and decision problems, and to remark that for all instances of the second formulation with fractional square side length there exists an equivalent instance of the CPS described by integer data.