Orbital Shrinking

Symmetry plays an important role in optimization. The usual approach to cope with symmetry in discrete optimization is to try to eliminate it by introducing artificial symmetry-breaking conditions into the problem, and/or by using an ad-hoc search strategy. In this paper we argue that symmetry is instead a beneficial feature that we should preserve and exploit as much as possible, breaking it only as a last resort. To this end, we outline a new approach, that we call orbital shrinking, where additional integer variables expressing variable sums within each symmetry orbit are introduces and used to "encapsulate" model symmetry. This leads to a discrete relaxation of the original problem, whose solution yields a bound on its optimal value. Encouraging preliminary computational experiments on the tightness and solution speed of this relaxation are presented.