Increasing Symmetry Breaking by Preserving Target Symmetries

Abstract

Breaking the exponential number of all symmetries of a constraint satisfaction problem is often too costly. In practice, we often aim at breaking a subset of the symmetries efficiently, which we call target symmetries. In static symmetry breaking, the goal is to post a set of constraints to break these target symmetries in order to reduce the solution set and thus also the search space. Symmetries of a problem are all intertwined. A symmetry breaking constraint intended for a particular symmetry almost always breaks more than just the intended symmetry as a side-effect. Different constraints for breaking the same target symmetry can have different side-effects. Conventional wisdom suggests that we should select a symmetry breaking constraint that has more side-effects by breaking more symmetries. While this wisdom is valid in many ways, we should be careful where the side-effects take place. A symmetry σ of a CSP $\mathcal{P} = (\mathcal{V},\mathcal{D},\mathcal{C})$ is preserved by a set of symmetry breaking constraints C^sb iff σ is a symmetry of $\mathcal{P}' = (\mathcal{V},\mathcal{D},\mathcal{C} \cup C^{sb})$. We give theorems and examples to demonstrate that it is beneficial to post symmetry breaking constraints that preserve the target symmetries and restrict the side-effects to only non-target symmetries as much as possible. The benefits are in terms of the number of symmetries broken and the extent to which a symmetry is broken (or eliminated), resulting in a smaller solution set and search space. Extensive experiments are also conducted to confirm the feasibility and efficiency of our proposal empirically.