Dynamic Symmetry Breaking in Constraint Programming and Linear Programming Hybrids: Still Life as a Case Study

The efficient solving of many CSPs has been achieved by hybrid methods that combine Constraint Programming (CP) and Linear Programming (LP). One such case is the ‘Maximum Density Still Life’ problem. An obstacle to overcome in order to solve this problem efficiently is the inherent symmetry. Symmetry in Constraint Satisfaction Problems (CSPs) can lead to redundant search, since subtrees may be explored which are symmetric to subtrees previously explored. In CP symmetry can be excluded by Symmetry Breaking During Search (SBDS), a dynamic method that adds constraints during search so that partial assignments symmetric to those already considered will not be visited. We outline a novel integration of SBDS with CP-LP hybrids which allows the ‘Maximum Density Still Life’ problem to be solved efficiently. We show that CP modelling techniques which improve the performance of the CP-LP hybrid can also be integrated with SBDS, possibly increasing their efficiency still further.