Symmetry Breaking Ordering Constraints

In a constraint satisfaction problem (CSP), symmetry involves the variables, values in the domains, or both, and maps each search state into an equivalent one. When searching for solutions, symmetrically equivalent (partial) assignments can dramatically increase the search space. Hence, elimination of symmetry is essential to avoid exploring equivalent branches in a search tree. An important class of symmetries in constraint programming arises from matrices of decison variables where rows and columns represent indistinguishable objects and are therefore symmetric. We can permute any two rows as well as two columns of a matrix with row and column symmetry without affecting any (partial) assignments. An n×m matrix with row and column symmetry exhibits n!.m! symmetries, which increase super-exponentially as the size of the matrix enlarges. This generates too many symmetric search states and thus it can be very expensive to visit all the symmetric branches of the search tree. In recent years, many techniques have been developed towards eliminating symmetry in CSPs. Unfortunately, none of the methods are yet able to deal with row and column symmetries effectively. Our main goal is to eliminate row and column symmetries of a matrix of decision variables effectively and efficiently. Towards this goal, our aims are (1) to investigate ordering constraints that can be posted on matrices to eliminate row and column symmetries without removing feasible solutions; (2) to devise global constraints to easily pose, and effectively and efficiently solve the ordering constraints; (3) to study the effectiveness of the ordering constraints in breaking row and column symmetries.