Breaking Symmetry with Different Orderings
This addresses the challenge of symmetry breaking in constraint solving for researchers and practitioners, though it is incremental as it builds on known methods with theoretical limitations but practical improvements.
The paper tackles the problem of breaking symmetry in constraint satisfaction by exploring alternative orderings beyond the intractable Lex-Leader method, proving that worst-case complexity cannot be reduced for certain orderings like Gray code and Snake-Lex, but showing experimentally that these can be beneficial in practice by aligning better with objective functions and branching heuristics.
We can break symmetry by eliminating solutions within each symmetry class. For instance, the Lex-Leader method eliminates all but the smallest solution in the lexicographical ordering. Unfortunately, the Lex-Leader method is intractable in general. We prove that, under modest assumptions, we cannot reduce the worst case complexity of breaking symmetry by using other orderings on solutions. We also prove that a common type of symmetry, where rows and columns in a matrix of decision variables are interchangeable, is intractable to break when we use two promising alternatives to the lexicographical ordering: the Gray code ordering (which uses a different ordering on solutions), and the Snake-Lex ordering (which is a variant of the lexicographical ordering that re-orders the variables). Nevertheless, we show experimentally that using other orderings like the Gray code to break symmetry can be beneficial in practice as they may better align with the objective function and branching heuristic.