Sets of Symmetry Breaking Constraints

[Puget, 2004] has shown that if the symmetry in a constraint satisfaction problem acts only on the variables and there is also an allDifferent constraint on the variables, the symmetry can be eliminated by adding a small number of constraints, linear in the number of variables. In this paper, Puget’s procedure for finding a set of symmetrybreaking constraints is extended to find all possible distinct sets. It is shown that there can be exponentially many distinct sets of symmetry breaking constraints, leading to the need to choose between them. The choice depends on how the problem will be solved, and specifically on the variable order. Since a variable order can lead to a choice of symmetry-breaking constraints, it seems plausible that the same variable order should be used during search; however, experiments with a graceful graph problem do not show that pairing the symmetrybreaking constraints with a compatible variable order in this way leads to reduced search.