Breaking Variable Symmetry in Almost Injective Problems

Lexicographic constraints are commonly used to break variable symmetries. In the general case, the number of constraint to be posted is potentially exponential in the number of variables. For injective problems (AllDiff), Puget’s method[11] breaks all variable symmetries with a linear number of constraints. In this paper we assess the number of constraints for “almost” injective problems. We propose to characterize them by a parameter μ based on Global Cardinality Constraint as a generalization of the AllDiff constraint. We show that for almost injective problems, variable symmetries can be broken with no more than ` n μ ́ constraints which is XP in the framework of parameterized complexity. When only ν variables can take duplicated values, the number of constraints is FPT in μ and ν.