Variable symmetry breaking in numerical constraint problems

Symmetry breaking has been a hot topic of research in the past years, leading to many theoretical developments as well as strong scaling strategies for dealing with hard applications. Most of the research has however focused on discrete, combinatorial, problems, and only few considered also continuous, numerical, problems. While part of the theory applies in both contexts, numerical problems have specificities that make most of the technical developments inadequate. In this paper, we present the rlex constraints, partial symmetry-breaking inequalities corresponding to a relaxation of the famous lex constraints extensively studied in the discrete case. They allow (partially) breaking any variable symmetry and can be generated in polynomial time. Contrarily to lex constraints that are impractical in general (due to their overwhelming number) and inappropriate in the continuous context (due to their form), rlex constraints can be efficiently handled natively by numerical constraint solvers. Moreover, we demonstrate their pruning power on continuous domains is almost as strong as that of lex constraints, and they subsume several previous work on breaking specific symmetry classes for continuous problems. Their experimental behavior is assessed on a collection of standard numerical problems and the factors influencing their impact are studied. The results confirm rlex constraints are a dependable counterpart to lex constraints for numerical problems.Peer ReviewedPostprint (author's final draft

Detection of permutation symmetries in numerical constraint satisfaction problems

Technical reportThis technical report summarizes the work done by Ieva Dauzickaite during her traineeship at IRI from 2016-10-03 to 2017-03-31. The main objectives of the stay at IRI were to study detection of variable symmetries in numerical constraint satisfaction problems and to implement a method that detects such symmetries.Peer ReviewedPostprint (published version

Constraint symmetries

Constraint Symmetries have been suggested as an appropriate subgroup of solution symmetries easier to be more easily found that the complete group of symmetries. The concept is analyzed, and it becomes clear that a knowledge of the problem as deep as for finding the whole group is required to find that subset. Indeed, we show that the Microstructure Complement in most cases only find an small fraction of the subgroup of Constraint Symmetries. Moreover, not all the symmetries it finds are truly Constraint Symmetries. We show also that in the context of point symmetries (as opposed to literal symmetries), the subgroup of Constraint Symmetries coincide with the whole solution symmetry group.Preprin

Variable symmetry breaking in numerical constraint problems

Symmetry breaking has been a hot topic of research in the past years, leading to many theoretical developments as well as strong scaling strategies for dealing with hard applications. Most of the research has however focused on discrete, combinatorial, problems, and only few considered also continuous, numerical, problems. While part of the theory applies in both contexts, numerical problems have specificities that make most of the technical developments inadequate. In this paper, we present the rlex constraints, partial symmetry-breaking inequalities corresponding to a relaxation of the famous lex constraints extensively studied in the discrete case. They allow (partially) breaking any variable symmetry and can be generated in polynomial time. Contrarily to lex constraints that are impractical in general (due to their overwhelming number) and inappropriate in the continuous context (due to their form), rlex constraints can be efficiently handled natively by numerical constraint solvers. Moreover, we demonstrate their pruning power on continuous domains is almost as strong as that of lex constraints, and they subsume several previous work on breaking specific symmetry classes for continuous problems. Their experimental behavior is assessed on a collection of standard numerical problems and the factors influencing their impact are studied. The results confirm rlex constraints are a dependable counterpart to lex constraints for numerical problems.Peer Reviewe