Symmetry breaking in numeric constraint problems

Symmetry-breaking constraints in the form of inequalities between variables have been proposed for a few kind of solution symmetries in numeric CSPs. We show that, for the variable symmetries among those, the proposed inequalities are but a specific case of a relaxation of the well-known LEX constraints extensively used for discrete CSPs. We discuss the merits of this relaxation and present experimental evidences of its practical interest.Postprint (author’s final draft

Solving Linear Constraints in Elementary Abelian p-Groups of Symmetries

Symmetries occur naturally in CSP or SAT problems and are not very difficult to discover, but using them to prune the search space tends to be very challenging. Indeed, this usually requires finding specific elements in a group of symmetries that can be huge, and the problem of their very existence is NP-hard. We formulate such an existence problem as a constraint problem on one variable (the symmetry to be used) ranging over a group, and try to find restrictions that may be solved in polynomial time. By considering a simple form of constraints (restricted by a cardinality k) and the class of groups that have the structure of Fp-vector spaces, we propose a partial algorithm based on linear algebra. This polynomial algorithm always applies when k=p=2, but may fail otherwise as we prove the problem to be NP-hard for all other values of k and p. Experiments show that this approach though restricted should allow for an efficient use of at least some groups of symmetries. We conclude with a few directions to be explored to efficiently solve this problem on the general case.Comment: 18 page

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

Computer science: the hardware software and heart of IT

1st edition, 201