Using Symmetry of Global Constraints to Speed up the Resolution of Constraint Satisfaction Problems

1 LIP6, Boite 169, 4 Place Jussieu, 75 252 Paris Cedex 05, France. e-mail: roy@poleia.lip6.fr 2 SONY CSL-Paris, 6 rue Amyot, 75 005 Paris, France. e-mail: pachet@csl.sony.fr Abstract. Symmetry in constraint satisfaction problems (CSP) can be used to either compute only a subset of the total solution set, or to prune branches of the search tree. However, detecting symmetry in general is a difficult task. In this paper, we address the problem of detecting and exploiting a particular class of symmetry called intensional permutability, which is based on the notion of interchangeability between variables and can be detected with a very small overhead. This kind of symmetry is detected by collecting information on symmetrical properties of individual constraints. This method works particularly well on problems designed using global constraints. We show how intensional permutability dramatically reduces the search tree for some problems. We propose a simple method to exploit it, which can be implemented as a lightweight extension to most resolution algorithms based on backtracking. We illustrate the method on several symmetrical problems, such as a classical layout problem and the pigeonhole problem, stated with a global constraint. Finally, we extend the method to symmetries involving groups of variables.