The Symmetric Alldiff Constraint

The symmetric alldiff constraint is a particular case of the alldiff constraint, a case in which variables and values are defined from the same set 5. That is, every variable represents an element c of S and its values represent the elements of S that are compatible with c. This constraint requires that all the values taken by the variables are different (similar to the classical alldiff constraint) and that if the variable representing the element / is assigned to the value representing the element j, then the variable representing the element j is assigned to the value representing the element ?'. This constraint is present in many real-world problems , such sports scheduling where it expresses matches between teams. In this paper, we show how to compute the arc consistency of this constraint in , where n is the number of involved variables and the domain of the variable i. We also propose a filtering algorithm of less complexity (O(rn)). 1 Introduction Constraint Satisfaction Problems (CSPs) involve finding values for problem variables subject to constraints on which combinations are acceptable. They are more and more used in real-life applications, such as frequency allocation , crew scheduling, time tabling, car sequencing, etc. [Simonis, 1996]. The general task of finding a solution in a constraint network being NP-hard, many researchers have concentrated on improving the efficiency of solving a CSP. Currently, it seems that a look-ahead approach is the most promising way. The purpose of this technique is to look at the values of the variables that are not yet instantiated and to remove values that cannot lead to a solution w.r.t. the current partial instantiation. Thus, it anticipates the detection of some failures by using a particular treatment after each modification of domain variables. A filtering algorithm is one such particular treatment. With respect to a partial instantiation, it removes once and for all certain inconsistencies that would have been discovered several times otherwise. Techniques ba,sed on filtering algorithms are thus quite important. Particularly, arc consistency caught the attention of many researchers, who then discovered a large number of algorithms. Furthermore, it is necessary to deal directly with the arity of the constraints because nonbiliary constraints lose much of their semantics when encoded into a set of binary constraints. (See [Regin, 1994].) This encoding leads, for example, to behavior that prunes much less for filtering algorithms handling it. When the …