Ways of Maintaining Arc Consistency in Search using the Cartesian Representation Marius-C˘alin Silaghi, Djamila Sam-Haroud, and Boi Faltings Artificial Intelligence Lab Swiss Institute of Technology Lausanne Lausanne, Switzerland {silaghi,djamila,faltings}@lia.di.epfl.ch Abstract. The search space of constraint satisfaction problems (CSPs) can be reduced by using value interchangeability. This notion consists of aggregating subsets of values that behave similarly on the future branches of the search. BT-CPR [8], is a typical backtracking algorithm using value interchangeability. It uses the Cartesian product representation of the search space (CPR) which aggregates partial solutions and proves particularly useful for finding and representing all solutions of a CSP. It is assessed that maintaining arc-consistency (MAC) is the most effi- cient general algorithm for solving hard problems. A few work on com- bining MAC with CPR exists. In this paper we study comparatively two other possible alternatives of MAC-CPR. 1 Introduction A lot of real world problems can be cast as Constraint Satisfaction Problems (CSP). A CSP, (V,C,D), is classically defined as a set V of variables x 1 ,x 2 , ..., x n , taking their values respectively in a set D of domains D 1 ,D 2 , ..., D n and con- strained by a set of constraints C = {C 1 , ..., C m }. If a constraint C i links the variables x i1 ,x i2 , ..., x i k i then it is defined as a subset of the Cartesian product D i1 × D i2 × ... × D i k i . k i gives the arity of the constraint C i . Two variables are neighbors if there is a constraint linking them. A tuple t satisfies a constraint C i if the projection of t on the variables linked by C i belongs to C i . A tuple with values for all variables in V is a solution for (V,C,D) if it satisfies all the constraints in C. Depending on the original problem we may need one, several, or all possible solutions. The task of extracting such solutions is NP-complete in general. An intuitive way around this complexity barrier is to structure the search space so that the exploration algorithm operates on aggregated subsets of data rather than on individual possible instantiations. This is the idea behind the Cartesian product representation (CPR) which aggregates partial solutions during back- tracking. The use of CPR was shown to bring improvements, especially to the problem of finding all solutions. MAC is one of the most powerful general search algorithms. It consists of interleaving backtracking with a notion of local consistency called Arc Consis- tency (AC). In [7] is presented an algorithm called backtracking dp that combines