Extremal CSPs Nicolas Prcovic LSIS - Universit´ e Paul C´ ezanne Aix-Marseille III Facult´ e de St-J´ erˆ ome - Avenue Escadrille Normandie-Niemen - 13013 Marseille nicolas.prcovic@lsis.org Abstract. We present a new class of binary CSPs called extremal CSPs. The CSPs of this class are inconsistent but would become consistent if any pair of variable assignments among the forbidden ones was allowed. Being inconsistent, they cannot be solved by any local repair method. As they allow a great num- ber of partial (almost complete) solutions, they can be very hard to solve with tree search methods integrating domain filtering. We experiment that balanced extremal CSPs are much harder to solve than random CSPs of same size at the complexity peak. 1 Introduction Even if Constraint Satisfaction Problems are NP-complete problems, it is known that only a small part of the CSP set is hard to solve with the methods used until now. When the problem has a lot of solutions uniformly distributed in the search space, local repair methods are efficient. When a problem is much overconstrained, a tree search associated with a domain filtering technique can allow to prove quickly that the problem has no solution. Only the problems that resist to that two types of solving methods are usually considered as hard. Characterizations of hardness for CSPs have been found for random CSPs defined by some parameters (number of variables, domain size, graph density and constraint tightness). In this model, the CSPs for which parameters are such that they have prob- ability 0.5 to have a solution are longer to solve than the others. The drawback of this characterization is that it does not give a property that would ensure the hardness of the problem but only a stronger probability to be hard. In addition, some other general properties (e.g., backbone, minimal backdoor and unsatisfiable core size) for explaining problem hardness have been proposed. However, they do not allow a precise description of hard problem’s structure. In this paper, we propose a simple yet precise description of CSPs that are very hard to solve with the usual and standard solving techniques. 2 Resistant CSPs We place ourselves in the context of binary CSPs, where each problem is defined by a triple (X, D, C), where X={x 1 , ..., x n } is the variable set, D={D 1 , ..., D n } is the discrete and finite domain set (where each domain D i contains the values that can be assigned to variable x i ) and C is the constraint set that explicits the allowed variable