A new Evaluation of Forward Checking and its Consequences on Efficiency of Tools for Decomposition of CSPs Philippe J´ egou Samba Ndojh Ndiaye Cyril Terrioux LSIS - UMR CNRS 6168 Universit´ e Paul C´ ezanne (Aix-Marseille 3) Avenue Escadrille Normandie-Niemen 13397 Marseille Cedex 20 (France) {philippe.jegou, samba-ndojh.ndiaye, cyril.terrioux}@univ-cezanne.fr Abstract In this paper, a new evaluation of the complexity of For- ward Checking for solving non-binary CSPs with finite do- mains is proposed. Unlike what is done usually, it does not consider the size of domains, but the size of the relations associated to the constraints. It may lead sometimes to de- fine better complexity bounds. By using this first result, we show that the tractability hierarchy proposed in [6] which compares different methods based on a decomposition of constraint networks can be seen from a new viewpoint. 1 Introduction It is well known that the CSP formalism and its gener- alizations to Valued CSPs offer interesting frameworks to express and solve various problems in numerous fields. A CSP can be considered as the problem of checking if a fi- nite set X of variables can be assigned in their domains of values given by D, while satisfying simultaneously a set C of constraints. Such an assignment is a solution of the CSP. Then the problem is generally to find one solution. Unfor- tunately, checking the existence of a solution of a CSP is NP-complete. So, for solving CSPs, different classes of al- gorithms have been proposed, which combine backtracking and filtering. From a practical viewpoint, these algorithms can be frequently efficient. Precisely, it is the case when an adapted level for filtering is considered to help back- tracking. Generally, this trade-off uses arc-consistency to filter the domain of unassigned variables during the search. The good level of filtering is generally situated between FC (Forward Checking) [7], which is the most restricted form of arc-consistency (one pass and limited form of filtering with arc-consistency) and MAC [13] which maintains arc- consistency on the whole resulting problem. While these al- gorithms can be really efficient from a practical viewpoint, their time complexity is O(S.m n ) where S is the size of the considered CSP, n the number of variables and m the max- imum size of domains of variables. This evaluation is then clearly driven by the size of domains. Different approaches have been proposed to improve these bounds, for example by exploiting structural proper- ties that exist frequently in real life problems. The interest for the exploitation of structural properties was observed in numerous domains as in CSP [4], in constraint optimiza- tion (VCSPs) [14, 2], in SAT, in relational databases, or in Bayesian or probabilistic networks. Formally, complexity results based on topological features of problems have been proposed. Generally, they rely on the properties of a tree- decomposition [12] or a hypertree-decomposition [6] of the constraint network which formalizes the structure and con- sequently allows to express topological properties. Given a tree-decomposition of width w, the time com- plexity of the best structural approaches is O(S.m w+1 ), with the guarantee to have w<n, and in many cases, w ≪ n. Given a hypertree-decomposition of width h, the time complexity is then O(S.r h ), with r the maxi- mum size of relations (tables) associated to constraints. [6] has shown that hypertree-decomposition is better than tree- decomposition, since h ≤ w. The practical interest of such approaches has been proved in some recent works around (V)CSPs [9, 10, 11, 2]. These empirical observations seem to contradict the theoretical results, since they rely on tree-decomposition while, in our knowledge, no approach based on hypertree- decomposition has shown a practical interest yet. From a first analysis, we can think that it is due to the fact that the complexity bounds based on hypertree-decomposition are often reached to the detriment of the practical efficiency.