Some Untractable Classes of Constraint Networks

The SAT problem being NP-complete, some authors have partitioned its instances in classes described by simple parameters (e.g. the size of a clause). Some of these classes have been shown NP-complete and some polynomial. It is known that the problem of the existence of a solution in a Constraint Network (CN) is NP-complete. But, for CNs having a particular structure (e.g. trees), we know polynomial algorithms to answer that question. Nevertheless, the literature has never proposed a partition of CNs in classes described by simple parameters (like for the instances of SAT). This paper proposes a partition of CNs in classes described by two simple and essential parameters. Then, for each class, we say if the problem of the existence of solutions is polynomial or NP-complete.