The concept of local consistency-making global deductions from local infeasibility-is central to constraint programming. When reasoning about NP-complete constraints, however, since achieving a "complete" form of local consistency is often considered too hard, we need other tools to design and analyze propagation algorithms. In this paper, we argue that NP-complete constraints are an essential part of constraint programming , that designing dedicated methods has lead to, and will bring, significant breakthroughs, and that we need to carefully investigate methods to deal about a necessarily incomplete inference. In particular, we advocate the use of fixed-parameter tractability and kernelization to this purpose.