The registers constraints are usually taken into account during the scheduling pass of an acyclic data dependence graph (DAG): any schedule of the instructions inside a basic block must bound the register requirement under a certain limit. In this work, we show how to handle the register pressure before the instruction scheduling of a DAG. We mathematically study an approach which consists in managing the exact upper-bound of the register need for all the valid schedules of a considered DAG, independently of the functional unit constraints. We call this computed limit the register saturation (RS) of the DAG. Its aim is to detect possible obsolete register constraints, i.e., when RS does not exceed the number of available registers. If it does, we add some serial edges to the original DAG such that the worst register need does not exceed the number of available registers. We propose an appropriate mathematical formalism for this problem. Our generic processor model takes into account superscalar, VLIW and EPIC/IA64 architectures. Our deeper analysis of the problem and our formal methods enable us to provide nearly optimal heuristics and strategies for register optimization in the face of ILP.