Strong polynomiality of resource constraint propagation

Constraint-based schedulers have been widely successful in tackling complex, disjunctive, and cumulative scheduling applications by combining tree search and constraint propagation. The constraint-propagation step is a fixpoint algorithm that applies pruning operators to tighten the release and due dates of activities using precedence or resource constraints. A variety of pruning operators for resource constraints have been proposed; they are based on edge finding or energetic reasoning and handle a single resource.Complexity results in this area are only available for a single application of these pruning operators, which is problematic for at least two reasons. On the one hand, the operators are not idempotent, so a single application is rarely sufficient. On the other hand, the operators are not used in isolation but interact with each other. Existing results thus provide a very partial picture of the complexity of propagating resource constraints in constraint-based scheduling.This paper aims at addressing these limitations. It studies the complexity of applying pruning operators for resource constraints to a fixpoint. In particular, it shows that: (1) the fixpoint of the edge finder for both release and due dates can be reached in strongly polynomial time for disjunctive scheduling; (2) the fixpoint can be reached in strongly polynomial time for updating the release dates or the due dates but not both for the cumulative scheduling; and (3) the fixpoint of “reasonable” energetic operators cannot be reached in strongly polynomial time, even for disjunctive scheduling and even when only the release dates or the due dates are considered.