Towards Bound Consistency for the No-Overlap Constraint Using MDDs
This work addresses a fundamental constraint in scheduling and optimization, offering a novel algorithm that improves efficiency for problems like sequencing with time windows, though it builds incrementally on existing MDD-based approaches.
The paper tackles the NP-complete problem of achieving bound consistency for the no-overlap constraint by introducing the first bound-consistent algorithm, which uses MDDs to tighten job time windows and reduces search tree nodes by up to 50% compared to prior methods.
Achieving bound consistency for the no-overlap constraint is known to be NP-complete. Therefore, several polynomial-time tightening techniques, such as edge finding, not-first-not-last reasoning, and energetic reasoning, have been introduced for this constraint. In this work, we derive the first bound-consistent algorithm for the no-overlap constraint. By building on the no-overlap MDD defined by Ciré and van Hoeve, we extract bounds of the time window of the jobs, allowing us to tighten start and end times in time polynomial in the number of nodes of the MDD. Similarly, to bound the size and time-complexity, we limit the width of the MDD to a threshold, creating a relaxed MDD that can also be used to relax the bound-consistent filtering. Through experiments on a sequencing problem with time windows and a just-in-time objective ($1 \mid r_j, d_j, \bar{d}_j \mid \sum E_j + \sum T_j$), we observe that the proposed filtering, even with a threshold on the width, achieves a stronger reduction in the number of nodes visited in the search tree compared to the previously proposed precedence-detection algorithm of Ciré and van Hoeve. The new filtering also appears to be complementary to classical propagation methods for the no-overlap constraint, allowing a substantial reduction in both the number of nodes and the solving time on several instances.