Towards Bound Consistency for the No-Overlap Constraint Using MDDs

TL;DR

This paper introduces the first bound-consistent algorithm for the no-overlap constraint using MDDs, improving search efficiency in scheduling problems by tightening bounds and limiting MDD width.

Contribution

It develops a novel bound-consistent filtering algorithm for the no-overlap constraint based on MDDs, enhancing pruning and solving efficiency in scheduling.

Findings

Stronger reduction in search nodes compared to previous methods.

Effective even with MDD width thresholds, maintaining performance.

Complementary to classical methods, reducing solving time.

Abstract

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 Cir\'e 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 Cir\'e 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.