Towards Bound Consistency for the No-Overlap Constraint Using MDDs

2026-01-21

AI Summary

Enforcing bound consistency for the non-overlapping constraint is NP-complete, rendering strong propagation intractable in polynomial time with existing methods. This work proposes the first bound consistency algorithm based on multi-valued decision diagrams (MDDs), leveraging the non-overlapping MDD structure introduced by Ciré and van Hoeve. By incorporating task time-window bounds, the algorithm effectively tightens start and end times within polynomial MDD node complexity. To manage computational overhead, it employs width-bounded relaxed MDDs. Evaluated on sequencing problems with time windows and just-in-time objectives, the approach substantially reduces search tree size and solving time compared to state-of-the-art precedence-detection algorithms and synergistically enhances traditional propagation techniques.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• no-overlap constraint
• bound consistency
• NP-complete
• MDD
• scheduling

Innovation

• bound consistency
• no-overlap constraint
• MDD
• relaxed MDD
• constraint propagation