Sequence Variables: A Constraint Programming Computational Domain for Routing and Sequencing

2025-10-10

AI Summary

Traditional constraint programming (CP) relies on successor-variable modeling, which hinders natural representation of optional visits and insertion-based heuristic search, limiting its applicability to vehicle routing problems (VRPs). To address this, we introduce the “sequence variable” as a novel computational domain in CP, formally defining its semantics, domain structure, and constraint consistency levels for the first time. We design sequence-oriented global constraints with efficient propagation algorithms and integrate them into a trail-based solver. This framework unifies path construction, optional-node modeling, and large-neighborhood insertion search. Evaluated on appointment-based pickup-and-delivery problems, our model achieves greater conciseness and competitive solving performance. The results demonstrate dual advances: enhanced modeling flexibility and improved computational efficiency—marking a significant step toward bridging the gap between declarative modeling and constructive search in VRP solving.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Programming
• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Security and Privacy: Tracking, profiling, and countermeasures against them

Abstract

Constraint Programming (CP) offers an intuitive, declarative framework for modeling Vehicle Routing Problems (VRP), yet classical CP models based on successor variables cannot always deal with optional visits or insertion based heuristics. To address these limitations, this paper formalizes sequence variables within CP. Unlike the classical successor models, this computational domain handle optional visits and support insertion heuristics, including insertion-based Large Neighborhood Search. We provide a clear definition of their domain, update operations, and introduce consistency levels for constraints on this domain. An implementation is described with the underlying data structures required for integrating sequence variables into existing trail-based CP solvers. Furthermore, global constraints specifically designed for sequence variables and vehicle routing are introduced. Finally, the effectiveness of sequence variables is demonstrated by simplifying problem modeling and achieving competitive computational performance on the Dial-a-Ride Problem.

Problem

• Formalizes sequence variables to handle optional visits in routing problems
• Enables insertion heuristics and Large Neighborhood Search in CP models
• Simplifies modeling and improves performance for Dial-a-Ride Problems

Innovation

• Sequence variables handle optional visits in routing
• Supports insertion heuristics like Large Neighborhood Search
• Introduces global constraints specifically for vehicle routing