Dual Modelling of Permutation and Injection Problems

TL;DR

This paper investigates primal, dual, and combined modeling approaches for permutation and injection problems, demonstrating that multiple viewpoints with channelling constraints can improve problem-solving effectiveness through extensive theoretical and empirical analysis.

Contribution

It provides a comprehensive comparison of primal, dual, and combined models for permutation and injection problems, highlighting the benefits of multi-viewpoint modeling with channelling constraints.

Findings

Multiple viewpoints often outperform single-view models.

Channelling constraints help maintain consistency and improve solving efficiency.

The methodology aids in selecting effective constraint models.

Abstract

When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many values as variables, and each variable takes an unique value. In such problems, we can choose between a primal and a dual viewpoint. In the dual viewpoint, each dual variable represents one of the primal values, whilst each dual value represents one of the primal variables. Alternatively, by means of channelling constraints to link the primal and dual variables, we can have a combined model with both sets of variables. In this paper, we perform an extensive theoretical and empirical study of such primal, dual and combined models for two classes of problems: permutation problems and injection problems. Our results show that it often be advantageous to use multiple viewpoints, and to have constraints which channel between them to maintain consistency. They also illustrate a general methodology for comparing different constraint models.