Decomposition of the NVALUE constraint

TL;DR

This paper investigates the decomposition of the NVALUE global constraint, revealing that decompositions can match the time complexity of global propagators but often require more space, and shows how to enforce range consistency efficiently.

Contribution

It provides theoretical insights into the space and time complexity of decompositions of NVALUE and demonstrates how to enforce range consistency with the same complexity as bound consistency.

Findings

Decompositions can simulate global propagators with similar time complexity.

Range consistency can be enforced on NVALUE with the same worst-case time as bound consistency.

Decompositions can be encoded as linear inequalities for use in integer linear programming.

Abstract

We study decompositions of the global NVALUE constraint. Our main contribution is theoretical: we show that there are propagators for global constraints like NVALUE which decomposition can simulate with the same time complexity but with a much greater space complexity. This suggests that the benefit of a global propagator may often not be in saving time but in saving space. Our other theoretical contribution is to show for the first time that range consistency can be enforced on NVALUE with the same worst-case time complexity as bound consistency. Finally, the decompositions we study are readily encoded as linear inequalities. We are therefore able to use them in integer linear programs.