Decomposition of the NVALUE constraint

We study decompositions of NVALUE, a global constraint that can be used to model a wide range of problems where values need to be counted. Whilst decomposition typically hinders propagation, we identify one decomposition that maintains a global view as enforcing bound consistency on the decomposition achieves bound consistency on the original global NVALUE constraint. Such decompositions offer the prospect for advanced solving techniques like nogood learning and impact based branching heuristics. They may also help SAT and IP solvers take advantage of the propagation of global constraints.

💡 Research Summary

The paper investigates how to decompose the global NVALUE constraint—a constraint that links a set of variables X₁,…,Xₙ to a variable N representing the number of distinct values taken by the X‑variables—while preserving strong propagation properties. The authors begin by reviewing the difficulty of enforcing domain consistency (DC) on NVALUE, which is known to be NP‑hard, and note that bound consistency (BC) can be achieved in polynomial time.

A first, naïve decomposition introduces binary indicator variables Bⱼ (j ∈