On Global Warming (Softening Global Constraints)
  We describe soft versions of the global cardinality constraint and the regular constraint, with efficient filtering algorithms maintaining domain consistency. For both constraints, the softening is achieved by augmenting the underlying graph. The softened constraints can be used to extend the meta-constraint framework for over-constrained problems proposed by Petit, Regin and Bessiere. 
Willem Jan van Hoeve and Gilles Pesant and Louis-Martin Rousseau