Abstract Powerful consistency techniques, such as AC* and FDAC*, have been developed for Weighted Constraint Satisfaction Problems (WCSPs) to reduce the space in solution search, but are restricted to only unary and binary constraints. On the other hand, van Hoeve et al. developed effifi- cient graph-based algorithms for handling soft constraints as classical constraint optimization problems. We prove that naively incorporating van Hoeve’s method into the WCSP framework can enforce a strong form of ∅-Inverse Consistency, which can prune infeasible values and deduce good lower bound estimates. We further show how Van Hoeve’s method can be modifified so as to handle cost projection and extension to maintain the stronger AC* and FDAC* generalized for non-binary constraints. Using the soft allDifferent constraint as a testbed, preliminary results demonstrate that our proposal gives improvements up to an order of magnitude both in terms of time and pruning