We generalise the linear programming relaxation approach to Weighted CSP by Schlesinger and the max-sum diffusion algorithm by Koval and Kovalevsky twice: from Weighted CSP to Semiring CSP, and from binary networks to networks of arbitrary arity. This generalisation reveals a deep property of constraint networks on commutative semirings: by locally changing constraint values, any network can be transformed into an equivalent form in which all corresponding marginals of each constraint pair coincide. We call this state marginal consistency. It corresponds to a local minimum of an upper bound on the Semiring CSP. We further show that a hierarchy of gradually tighter bounds is obtained by adding neutral constraints with higher arity. We argue that marginal consistency is a fundamental concept to unify local consistency techniques in constraint networks on commutative semirings
