Minimization of locally-de ned submodular functions by optimal soft arc consistency

Submodular function minimization is a polynomially-solvable combinatorial problem. Unfortunately the best known general-purpose algorithms have high-order polynomial time complexity. In many applications the objective function is locally-de ned in that it is the sum of cost functions (also known as soft or valued constraints) whose arities are bounded by a constant. We prove that every Valued Constraint Satisfaction Problem with submodular cost functions has an equivalent instance on the same constraint scopes in which the actual minimum value of the objective function is rendered explicit. Such an equivalent instance is the result of establishing optimal soft arc consistency and can hence be found by solving a linear program. From a practical point of view, this provides us with an alternative algorithm for minimizing locally-de ned submodular functions. From a theoretical point of view, this brings to light a previously-unknown connection between submodularity and soft arc consistency.