Consistency techniques for polytime linear global cost functions in weighted constraint satisfaction

Lee and Leung make practical the consistency enforcement of global cost functions in Weighted Constraint Satisfaction Problems (WCSPs). The main idea of their approach lies in the derivation of polynomial time algorithms for the computation of the minimum cost of global cost functions. In this paper, we investigate how soft arc consistency can also be applied on global cost functions with no known efficient minimum cost computation algorithms. We propose polytime linear projection-safe (PLPS) cost functions, which have a polynomial size integer linear formulation and can maintain this good property across projection/extension operations. We observe that the minimum of the linear relaxation gives a good approximation to the minimum of the integer formulation. This is used as the basis for the enforcement of relaxed forms of existing soft arc consistencies. By using the linear formulations, we can easily enforce conjunctions of overlapping PLPS, which give stronger pruning power. We further propose polytime integral linear projection-safe (PILPS) cost functions, which are PLPS cost functions with guaranteed integral solutions to the linear relaxation. We prove theorems to compare the consistency strengths among PLPS, PILPS and their conjunctions. Extensive experimentations are conducted to compare our proposed algorithms against state of the art global cost functions consistency enforcement algorithms and integer programming. Empirical results agree with our theoretical predictions, and confirm orders of magnitude improvement in terms of pruning and runtime by our proposals.