Modeling Soft Global Constraints as Linear Programs in Weighted Constraint Satisfaction

The solving of Weighted CSP (WCSP) with global constraints relies on powerful consistency techniques, but enforcing these consistencies on soft global constraints is not a trivial task. Lee and Leung suggest that a soft global constraint can be used practically if we can find its minimum cost and perform projections/extensions on it in polynomial time, at the same time projections and extensions should not destroy those conditions. However, there are many useful constraints, whose minimum costs cannot be found in polynomial time. In this paper, we propose a special class of soft global constraints which can be modeled as integer linear programs. We show that they are soft linear projection-safe and their minimum cost can be computed by integer programming. By linear relaxation we can avoid the exponential time taken to solve the integer programs, as the approximation of their actual minimum costs can be obtained to serve as a good lower bound in enforcing the approximated consistency notions. While less pruning can be done, our approach allows much more efficient consistency enforcement, and we demonstrate the efficiency of such approaches experimentally.