Encoding partial constraint satisfaction in the semiring-based framework for soft constraints

The partial constraint satisfaction paradigm focuses on solving relaxations of problems that either do not admit solutions, or that are either impractical or impossible to solve completely. The semiring-based framework for soft constraints is a unifying model for a variety of extensions of the constraint satisfaction formalism. For example, the semiring-based framework can represent weighted, fuzzy, probabilistic and set-based constraint satisfaction problems. We discuss how the semiring-based framework for soft constraints can be used to model partial constraint satisfaction problems. We show how the semiring framework can be used to capture a notion of distance between a solution and a problem based on the known distance metrics used in the partial constraint satisfaction literature. These solution-problem distance metrics can be seen as providing lower-bounds on the distance between a problem and its relaxation.