Polynomially Decomposable Global Cost Functions in Weighted Constraint Satisfaction.

In maintaining consistencies, such as GAC*, FDGAC* and weak EDGAC*, for global cost functions, Weighted CSP (WCSP) solvers rely on the projection and extension operations, which entail the computation of the cost functions' minima. Tractability of this minimum computation is essential for efficient execution. Since projections/extensions modify the cost functions, an important issue is tractable projection-safety, concerning whether minimum cost computation remains tractable after projections/extensions. In this paper, we prove that tractable projection-safety is always possible for projections/extensions to/from the nullary cost function (W0), and always impossible for projections/extensions to/from n-ary cost functions for n > = 2. When n = 1, the answer is indefinite. We give a simple negative example, while Lee and Leung's flow-based projection-safe cost functions are also tractable projection-safe. We propose polynomially decomposable cost functions, which are amenable to tractable minimum computation. We further prove that the polynomial decomposability property is unaffected by projections/extensionsto/from unary cost functions. Thus, polynomially decomposable cost functions are tractable projection-safe. We show that the SOFT_AMONG, SOFT_REGULAR, SOFT_GRAMMAR and MAX_WEIGHT/MIN_WEIGHT are polynomially decomposable. They are embedded in a WCSP solver for extensive experiments to confirm the feasibility and efficiency of our proposal.