Soft constraint abstraction based on semiring homomorphism

The semiring-based constraint satisfaction problems (semiring CSPs), proposed by Bistarelli, Montanari and Rossi [S. Bistarelli, U. Montanari, F. Rossi, Semiring-based constraints solving and optimization, Journal of the ACM 44 (2) (1997) 201-236], is a very general framework of soft constraints. In this paper we propose an abstraction scheme for soft constraints that uses semiring homomorphism. To find optimal solutions of the concrete problem, we first work in the abstract problem and find its optimal solutions, and then use them to solve the concrete problem. In particular, we show that a mapping preserves optimal solutions if and only if it is an order-reflecting semiring homomorphism. Moreover, for a semiring homomorphism @a and a problem P over S, if t is optimal in @a(P), then there is an optimal solution [email protected]? of P such that [email protected]? has the same value as t in @a(P).

[1]  D. Dubois,et al.  The calculus of fuzzy restrictions as a basis for flexible constraint satisfaction , 1993, [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems.

[2]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[3]  Francesca Rossi,et al.  Abstracting soft constraints: Framework, properties, examples , 2002, Artif. Intell..

[4]  Francesca Rossi,et al.  Semiring-based constraint satisfaction and optimization , 1997, JACM.

[5]  Ugo Montanari,et al.  Networks of constraints: Fundamental properties and applications to picture processing , 1974, Inf. Sci..

[6]  R. Dechter.  to Constraint Satisfaction , 1991 .

[7]  Richard J. Wallace,et al.  Partial Constraint Satisfaction , 1989, IJCAI.

[8]  Azriel Rosenfeld,et al.  Scene Labeling by Relaxation Operations , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[9]  Thomas Schiex,et al.  Semiring-Based CSPs and Valued CSPs: Basic Properties and Comparison , 1995, Over-Constrained Systems.

[10]  Simon de Givry,et al.  Bounding the Optimum of Constraint Optimization Problems , 1997, CP.

[11]  Jf Baldwin,et al.  Encyclopedia of AI , 1992 .

[12]  Jérôme Lang,et al.  Uncertainty in Constraint Satisfaction Problems: a Probalistic Approach , 1993, ECSQARU.

[13]  Robert M. Haralick,et al.  Structural Descriptions and Inexact Matching , 1981, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[14]  Thomas Schiex,et al.  Possibilistic Constraint Satisfaction Problems or "How to Handle Soft Constraints?" , 1992, UAI.

[15]  Thomas Schiex,et al.  Valued Constraint Satisfaction Problems: Hard and Easy Problems , 1995, IJCAI.

[16]  Z. Ruttkay.  Fuzzy constraint satisfaction , 1994, Proceedings of 1994 IEEE 3rd International Fuzzy Systems Conference.

[17]  Francesca Rossi,et al.  Semiring-based constraint solving and optimization , 1997 .