Arc Consistency on n-ary Monotonic and Linear Constraints

Many problems and applications can be naturally modelled and solved using constraints with more than two variables. Such n-ary constraints, in particular, arithmetic constraints are provided by many finite domain constraint programming systems. The best known worst case time complexity of existing algorithms (GAC-schema) for enforcing arc consistency on general CSPs is O(edn) where d is the size of domain, e is the number of constraints and n is the maximum number of variables in a single constraint. We address the question of efficient consistency enforcing for n-ary constraints. An observation here is that even with a restriction of n-ary constraints to linear constraints, arc consistency enforcing is NP-complete. We identify a general class of monotonic n-ary constraints (which includes linear inequalities as a special case). Such monotonic constraints can be made arc consistent in time O(en3d). The special case of linear inequalities can be made arc consistent in time O(en2d) using bounds-consistency which exploits special properties of the projection function.

[1]  Pascal Van Hentenryck.  Constraint satisfaction in logic programming , 1989, Logic programming.

[2]  Yahia Lebbah,et al.  Acceleration methods of numeric CSPc , 1998, AAAI 1998.

[3]  Pascal Van Hentenryck,et al.  Numerica: A Modeling Language for Global Optimization , 1997, IJCAI.

[4]  David L. Waltz,et al.  Generating Semantic Descriptions From Drawings of Scenes With Shadows , 1972 .

[5]  Philippe Codognet,et al.  Compiling Constraints in clp(FD) , 1996, J. Log. Program..

[6]  Rina Dechter,et al.  Temporal Constraint Networks , 1989, Artif. Intell..

[7]  Rina Dechter,et al.  Tree Clustering for Constraint Networks , 1989, Artif. Intell..

[8]  André Vellino,et al.  Constraint Arithmetic on Real Intervals , 1993, WCLP.

[9]  Michael J. Maher,et al.  Beyond Finite Domains , 1994, PPCP.

[10]  Ernest Davis,et al.  Constraint Propagation with Interval Labels , 1987, Artif. Intell..

[11]  Christian Bessiere,et al.  Arc-Consistency and Arc-Consistency Again , 1993, Artif. Intell..

[12]  A. Monfroglio.  Constraint satisfaction and parallel logic programming , 1986 .

[13]  Christos H. Papadimitriou,et al.  On the complexity of integer programming , 1981, JACM.

[14]  Olivier Lhomme,et al.  Consistency Techniques for Numeric CSPs , 1993, IJCAI.

[15]  Eero Hyvönen,et al.  Constraint Reasoning Based on Interval Arithmetic: The Tolerance Propagation Approach , 1992, Artif. Intell..

[16]  Christian Bessiere,et al.  MAC and Combined Heuristics: Two Reasons to Forsake FC (and CBJ?) on Hard Problems , 1996, CP.

[17]  Jean-Louis Laurière,et al.  A Language and a Program for Stating and Solving Combinatorial Problems , 1978, Artif. Intell..

[18]  Pascal Van Hentenryck,et al.  A Generic Arc-Consistency Algorithm and its Specializations , 1992, Artif. Intell..

[19]  Peter van Beek,et al.  On the conversion between non-binary constraint satisfaction problems , 1998, AAAI 1998.

[20]  Alan K. Mackworth.  On Reading Sketch Maps , 1977, IJCAI.

[21]  Pascal Van Hentenryck,et al.  CLP(Intervals) Revisited , 1994, ILPS.

[22]  Alan K. Mackworth.  Consistency in Networks of Relations , 1977, Artif. Intell..

[23]  Peter van Beek,et al.  On the minimality and global consistency of row-convex constraint networks , 1995, JACM.

[24]  Peter van Beek,et al.  On the Conversion between Non-Binary and Binary Constraint Satisfaction Problems , 1998, AAAI/IAAI.

[25]  Christian Bessiere,et al.  Arc Consistency for General Constraint Networks: Preliminary Results , 1997, IJCAI.

[26]  Jean-Charles Régin,et al.  Generalized Arc Consistency for Global Cardinality Constraint , 1996, AAAI/IAAI, Vol. 1.

[27]  Yahia Lebbah,et al.  Acceleration Methods for Numeric CSPs , 1998, AAAI/IAAI.

[28]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[29]  Charles J. Petrie,et al.  On the Equivalence of Constraint Satisfaction Problems , 1990, ECAI.

[30]  Frédéric Benhamou,et al.  Applying Interval Arithmetic to Real, Integer, and Boolean Constraints , 1997, J. Log. Program..

[31]  Roger Mohr,et al.  Good Old Discrete Relaxation , 1988, ECAI.