The power of propagation: when GAC is enough

Considerable effort in constraint programming has focused on the development of efficient propagators for individual constraints. In this paper, we consider the combined power of such propagators when applied to collections of more than one constraint. In particular we identify classes of constraint problems where such propagators can decide the existence of a solution on their own, without the need for any additional search. Sporadic examples of such classes have previously been identified, including classes based on restricting the structure of the problem, restricting the constraint types, and some hybrid examples. However, there has previously been no unifying approach which characterises all of these classes: structural, language-based and hybrid. In this paper we develop such a unifying approach and embed all the known classes into a common framework. We then use this framework to identify a further class of problems that can be solved by propagation alone.

[1]  Marc Gyssens,et al.  Closure properties of constraints , 1997, JACM.

[2]  Toby Walsh,et al.  Decompositions of All Different, Global Cardinality and Related Constraints , 2009, IJCAI.

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

[4]  Mats Carlsson,et al.  Global Constraint Catalogue: Past, Present and Future , 2007, Constraints.

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

[6]  Peter Jeavons,et al.  An Algebraic Approach to Multi-sorted Constraints , 2003, CP.

[7]  Charles J. Colbourn,et al.  The complexity of completing partial Latin squares , 1984, Discret. Appl. Math..

[8]  Alexandra Silva,et al.  Non-Deterministic Kleene Coalgebras , 2010, Log. Methods Comput. Sci..

[9]  Willem Jan van Hoeve,et al.  Global Constraints , 2006, Handbook of Constraint Programming.

[10]  Hubie Chen,et al.  Constraint satisfaction with succinctly specified relations , 2010, J. Comput. Syst. Sci..

[11]  Marko Samer,et al.  Tractable cases of the extended global cardinality constraint , 2009, Constraints.

[12]  Joachim Schimpf,et al.  ECLiPSe: A Platform for Constraint Logic Programming , 1997 .

[13]  David A. Cohen,et al.  A New Classs of Binary CSPs for which Arc-Constistency Is a Decision Procedure , 2003, CP.

[14]  David A. Cohen,et al.  Tractability by Approximating Constraint Languages , 2003, CP.

[15]  Georg Gottlob,et al.  Optimization Methods for the Partner Units Problem , 2011, CPAIOR.

[16]  Mark Wallace,et al.  Practical applications of constraint programming , 2004, Constraints.

[17]  Peter Jeavons,et al.  Classifying the Complexity of Constraints Using Finite Algebras , 2005, SIAM J. Comput..

[18]  Martin C. Cooper,et al.  Broken triangles: From value merging to a tractable class of general-arity constraint satisfaction problems , 2016, Artif. Intell..

[19]  Catriel Beeri,et al.  On the Desirability of Acyclic Database Schemes , 1983, JACM.

[20]  Joxan Jaffar,et al.  Principles and Practice of Constraint Programming – CP’99 , 1999, Lecture Notes in Computer Science.

[21]  Dániel Marx,et al.  The Complexity of Global Cardinality Constraints , 2009, 2009 24th Annual IEEE Symposium on Logic In Computer Science.

[22]  Xavier Lorca,et al.  Bin Repacking Scheduling in Virtualized Datacenters , 2011, CP.

[23]  Marc Gyssens,et al.  Decomposing Constraint Satisfaction Problems Using Database Techniques , 1994, Artif. Intell..

[24]  Georg Gottlob,et al.  A Comparison of Structural CSP Decomposition Methods , 1999, IJCAI.

[25]  Eugene C. Freuder.  A Sufficient Condition for Backtrack-Free Search , 1982, JACM.

[26]  Eugene C. Freuder.  Complexity of K-Tree Structured Constraint Satisfaction Problems , 1990, AAAI.

[27]  Toby Walsh,et al.  The Complexity of Reasoning with Global Constraints , 2007, Constraints.

[28]  Justin Pearson,et al.  Closure Functions and Width 1 Problems , 1999, CP.

[29]  Marc Gyssens,et al.  A unified theory of structural tractability for constraint satisfaction problems , 2008, J. Comput. Syst. Sci..

[30]  Martin C. Cooper,et al.  The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns* , 2016, 2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS).

[31]  Tomás Feder,et al.  The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory , 1999, SIAM J. Comput..

[32]  Martin C. Cooper,et al.  Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination , 2010, Artif. Intell..

[33]  Nina Narodytska,et al.  Constraint satisfaction problems: Convexity makes AllDifferent constraints tractable , 2013, Theor. Comput. Sci..

[34]  Ian P. Gent,et al.  Minion: A Fast Scalable Constraint Solver , 2006, ECAI.

[35]  Toby Walsh,et al.  Propagating Conjunctions of AllDifferent Constraints , 2010, AAAI.

[36]  D. Shmoys,et al.  The Promise of LP to Boost CSP Techniques for Combinatorial Problems , 2002 .

[37]  T. K. Satish Kumar,et al.  A Framework for Hybrid Tractability Results in Boolean Weighted Constraint Satisfaction Problems , 2008, CP.

[38]  Phokion G. Kolaitis,et al.  Constraint Satisfaction, Bounded Treewidth, and Finite-Variable Logics , 2002, CP.

[39]  Fahiem Bacchus,et al.  GAC Via Unit Propagation , 2007, CP.

[40]  Evgenij Thorstensen,et al.  Tractable Combinations of Global Constraints , 2013, CP.

[41]  Nicolas Beldiceanu,et al.  Global Constraint Catalog , 2005 .

[42]  David A. Cohen,et al.  Tractable Decision for a Constraint Language Implies Tractable Search , 2004, Constraints.

[43]  Martin C. Cooper,et al.  Tractable Constraints on Ordered Domains , 1995, Artif. Intell..

[44]  Christopher Jefferson,et al.  Structural Tractability of Propagated Constraints , 2008, CP.

[45]  Toby Walsh,et al.  Handbook of Constraint Programming , 2006, Handbook of Constraint Programming.

[46]  Martin Grohe,et al.  The complexity of homomorphism and constraint satisfaction problems seen from the other side , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..