Tractability by Approximating Constraint Languages

A constraint satisfaction problem instance consists of a collection of variables that need to have values assigned to them. The assignments are limited by constraints that force the values taken by certain collections of variables (the constraint scopes) to satisfy specified properties (the constraint relations). As the general CSP problem is NP-hard there has been significant effort devoted to discovering tractable subproblems of the CSP. The structure of a CSP instance is defined to be the hypergraph formed by the constraint scopes. Restricting the possible structure of the CSP instances has been a successful way of identifying tractable subproblems. The language of a CSP instance is defined to be the set of constraint relations of the instance. Restricting the language allowed for CSP instances has also yielded many interesting tractable subproblems. Almost all known tractable subproblems are either structural or relational. In this paper we construct tractable subproblems of the general CSP that are neither defined by structural nor relational properties. These new tractable classes are related to tractable languages in much the same way that general decompositions (cutset, tree-clustering, etc.) are related to acyclic decompositions. It may well be that our results will begin to make language based tractability of more practical interest. We show that our theory allows us to properly extend the binary max-closed language based tractable class, which is maximal as a tractable binary constraint language. Our theory also explains the tractability of the constraint representation of the Stable Marriage Problem which has not been amenable to existing explanations of tractability. In fact we provide a uniform explanation for the tractability of the class of max-closed CSPs and the SMP. There has been much work done on so called renamable HORN theories which are a tractable subproblem of SAT. It has been shown that renamable HORN theories are tractably identifiable and solvable. It has also been shown that finding the largest sub-theory that is renamable HORN is NP-hard. These results also follow immediately from our theory.

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

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

[3]  David Manlove,et al.  A Constraint Programming Approach to the Stable Marriage Problem , 2001, CP.

[4]  Nadia Creignou,et al.  A Dichotomy Theorem for Maximum Generalized Satisfiability Problems , 1995, J. Comput. Syst. Sci..

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

[6]  Thomas J. Schaefer,et al.  The complexity of satisfiability problems , 1978, STOC.

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

[8]  Endre Boros.  Maximum Renamable Horn sub-CNFs , 1999, Discret. Appl. Math..

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

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

[11]  Alvaro del Val.  On 2-SAT and Renamable Horn , 2000, AAAI/IAAI.

[12]  Bengt Aspvall,et al.  Recognizing Disguised NR(1) Instances of the Satisfiability Problem , 1980, J. Algorithms.

[13]  Ronald Fagin,et al.  Degrees of acyclicity for hypergraphs and relational database schemes , 1983, JACM.

[14]  L. Shapley,et al.  College Admissions and the Stability of Marriage , 1962 .

[15]  Harry R. Lewis,et al.  Renaming a Set of Clauses as a Horn Set , 1978, JACM.

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

[17]  Peter L. Hammer,et al.  On renamable Horn and generalized Horn functions , 2005, Annals of Mathematics and Artificial Intelligence.

[18]  Rina Dechter,et al.  Network-based heuristics for constraint satisfaction problems , 1988 .

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

[20]  John N. Hooker,et al.  Extended Horn sets in propositional logic , 1991, JACM.

[21]  Georg Gottlob,et al.  Hypertree Decompositions: A Survey , 2001, MFCS.

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

[23]  Peter Bro Miltersen,et al.  On Pseudorandom Generators in NC , 2001, MFCS.

[24]  Robert E. Tarjan,et al.  A Linear-Time Algorithm for Testing the Truth of Certain Quantified Boolean Formulas , 1979, Inf. Process. Lett..