Constraint propagation is one of the techniques central to the success of constraint programming. To reduce search, fast algorithms associated with each constraint prune the domains of variables. With global (or non-binary) constraints, the cost of such propagation may be much greater than the quadratic cost for binary constraints. We therefore study the computational complexity of reasoning with global constraints. We first characterise a number of important questions related to constraint propagation. We show that such questions are intractable in general, and identify dependencies between the tractability and intractability of the different questions. We then demonstrate how the tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when constraints can be safely generalized, when decomposing constraints will reduce the amount of pruning, and when combining constraints is tractable.
Analysis of tractability and intractability is not new in constraint programming. Identifying properties under which a constraint satisfaction problem is tractable has been studied for a long time. For example, Freuder @cite_12 , Dechter and Pearl @cite_13 @cite_33 or @cite_17 gave increasingly general conditions on the structure of the underlying (hyper)graph to obtain a backtrack-free resolution of a problem. van Beek and Dechter @cite_30 and @cite_39 presented conditions on the semantics of the individual constraints that make the problem tractable. Finally, @cite_36 showed that when the constraints composing a problem are defined as disjunctions of other constraints of specified types, then the whole problem is tractable. However, these lines of research are concerned with a constraint satisfaction problem as a whole, and do not say much about individual particular constraints.
@cite_30 @cite_33 @cite_36 @cite_39 @cite_13 @cite_12 @cite_17
2026753685 1861000399 2016639451 2951647103
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra. We introduce a notion of "tractable algebra," and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer's dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem. Many combinatorial search problems can be expressed as 'constraint satisfaction problems', and this class of problems is known to be NP-complete in general. In this paper we investigate 'disjunctive constraints', that is, constraints which have the form of the disjunction of two constraints of specified types. We show that when the constraint types involved in the disjunction have a certain property, which we call 'independence', and when a certain restricted class of problems is tractable, then the class of all problems involving these disjunctive constraints is tractable. We give examples to show that many known examples of tractable constraint classes arise in this way, and derive new tractable classes which have not previously been identified. Abstract Finding solutions to a constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NP-complete. We consider discrete pairwise energy minimization problem (weighted constraint satisfaction, max-sum labeling) and methods that identify a globally optimal partial assignment of variables. When finding a complete optimal assignment is intractable, determining optimal values for a part of variables is an interesting possibility. Existing methods are based on different sufficient conditions. We propose a new sufficient condition for partial optimality which is: (1) verifiable in polynomial time (2) invariant to reparametrization of the problem and permutation of labels and (3) includes many existing sufficient conditions as special cases. We pose the problem of finding the maximum optimal partial assignment identifiable by the new sufficient condition. A polynomial method is proposed which is guaranteed to assign same or larger part of variables than several existing approaches. The core of the method is a specially constructed linear program that identifies persistent assignments in an arbitrary multi-label setting.