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.
Constraint networks have been shown to be useful in formulating such diverse problems as scene labeling, natural language parsing, and temporal reasoning. Given a constraint network, we often wish to (i) find a solution that satisfies the constraints and (ii) find the corresponding minimal network where the constraints are as explicit as possible. Both tasks are known to be NP-complete in the general case. Task (1) is usually solved using a backtracking algorithm, and task (ii) is often solved only approximately by enforcing various levels of local consistency. In this paper, we identify a property of binary constraint called row convexity and show its usefulness in deciding when a form of local consistency called path consistency is sufficient to guarantee that a network is both minimal and globally consistent. Globally consistent networks have the property that a solution can be found without backtracking. We show that one can test for the row convexity property efficiently and we show, by examining applications of constraint networks discussed in the literature, that our results are useful in practice. Thus, we identify a class of binary constraint networks for which we can solve both tasks (i) and (ii) efficiently. Finally, we generalize the results for binary constraint networks to networks with nonbinary constraints. Abstract The paper offers a systematic way of regrouping constraints into hierarchical structures capable of supporting search without backtracking. The method involves the formation and preprocessing of an acyclic database that permits a large variety of queries and local perturbations to be processed swiftly, either by sequential backtrack-free procedures, or by distributed constraint propagation processes. 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. This paper studies constraint satisfaction over connected row-convex (CRC) constraints. It shows that CRC constraints are closed under composition, intersection, and transposition, the basic operations of path-consistency algorithms. This establishes that path consistency over CRC constraints produces a minimal and decomposable network and is thus a polynomial-time decision procedure for CRC networks. This paper also presents a new path-consistency algorithm for CRC constraints running in time O(n(3)d(2)) and space O(n(2)d), where n is the number of variables and d is the size of the largest domain, improving the traditional time and space complexity by orders of magnitude. The paper also shows how to construct CRC constraints by conjunction and disjunction of a set of basic CRC constraints, highlighting how CRC constraints generalize monotone constraints and presenting interesting subclasses of CRC constraints. Experimental results show that the algorithm behaves well in practice. (C) 1999 Elsevier Science B.V. All rights reserved.   Abstract We compare tractable classes of constraint satisfaction problems (CSPs). We first give a uniform presentation of the major structural CSP decomposition methods. We then introduce a new class of tractable CSPs based on the concept of hypertree decomposition recently developed in Database Theory, and analyze the cost of solving CSPs having bounded hypertree-width. We provide a framework for comparing parametric decomposition-based methods according to tractability criteria and compare the most relevant methods. We show that the method of hypertree decomposition dominates the others in the case of general CSPs (i.e., CSPs of unbounded arity). We also make comparisons for the restricted case of binary CSPs. Finally, we consider the application of decomposition methods to the dual graph of a hypergraph. In fact, this technique is often used to exploit binary decomposition methods for nonbinary CSPs. However, even in this case, the hypertree-decomposition method turns out to be the most general method.