Tractable Combinations of Global Constraints

We study the complexity of constraint satisfaction problems involving global constraints, i.e., special-purpose constraints provided by a solver and represented implicitly by a parametrised algorithm. Such constraints are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. Previous work has focused on the development of efficient propagators for individual constraints. In this paper, we identify a new tractable class of constraint problems involving global constraints of unbounded arity. To do so, we combine structural restrictions with the observation that some important types of global constraint do not distinguish between large classes of equivalent solutions.Comment: To appear in proceedings of CP'13, LNCS 8124. arXiv admin note: text overlap with arXiv:1307.179

Structural Decompositions for Problems with Global Constraints

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed combinations of values, or implicitly, by special-purpose algorithms provided by a solver. Such implicitly represented constraints, known as global constraints, are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. In recent years, a variety of restrictions on the structure of CSP instances have been shown to yield tractable classes of CSPs. However, most such restrictions fail to guarantee tractability for CSPs with global constraints. We therefore study the applicability of structural restrictions to instances with such constraints. We show that when the number of solutions to a CSP instance is bounded in key parts of the problem, structural restrictions can be used to derive new tractable classes. Furthermore, we show that this result extends to combinations of instances drawn from known tractable classes, as well as to CSP instances where constraints assign costs to satisfying assignments.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s10601-015-9181-

Propagators and Solvers for the Algebra of Modular Systems

To appear in the proceedings of LPAR 21. Solving complex problems can involve non-trivial combinations of distinct knowledge bases and problem solvers. The Algebra of Modular Systems is a knowledge representation framework that provides a method for formally specifying such systems in purely semantic terms. Formally, an expression of the algebra defines a class of structures. Many expressive formalism used in practice solve the model expansion task, where a structure is given on the input and an expansion of this structure in the defined class of structures is searched (this practice overcomes the common undecidability problem for expressive logics). In this paper, we construct a solver for the model expansion task for a complex modular systems from an expression in the algebra and black-box propagators or solvers for the primitive modules. To this end, we define a general notion of propagators equipped with an explanation mechanism, an extension of the alge- bra to propagators, and a lazy conflict-driven learning algorithm. The result is a framework for seamlessly combining solving technology from different domains to produce a solver for a combined system.Comment: To appear in the proceedings of LPAR 2

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

Structural Tractability of Propagated Constraints

Modern constraint solvers do trot require constraints to l), represented using ally particular data structure. Instead, constraints rue given as black boxes known as propagators. Propagators are given a. list of current domains for variables and are :allowed to prune values not. consistent with these current domains.Using propagation as the only primitive operation on constraints imposes restrictions on the operations that can be performed in polynomial time. In the extensional representation of constraints (so-called positive table constraints) join and project are primitive polynomial-time operations. This is not true for propagated constraints.The question we pose ill this paper is: If propagation is the only primitive operation, what are: the: structurally tractable classes of constraint programs (whose instances can he solved in polynomial time)?We consider a hierarchy of propagators: arbitrary propagators, whose only ability is consistency checking; partial assignment membership propagators, which allow us to check partial assignments; and generalised arc consistency propagators, the strongest, form of propagator.In the first two cases, we answer the posed question by establishing dichotomies. In the case of generalised are consistency propagators, we achieve a useful dichotomy in the restricted case: of acyclic structures.</p

Structural Tractability of Propagated Constraints

Modern constraint solvers do trot require constraints to l), represented using ally particular data structure. Instead, constraints rue given as black boxes known as propagators. Propagators are given a. list of current domains for variables and are :allowed to prune values not. consistent with these current domains.Using propagation as the only primitive operation on constraints imposes restrictions on the operations that can be performed in polynomial time. In the extensional representation of constraints (so-called positive table constraints) join and project are primitive polynomial-time operations. This is not true for propagated constraints.The question we pose ill this paper is: If propagation is the only primitive operation, what are: the: structurally tractable classes of constraint programs (whose instances can he solved in polynomial time)?We consider a hierarchy of propagators: arbitrary propagators, whose only ability is consistency checking; partial assignment membership propagators, which allow us to check partial assignments; and generalised arc consistency propagators, the strongest, form of propagator.In the first two cases, we answer the posed question by establishing dichotomies. In the case of generalised are consistency propagators, we achieve a useful dichotomy in the restricted case: of acyclic structures.</p