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-

A maximal tractable class of soft constraints

Many researchers in artificial intelligence are beginning to explore the use of soft constraints to express a set of (possibly conflicting) problem requirements. A soft constraint is a function defined on a collection of variables which associates some measure of desirability with each possible combination of values for those variables. However, the crucial question of the computational complexity of finding the optimal solution to a collection of soft constraints has so far received very little attention. In this paper we identify a class of soft binary constraints for which the problem of finding the optimal solution is tractable. In other words, we show that for any given set of such constraints, there exists a polynomial time algorithm to determine the assignment having the best overall combined measure of desirability. This tractable class includes many commonly-occurring soft constraints, such as 'as near as possible' or 'as soon as possible after', as well as crisp constraints such as 'greater than'. Finally, we show that this tractable class is maximal, in the sense that adding any other form of soft binary constraint which is not in the class gives rise to a class of problems which is NP-hard

Characterising the Complexity of Constraint Satisfaction Problems Defined by 2-Constraint Forbidden Patterns

International audienceAlthough the CSP (constraint satisfaction problem) is NP-complete, even in the case when all constraints are binary, certain classes of instances are tractable. We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes. The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem. We demonstrate a dichotomy in the case of forbidden patterns consisting of either one or two constraints. This has allowed us to discover several new tractable classes including, for example, a novel generalisation of 2SAT. We then extend this dichotomy to existential patterns whic hare only forbidden on specific domain values