Martin C. Cooper

Constraints, consistency and closure

Although the constraint satisfaction problem is NP-complete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper we describe a simple algebraic property which characterises all possible constraint types for which strong k-consistency is sufficient to ensure global consistency, for each k > 2. We give a number of examples to illustrate the application of this result.

Tractable constraints on ordered domains

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.

Arc consistency for soft constraints

The notion of arc consistency plays a central role in constraint satisfaction [R. Dechter, Constraint Processing, Morgan Kaufmann, San Mateo, CA, 2003]. It is known since the introduction of valued and semi-ring constraint networks in 1995 that the notion of local consistency can be extended to constraint optimisation problems defined by soft constraint frameworks based on an idempotent cost combination operator. This excludes non-idempotent operators such as + which define problems which are very important in practical applications such as MAX-CSP, where the aim is to minimise the number of violated constraints.In this paper, we show that using a weak additional axiom satisfied by most existing soft constraints proposals, it is possible to define a notion of soft arc consistency that extends the classical notion of arc consistency and this even in the case of non-idempotent cost combination operators. A polynomial time algorithm for enforcing this soft arc consistency exists and its space and time complexities are identical to that of enforcing arc consistency in CSPs when the cost combination operator is strictly monotonic (for example MAX-CSP).A directional version of arc consistency, first introduced by M.C. Cooper [Reduction operations in fuzzy or valued constraint satisfaction, Fuzzy Sets and Systems 134 (3) (2003) 311-342] is potentially even stronger than the non-directional version, since it allows non-local propagation of penalties. We demonstrate the utility of directional arc consistency by showing that it not only solves soft constraint problems on trees, but that it also implies a form of local optimality, which we call arc irreducibility.

The complexity of soft constraint satisfaction

Over the past few years there has been considerable progress in methods to systematically analyse the complexity of constraint satisfaction problems with specified constraint types. One very powerful theoretical development in this area links the complexity of a set of constraints to a corresponding set of algebraic operations, known as polymorphisms. In this paper we extend the analysis of complexity to the more general framework of combinatorial optimisation problems expressed using various forms of soft constraints. We launch a systematic investigation of the complexity of these problems by extending the notion of a polymorphism to a more general algebraic operation, which we call a multimorphism. We show that many tractable sets of soft constraints, both established and novel, can be characterised by the presence of particular multimorphisms. We also show that a simple set of NP-hard constraints has very restricted multimorphisms. Finally, we use the notion of multimorphism to give a complete classification of complexity for the Boolean case which extends several earlier classification results for particular special cases.

Characterising tractable constraints

Abstract Finding solutions to a binary 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. This paper considers restricted sets of constraints which are closed under permutation of the labels. We identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems, and for finding the equivalent minimal network. We also prove that the class of problems generated by any set of constraints not contained in this restricted set is NP-complete.

Soft arc consistency revisited

The Valued Constraint Satisfaction Problem (VCSP) is a generic optimization problem defined by a network of local cost functions defined over discrete variables. It has applications in Artificial Intelligence, Operations Research, Bioinformatics and has been used to tackle optimization problems in other graphical models (including discrete Markov Random Fields and Bayesian Networks). The incremental lower bounds produced by local consistency filtering are used for pruning inside Branch and Bound search. In this paper, we extend the notion of arc consistency by allowing fractional weights and by allowing several arc consistency operations to be applied simultaneously. Over the rationals and allowing simultaneous operations, we show that an optimal arc consistency closure can theoretically be determined in polynomial time by reduction to linear programming. This defines Optimal Soft Arc Consistency (OSAC). To reach a more practical algorithm, we show that the existence of a sequence of arc consistency operations which increases the lower bound can be detected by establishing arc consistency in a classical Constraint Satisfaction Problem (CSP) derived from the original cost function network. This leads to a new soft arc consistency method, called, Virtual Arc Consistency which produces improved lower bounds compared with previous techniques and which can solve submodular cost functions. These algorithms have been implemented and evaluated on a variety of problems, including two difficult frequency assignment problems which are solved to optimality for the first time. Our implementation is available in the open source toulbar2 platform.

Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination

The Constraint Satisfaction Problem (CSP) is a central generic problem in artificial intelligence. Considerable progress has been made in identifying properties which ensure tractability in such problems, such as the property of being tree-structured. In this paper we introduce the broken-triangle property, which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure. We show that the broken-triangle property is conservative (i.e., it is preserved under domain reduction and hence under arc consistency operations) and that there is a polynomial-time algorithm to determine an ordering of the variables for which the broken-triangle property holds (or to determine that no such ordering exists). We also present a non-conservative extension of the broken-triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time. We show that both the broken-triangle property and its extension can be used to eliminate variables, and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency. Finally, we also discuss the possibility of using the broken-triangle property in variable-ordering heuristics.

Reduction operations in fuzzy or valued constraint satisfaction

In the constraint satisfaction problem (CSP) knowledge about a relation on n variables is given in the form of constraints on subsets of the variables. A solution to the CSP is simply an instantiation of the n variables which satisfies all the constraints. The fuzzy constraint satisfaction problem (FCSP) is a generalisation of the CSP to the case in which the constraints are not categorical but represent preferences in the form of fuzzy sets. The FCSP is then an optimisation problem in the space of all possible instantiations of the n variables. It has potentially many more applications in real-world problems than the CSP.When the aim is to maximise the value of the most violated constraint, then the FCSP can be solved by solving a logarithmic number of CSPs. Fuzzy k-consistency can even be established by an algorithm whose worst-case complexity is identical to that of an optimal k-consistency algorithm for CSPs.When the operator used to aggregate constraint values is strictly monotonic (as is the case in MAX-CSP, for example) the classic operation of arc consistency has to be completely redefined. We show that an ordered version of arc consistency exists for all FCSPs with a strictly monotonic aggregation operator, and that, in the case of MAX-CSP, it can be established by an algorithm whose worst-case complexity is identical to that of the optimal arc consistency algorithm for CSPs.Neighbourhood substitution in CSPs can be generalised to fuzzy neighbourhood substitution in FCSPs, whether the aggregation operator is strictly monotonic or idempotent. We give an algorithm for applying fuzzy neighbourhood substitutions until convergence based on the corresponding algorithm for CSPs.

Generalising submodularity and horn clauses: Tractable optimization problems defined by tournament pair multimorphisms

The submodular function minimization problem (SFM) is a fundamental problem in combinatorial optimization and several fully combinatorial polynomial-time algorithms have recently been discovered to solve this problem. The most general versions of these algorithms are able to minimize any submodular function whose domain is a set of tuples over any totally-ordered finite set and whose range includes both finite and infinite values. In this paper we demonstrate that this general form of SFM is just one example of a much larger class of tractable discrete optimization problems defined by valued constraints. These tractable problems are characterized by the fact that their valued constraints have an algebraic property which we call a tournament pair multimorphism. This larger tractable class also includes the problem of satisfying a set of Horn clauses (Horn-SAT), as well as various extensions of this problem to larger finite domains.

An algebraic characterisation of complexity for valued constraint

Classical constraint satisfaction is concerned with the feasibility of satisfying a collection of constraints. The extension of this framework to include optimisation is now also being investigated and a theory of so-called soft constraints is being developed. In this extended framework, tuples of values allowed by constraints are given desirability weightings, or costs, and the goal is to find the most desirable (or least cost) assignment. The complexity of any optimisation problem depends critically on the type of function which has to be minimized. For soft constraint problems this function is a sum of cost functions chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are rational numbers or infinite the complexity of a soft constraint problem is determined by certain algebraic properties of the valued constraint language, which we call feasibility polymorphisms and fractional polymorphisms. As an immediate application of these results, we show that the existence of a non-trivial fractional polymorphism is a necessary condition for the tractability of a valued constraint language with rational or infinite costs over any finite domain (assuming P ≠ NP).