Toby Walsh

We describe how the propagator for the ALL-DIFFERENT constraint can be generalized to prune variables whose domains are not just simple finite domains. We show, for example, how it can be used to propagate set variables, multiset variables and variables which represent tuples of values. We also describe how the propagator for the global cardinality constraint (which is a generalization of the ALL-DIFFERENT constraint) can be generalized in a similar way. Experiments show that such propagators can be beneficial in practice, especially when the domains are large.

To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non-stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language (Van Hentenryck, P., Michael, L., Perron, L., & Regin, J.-C. (1999) Constraint programming in OPL. In G. Nadathur, (Ed.), Principles and practice of declarative programming, LNCS 1702, (pp. 97-116). Springer-Verlag). To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.

Finite-domain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finite-domain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finite-domain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq. (c) 2006 Elsevier B.V. All rights reserved.

Many real life optimization problems are defined in terms of both hard and soft constraints, and qualitative conditional preferences. However, there is as yet no single framework for combined reasoning about these three kinds of information. In this paper we study how to exploit classical and soft constraint solvers for handling qualitative preference statements such as those captured by the CP-nets model. In particular, we show how hard constraints are sufficient to model the optimal outcomes of a possibly cyclic CP-net, and how soft constraints can faithfully approximate the semantics of acyclic conditional preference statements whilst improving the computational efficiency of reasoning about these statements.

We describe some new propagators for breaking symmetries in constraint satisfaction problems. We also introduce symmetry breaking constraints to deal with symmetries acting simultaneously on variables and values, conditional symmetries, as well as symmeties acting on set and other types of variables. © Springer-Verlag Berlin Heidelberg 2006.

AMONG, COMMON and DISJOINT are global constraints useful in modelling problems involving resources. We study a number of variations of these constraints over integer and set variables. We show how computational complexity can be used to determine whether achieving the highest level of consistency is tractable. For tractable constraints, we present a polynomial propagation algorithm and compare it to logical decompositions with respect to the amount of constraint propagation. For intractable cases, we show in many cases that a propagation algorithm can be adapted from a propagation algorithm of a similar tractable one.