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.

Decision tree induction techniques attempt to find small trees that fit a training set of data. This preference for smaller trees, which provides a learning bias, is often justified as being consistent with the principle of Occam's Razor. Informally, this principle states that one should prefer the simpler hypothesis. In this paper we take this principle to the extreme. Specifically, we formulate decision tree induction as a combinatorial optimisation problem in which the objective is to minimise the number of nodes in the tree. We study alternative formulations based on satisfiability, constraint programming, and hybrids with integer linear programming. We empirically compare our approaches against standard induction algorithms, showing that the decision trees we obtain can sometimes be less than half the size of those found by other greedy methods. Furthermore, our decision trees are competitive in terms of accuracy on a variety of well-known benchmarks, often being the most accurate. Even when post-pruning of greedy trees is used, our constraint-based approach is never dominated by any of the existing techniques.

Abstract.To improve solution robustness, we introduce the concept of super solutions to constraint programming. An (a, b)-super solution is one in which if a variables lose their values, the solution can be repaired by assigning these variables with a new values and at most b other variables. Super solutions are a generalization of supermodels in propositional satisfiability. We focus in this paper on (1,0)-super solutions, where if one variable loses its value, we can find another solution by re-assigning this variable with a new value. To find super solutions, we explore methods based both on reformulation and on search. Our reformulation methods transform the constraint satisfaction problem so that the only solutions are super solutions. Our search methods are based on a notion of super consistency. Experiments show that super MAC, a novel search-based method shows considerable promise. When super solutions do not exist, we show how to find the most robust solution. Finally, we extend our approach from robust solutions of constraint satisfaction problems to constraint optimization problems.

In this chapter we present the recent results on constraint acquisition obtained by the Coconut team and their collaborators. In a first part we show how to learn constraint networks by asking the user partial queries. That is, we ask the user to classify assignments to subsets of the variables as positive or negative. We provide an algorithm, called QUACQ, that, given a negative example, finds a constraint of the target network in a number of queries logarithmic in the size of the example. In a second part, we show that using some background knowledge may improve the acquisition process a lot. We introduce the concept of generalization query based on an aggregation of variables into types. We propose a generalization algorithm together with several strategies that we incorporate in QUACQ. Finally we evaluate our algorithms on some benchmarks.

We present a variety of approaches for solving the post enrolment-based course timetabling problem, which was proposed as Track 2 of the 2007 International Timetabling Competition. We approach the problem using local search and constraint programming techniques. We show how to take advantage of a list-colouring relaxation of the problem. Our local search approach won Track 2 of the 2007 competition. Our best constraint programming approach uses an original problem decomposition. Incorporating this into a large neighbourhood search scheme seems promising, and provides motivation for studying complete approaches in further detail.

The NValue constraint counts the number of different values assigned to a vector of variables. Propagating generalized arc consistency on this constraint is NP-hard. We show that computing even the lower bound on the number of values is NP-hard. We therefore study different approximation heuristics for this problem. We introduce three new methods for computing a lower bound on the number of values. The first two are based on the maximum independent set problem and are incomparable to a previous approach based on intervals. The last method is a linear relaxation of the problem. This gives a tighter lower bound than all other methods, but at a greater asymptotic cost.

Numberjack is a modelling package written in Python for embedding constraint programming and combinatorial optimisation into larger applications. It has been designed to seamlessly and efficiently support a number of underlying combinatorial solvers. This paper illustrates many of the features of Numberjack through the use of several combinatorial optimisation problems.

We introduce some new mappings of constraint satisfaction problems into propositional satisfiability. These encodings generalize most of the existing encodings. Unit propagation on those encodings is the same as establishing relational k-arc consistency on the original problem. They can also be used to establish (i,j)-consistency on binary constraints. Experiments show that these encodings are an effective method for enforcing such consistencies, that can lead to a reduction in runtimes at the phase transition in most cases. Compared to the more traditional (direct) encoding, the search tree can be greatly pruned.