We study decompositions of NVALUE, a global constraint that can be used to model a wide range of problems where values need to be counted. Whilst decomposition typically hinders propagation, we identify one decomposition that maintains a global view as enforcing bound consistency on the decomposition achieves bound consistency on the original global NVALUE constraint. Such decompositions offer the prospect for advanced solving techniques like nogood learning and impact based branching heuristics. They may also help SAT and IP solvers take advantage of the propagation of global constraints.

Constraint Programming is an attractive approach for solving AI planning problems by modelling them as Constraint Satisfaction Problems (CSPs). However, formulating effective constraint models of complex planning problems is challenging,  and CSPs resulting from standard approaches often require further enhancement to perform well. Common subexpression elimination is a computationally cheap and  general technique for improving CSPs, which can lead to a great reduction in instance size, solving time and search space. In this work we identify general causes of common subexpressions from three modelling techniques often used to encode planning problems into   constraints. We present four case studies of constraint  models of AI planning problems. In each, we describe the constraint model, highlight the sources of common subexpressions, and present an empirical analysis of the effects of eliminating common subexpressions.

We study novel approaches for solving of hard combinatorial problems by translation to Boolean Satisfiability (SAT). Our focus is on combinatorial problems that can be represented as a permutation of n objects, subject to additional constraints. In the case of the Hamiltonian Cycle Problem (HCP), these constraints are that two adjacent nodes in a permutation should also be neighbors in the original graph for which we search for a Hamiltonian cycle. We use the absolute SAT encoding of permutations, where for each of the n objects and each of its pos- sible positions in a permutation, a predicate is defined to indicate whether the object is placed in that position. For implementation of this predicate, we compare the direct and logarithmic encodings that have been used previously, against 16 hierarchical parameterizable encodings of which we explore 416 instantiations. We propose the use of enumerative adjacency constraints—that enumerate the possible neighbors of a node in a permutation — instead of, or in addition to the exclusivity adjacency constraints — that exclude impossible neighbors, and that have been applied previously. We study 11 heuristics for efficiently choosing the first node in the Hamiltonian cycle, as well as 8 heuristics for static CNF variable ordering. We achieve at least 4 orders of magnitude average speedup on HCP benchmarks from the phase transition region, relative to the previously used encodings for solving of HCPs via SAT, such that the speedup is increasing with the size of the graphs.

Complete row and column symmetry breaking in constraint satisfaction problems using the lex leader method is generally prohibitively costly. Double lex, which is derived from lex leader, is commonly used in practice as an incomplete symmetry-breaking method for row and column symmetries. This technique uses a row-wise ordering to construct the lex leader. For this reason, it is generally counterproductive to choose a search ordering that is not also row-wise. It seems logical that the search order should be used to pick the symmetry breaking technique, rather than the other way around. This paper surveys other possible orderings and investigates one particular ordering, snake ordering. From this we derive a corresponding incomplete set of symmetry breaking constraints, snake lex. We present experimental data comparing double lex and the snake lex, showing that snake lex is substantially faster than double lex in many cases.

MINLP problems are hard constrained optimization problems, with nonlinear constraints and mixed discrete continuous variables. They can be solved using a Branch-and-Bound scheme combining several methods, such as linear programming, interval analysis, and cutting methods. Our goal is to integrate constraint programming techniques in this framework. Firstly, global constraints can be introduced to reformulate MINLP problems thus leading to clean models and more precise computations. Secondly, interval-based approximation techniques for nonlinear constraints can be improved by taking into account the integrality of variables early. These methods have been implemented in an interval solver and we present experimental results from a set of MINLP instances.

An important challenge in constraint programming is to rewrite constraint models into executable programs calculating the solutions. This phase of constraint processing may require translations between constraint programming languages, transformations of constraint representations, model optimizations, and tuning of solving strategies. In this paper, we introduce a pivot metamodel describing the common features of constraint models including different kinds of constraints, statements like conditionals and loops, and other first-class elements like object classes and predicates. This metamodel is general enough to cope with the constructions of many languages, from object-oriented modeling languages to logic languages, but it is independent from them. The rewriting operations manipulate metamodel instances apart from languages. As a consequence, the rewriting operations apply whatever languages are selected and they are able to manage model semantic information. A bridge is created between the metamodel space and languages using parsing techniques. Tools from the software engineering world can be useful to implement this framework.

Equidistant Frequency Permutation Arrays are combinatorial objects of interest in coding theory. A frequency permutation array is a type of constant composition code in which each symbol occurs the same number of times in each codeword. The problem is to find a set of codewords such that any pair of codewords are a given uniform Hamming distance apart. The equidistant case is of special interest given the result that any optimal constant composition code is equidistant. This paper presents, compares and combines a number of different constraint formulations of this problem class, including a new method of representing permutations with constraints. Using these constraint models, we are able to establish several new results, which are contributing directly to mathematical research in this area.

Arc consistency algorithms are widely used to prune the search space of Constraint Satisfaction Problems (CSPs). Since many researchers associate arc consistency with binary normalized CSPs, there is a confusion between the notion of arc consistency and 2-consistency. 2-consistency guarantees that any instantiation of a value to a variable can be consistently extended to any second variable. Thus, 2-consistency can be stronger than arc-consistency in binary CSPs. In this paper, we present a new algorithm, called 2-C3, which achieves 2-consistency in binary and non-normalized CSPs. This algorithm is a reformulation of the well-known AC3 algorithm. The evaluation section shows that 2-C3 is able to prune more search space than AC3 and AC4.

Tailoring solver-independent constraint instances to target solvers is an important component of automated constraint modelling. We augment the tailoring process by a set of enhancement techniques of which many are successfully established in related fields, such as common subexpression elimination. Our aim is to apply these techniques in an efficient fashion,  since we tailor instance-wise, and not whole problem classes. We integrate automated enhancement into the tailoring procedure, which creates a novel setup with great potential, as our empirical analysis confirms: impressive speedups, additional propagation and instance  reduction, all for investing little computational effort.

In a constraint satisfaction problem, the tightness of an individual constraint only describes the influence that the variables within its scope have on one another. Clusters provide a broader view; they are dense, tight subproblems within a problem. A set of clusters for a problem and the links between them provide an abstraction of it. That abstraction can be used to guide search, to curtail inference, and to provide explanations to the user. This work is a hybrid of global and local search, where local search creates an abstraction and then global search exploits it. Heuristics reference clusters to order variables and to propagate more thoughtfully with respect to them. Results are provided on a variety of challenging benchmark problems.