A major difficulty in evaluating incomplete local search style algorithms for constraint satisfaction problems is the need for a source of hard problem instances that are guaranteed to be satisfiable. A standard approach to evaluate incomplete search methods has been to use a general problem generator and a complete search method to filter out the unsatisfiable instances. Unfortunately, this approach cannot be used to create problem instances that are beyond the reach of complete search methods. So far, it has proven to be surprisingly difficult to develop a direct generator for satisfiable instances only. In this paper, we propose a generator that only outputs satisfiable problem instances. We also show how one can finely control the hardness of the satisfiable instances by establishing a connection between problem hardness and a new kind of phase transition phenomenon in the space of problem instances. Finally, we use our problem distribution to show the easy-hard-easy pattern in search complexity for local search procedures, analogous to the previously reported pattern for complete search methods.

When writing a constraint program, we have to decide what to make the decision variable, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. For example, with permutation problems, we can choose between a primal and a dual representation. In the dual representation, dual variables stand for the primal values, whilst dual values stand for the primal variables. By means of channelling constraints, a combined model can have both primal and dual variables. In this paper, we perform an extensive theoretical and empirical study of these different models. Our results will aid constraint programmers to choose a model for a permutation problem. They also illustrate a general methodology for comparing different constraint models.

In constraint programming one models a problem by stating constraints on acceptable solutions. The constraint model is then usually solved by interleaving backtracking search and constraint propagation. Previous studies have demonstrated that designing special purpose constraint propagators for commonly occurring constraints can significantly improve the efficiency of a constraint programming approach. In this paper we present a fast, simple algorithm for bounds consistency propagation of the alldifferent constraint. The algorithm has the same worst case behavior as the previous best algorithm but is much faster in practice. Using a variety of benchmark and random problems, we show that our algorithm outperforms existing bounds consistency algorithms and also outperforms—on problems with an easily identifiable property—state-ofthe-art commercial implementations of propagators for stronger forms of local consistency.

The constraint of difference is known to the constraint programming community since Lauriere introduced Alice in 1978. Since then, several strategies have been designed to solve the alldifferent constraint. This paper surveys the most important developments over the years regarding the alldifferent constraint. First we summarize the underlying concepts and results from graph theory and integer programming. Then we give an overview and an abstract comparison of different solution strategies. In addition, the symmetric alldifferent constraint is treated. Finally, we show how to apply cost-based filtering to the alldifferent constraint.

. The Golomb ruler problem has been proposed as a challenging constraint satisfaction problem. We consider a large number of different models of this problem, both binary and non-binary. The problem can be modelled using quaternary constraints, but in practice using a set of auxiliary variables and ternary constraints gives better results. A binary encoding of the problem gives a smaller search tree, but is impractical because it takes far longer to run. We compare variable ordering heuristics and consider the use of implied constraints to improve propagation. We believe that more case studies such as this are essential to reduce the skill currently required for successful modelling. 1 Introduction In his AAAI-98 invited talk, Gene Freuder identified modelling as one of the major hurdles preventing the uptake of constraint satisfaction technology. The availability of non-binary constraints can increase the number of possible models of a problem amnd so makes modelling still more diffi...

We perform an extensive theoretical and empirical analysis of the use of auxiliary variables and implied constraints in modelling a class of non-binary constraint satisfaction problems called problems of distance. This class of problems include 1-d, 2-d and circular Golomb rulers. We identify a large number of different models, both binary and non-binary, and compare theoretically the level of consistency achieved by generalized arc consistency on them. Our experiments show that the introduction of auxiliary variables and implied constraints can significantly reduce the size of the search space. For instance, our final models reduce the time to find an optimal 10-mark Golomb ruler 50-fold.

Abstract. We study the global cardinality constraint (gcc) and propose an O(n

Previous studies have demonstrated that designing special purpose constraint propagators can significantly improve the efficiency of a constraint programming approach. In this paper we present an efficient algorithm for bounds consistency propagation of the generalized cardinality constraint (gcc). Using a variety of benchmark and random problems, we show that our bounds consistency algorithm is competitive with and can dramatically outperform existing state-of-the-art commercial implementations of constraint propagators for the gcc. We also present a new algorithm for domain consistency propagation of the gcc which improves on the worst-case performance of the best previous algorithm for problems that occur often in applications.

Many constraint satisfaction problems can be naturally and efficiently modelled using non-binary constraints like the "all-different" and "global cardinality" constraints. Certain classes of these non-binary constraints are "network decomposable" as they can be represented by binary constraints on the same set of variables. We compare theoretically the levels of consistency which are achieved on non-binary constraints to those achieved on their binary decomposition. We present many new results about the level of consistency achieved by the forward checking algorithm and its various generalizations to non-binary constraints. We also compare the level of consistency achieved by arc-consistency and its generalization to non-binary constraints, and identify special cases of non-binary decomposable constraints where weaker or stronger conditions, than in the general case, hold. We also analyze the cost, in consistency checks, required to achieve certain levels of consistency.

Constraint programming (CP) is mainly based on filtering algorithms; their association with global constraints is one of the main strengths of CP. This chapter is an overview of these two techniques. Some of the most frequently used global constraints are presented. In addition, the filtering algorithms establishing arc consistency for two useful constraints, the alldiff and the global cardinality constraints, are fully detailed. Filtering algorithms are also considered from a theoretical point of view: three different ways to design filtering algorithms are described and the quality of the filtering algorithms studied so far is discussed. A categorization is then proposed. Over-constrained problems are also mentioned and global soft constraints are introduced.