The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effec-tive fashion. There are two classes of propagation algorithms for general constraints: fine-grained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables, and coarse-grained algorithms where the removal of a value will be propagated to the related variables. One big advantage of coarse-grained algorithms, like AC-3, over fine-grained algorithms, like AC-4, is the ease of integration when implementing an algorithm in a constraint solver. How-ever, fine-grained algorithms usually have optimal worst case time complexity while coarse-grained algorithms don’t. For example, AC-3 is an algorithm with non-optimal worst case complexity although it is simple, efficient in practice, and widely used. In this paper we propose a coarse-grained algorithm, AC2001/3.1, that is worst case op-timal and preserves as much as possible the ease of its integration into a solver (no heavy data structure to be maintained during search). Experimental results show that AC2001/3.1 is competitive with the best fine-grained algorithms such as AC-6. The idea behind the new algorithm can immediately be applied to obtain a path consis-tency algorithm that has the best-known time and space complexity. The same idea is then extended to non-binary constraints. Preliminary versions of this paper appeared in [BR01, ZY01].

Recently, a general definition of arc consistency (AC) for soft constraint frameworks has been proposed [1]. In this paper we specialize this definition to weighted CSP and introduce two O(ed³) enforcing algorithms. Then, we refine the definition and introduce a stronger form of arc consistency (AC*) along with two O(n²d² + ed³) algorithms. As in the CSP case, an important application of AC is to combine it with search. We empirically demonstrate that a branch and bound algorithm that maintains either AC or AC* is a state-of-the-art general solver for weighted CSP. Our experiments cover binary Max-CSP and Max-SAT problems.

The weighted CSP framework is a soft constraint framework with a wide range of applications. Most current state-of-the-art complete solvers can be described as a basic depth-first branch and bound search that maintain some form of arc consistency during the search. In this paper we introduce a new stronger form of arc consistency, that we call existential directional arc consistency and we provide an algorithm to enforce it. The efficiency of the algorithm is empirically demonstrated in a variety of domains. 1

Solving non-binary constraint satisfaction problems, a crucial challenge for the next years, can be tackled in two different ways: translating the non-binary problem into an equivalent binary one, or extending binary search algorithms to solve directly the original problem. The latter option raises some issues when we want to extend denitions written for the binary case. This paper focuses on the well-known forward checking algorithm, and shows that it can be generalized to several non-binary versions, all tting its binary denition. The classical version, proposed by Van Hentenryck, is only one of these generalizations.

Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent

The weighted CSP (WCSP) framework is a soft constraint framework with a wide range of applications. In this paper, we consider the problem of maintaining local consistency during search for solving WCSP. We first refine the notions of directional arc consistency (DAC) and full directional arc consistency (FDAC) introduced in [Cooper, 2003] for binary WCSP, define algorithms that enforce these properties and study their complexities. We then consider algorithms that maintain either arc consistency (AC), DAC or FDAC during search. The efficiency of these algorithms is empirically studied. It appears that despite its high theoretical cost, the strongest FDAC property is the best choice. 1

We propose a framework for solving CSPs based both on backtracking techniques and on the notion of tree-decomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the tree-width of the constraint networks. Finally, experimental results allow us to show the advantages of this approach.

We perform a detailed theoretical and empirical comparison of the dual and hidden variable encodings of non-binary constraint satisfaction problems. We identify a simple relationship between the two encodings by showing how we can translate between the two by composing or decomposing relations. This translation

Singleton arc consistency (SAC) enhances the pruning capability of arc consistency by ensuring that the network cannot become arc inconsistent af-ter the assignment of a value to a variable. Algo-rithms have already been proposed to enforce SAC, but they are far from optimal time complexity. We give a lower bound to the time complexity of en-forcing SAC, and we propose an algorithm that achieves this complexity, thus being optimal. How-ever, it can be costly in space on large problems. We then propose another SAC algorithm that trades time optimality for a better space complexity. Nev-ertheless, this last algorithm has a better worst-case time complexity than previously published SAC al-gorithms. An experimental study shows the good performance of the new algorithms. 1

In this paper, we present a general algorithm which gives an uniform view of several state-of-the-art systematic backtracking search algorithms for solving both binary and nonbinary CSP instances. More precisely, this algorithm integrates the most usual or/and sophisticated look-back and look-ahead schemes. By means of this algorithm, our purpose is then to study the interest of backjump-based techniques with respect to conflict-directed variable ordering heuristics. 1