A global cardinality constraint (gcc) is specified in terms of a set of variables X = fx1 ; :::; xpg which take their values in a subset of V = fv1 ; :::; vdg. It constrains the number of times a value v i 2 V is assigned toavariable in X to be in an interval (l i ;c i ). Cardinality constraints have proved very useful in many real-life problems, suchas scheduling, timetabling, or resource allocation. A gcc is more general than a constraint of difference, which requires each interval to be #0; 1#. In this paper, we present an efficient way of implementing generalized arc consistency for a gcc. The algorithm we propose is based on a new theorem of flow theory. Its space complexity is O(#Xj#jVj) and its time complexity is O(jXj 2 #jVj). We also show how this algorithm can efficiently be combined with other filtering techniques.

Constraint networks are used more and more to solve combinatorial problems in real-life applications. Much activity is concentrated on improving the efficiency of finding a solution in a constraint network (the constraint satisfaction problem, CSP). Particularly, arc consistency caught many researchers' attention, involving the discovery of a large number of algorithms. And, for the last two years, it has been shown that maintaining arc consistency during search is a worthwhile approach. However, results on CSPs and on arc consistency are almost always limited to binary constraint networks. The CSP is no longer an academic problem, and it is time to deal with non-binary CSPs, as widely required in real world constraint solvers. This paper proposes a general schema to implement arc consistency on constraints of any arity when no specific algorithm is known. A first instantiation of the schema is presented here, which deals with constraints given by a predicate, by the set of forbidden c...

Propagating constraints is the main feature of any constraint solver. This is thus of prime importance to manage constraint propagation as efficiently as possible, justifying the use of the best algorithms. But the ease of integration is also one of the concerns when implementing an algorithm in a constraint solver. This paper focuses on AC-3, which is the simplest arc consistency algorithm known so far. We propose two refinements that preserve as much as possible the ease of integration into a solver (no heavy data structure to be maintained during search), while giving some noticeable improvements in efficiency. One of the proposed refinements is analytically compared to AC-6, showing interesting properties, such as optimality of its worst-case time complexity. 1

This paper describes our experience with a simple modeling and programming approach for increasing the amount of constraint propagation in the constraint solving process. The idea, although similar to redundant constraints, is based on the concept of redundant modeling. We introduce the notions of CSP model and model redundancy, and show how mutually redundant models can be combined and connected using channeling constraints. The combined model contains the mutually redundant models as sub-models. Channeling constraints allow the sub-models to cooperate during constraint solving by propagating constraints freely amongst the sub-models. This extra level of pruning and propagation activities becomes the source of execution speedup. We perform two case studies to evaluate the effectiveness and efficiency of our method. The first case study is based on the simple and well-known n-queens problem, while the second case study applies our method in the design and construction of a real-life ...

Constraint satisfaction problems are widely used in artificial intelligence. They involve finding values for problem variables subject to constraints that specify which combinations of values are consistent. Knowledge about properties of the constraints can permit inferences that reduce the cost of consistency checking. In particular, such inferences can be used to reduce the number of constraint checks required in establishing arc consistency, a fundamental constraint-based reasoning technique. A general AC-Inference schema is presented and various forms of inference discussed. A specific algorithm, AC-7, is presented, which takes advantage of a simple property common to all binary constraints to eliminate constraint checks that other arc consistency algorithms perform. The effectiveness of this approach is demonstrated analytically, and experimentally on real-world problems.

Search algorithms for solving CSP (Constraint Satisfaction Problems) usually fall into one of two main families: local search algorithms and systematic algorithms. Both families have their advantages. Designing hybrid approaches seems promising since those advantages may be combined into a single approach. In this paper, we present a new hybrid technique. It performs a local search over partial assignments instead of complete assignments, and uses filtering techniques and conflict-based techniques to efficiently guide the search. This new technique benefits from both classical approaches: aprioripruning of the search space from filtering-based search and possible repair of early mistakes from local search. We focus on a specific version of this technique: tabu decision-repair.Experiments done on open-shop scheduling problems show that our approach competes well with the best highly specialized algorithms. 2002 Elsevier Science B.V. All rights reserved.

The use of constraint propagation is the main feature of any constraint solver. It is thus of

The Constraint Satisfaction Problem is a type of combinatorial search problem of much interest in Artificial Intelligence and Operations Research. The simplest algorithm for solving such a problem is chronological backtracking, but this method suffers from a malady known as "thrashing," in which essentially the same subproblems end up being solved repeatedly. Intelligent backtracking algorithms, such as backjumping and dependency-directed backtracking, were designed to address this difficulty, but the exact utility and range of applicability of these techniques have not been fully explored. This dissertation describes an experimental and theoretical investigation into the power of these intelligent backtracking algorithms. We compare the empirical performance of several such algorithms on a range of problem distributions. We show that the more sophisticated algorithms are especially useful on those problems with a small number of constraints that happen to be difficult for chronologica...

Consistency techniques can significantly reduce the search space of constraint satisfaction problems (CSP). In particular, arc-consistency algorithms, such as AC-3 [7], AC-4 [8] and AC-6 [2], have been designed. In [9], we presented DisAC-4, a coarse-grained parallel algorithm designed for distributed memory computer using message passing, which is a distributed version of AC-4. We extend here this result. We design DisAC-3 and DisAC-6. The communication scheme is also extended to allow communication during the propagation step of the consistency algorithms. All these algorithms were systematically experimented. An analysis of the different experiments shows that, as in the sequential case, DisAC-6 provides the best performance and that DisAC-3 outperforms DisAC-4 on most tests. All the distributed algorithms shows a linear speedup. This lead to the conclusion that DisAC-6 is a good candidate for distributed arc-consistency.

. We show that the same methodology used to study phase transition behaviour in NP-complete problems works with a polynomial problem class: establishing arc consistency. A general measure of the constrainedness of an ensemble of problems, used to locate phase transitions in random NP-complete problems, predicts the location of a phase transition in establishing arc consistency. A complexity peak for the AC3 algorithm is associated with this transition. Finite size scaling models both the scaling of this transition and the computational cost. On problems at the phase transition, this model of computational cost agrees with the theoretical worst case. As with NP-complete problems, constrainedness -- and proxies for it which are cheaper to compute -- can be used as a heuristic for reducing the number of checks needed to establish arc consistency in AC3. 1 Introduction Following [4] there has been considerable research into phase transition behaviour in NP-complete problems. Problems from...