This report introduces a classification scheme for the global constraints. This classification is based on four basic ingredients from which one can generate almost all existing global constraints and come up with new interesting constraints. Global constraints are defined in a very concise way, in term of graph properties that have to hold, where the graph is a structured network of same elementary constraints. Since this classification is based on the internal structure of the global constraints it is also a strong hint for the pruning algorithms of the global constraints. Keywords Constraint, finite domain, global constraint, classification, resource constraint scheduling, graph partitioning, timetabling. 2 Table of contents Table of contents ....................................................................................................................................................... 2 Table of figures.........................................................................

Abstract. Experts at modelling constraint satisfaction problems (CSPs) carefully choose model transformations to reduce greatly the amount of effort that is required to solve a problem by systematic search. It is a considerable challenge to automate such transformations and to identify which transformations are useful. Transformations include adding constraints that are implied by other constraints, adding constraints that eliminate symmetrical solutions, removing redundant constraints and replacing constraints with their logical equivalents. This paper describes the CGRASS (Constraint Generation And Symmetry-breaking) system that can improve a problem model by automatically performing transformations of these kinds. We focus here on transforming individual CSP instances. Experiments on the Golomb ruler problem suggest that producing good problem formulations solely by transforming problem instances is, generally, infeasible. We argue that, in certain cases, it is better to transform the problem class than individual instances and, furthermore, it can sometimes be better to transform formulations of a problem that are more abstract than a CSP. 1

Systematic backtracking is used in many constraint solvers and combinatorial optimisation algorithms. It is complete and can be combined with powerful search pruning techniques such as branch-and-bound, constraint propagation and dynamic variable ordering. However, it often scales poorly to large problems. Local search is incomplete, and has the additional drawback that it cannot exploit pruning techniques, making it uncompetitive on some problems. Nevertheless its scalability makes it superior for many large applications. This paper describes a hybrid approach called Incomplete Dynamic Backtracking, a very flexible form of backtracking that sacrifices completeness to achieve the scalability of local search. It is combined with forward checking and dynamic variable ordering, and evaluated on three combinatorial problems: on the n-queens problem it out-performs the best local search algorithms; it finds large optimal Golomb rulers much more quickly than a constraint-based backtracker, and better rulers than a genetic algorithm; and on benchmark graphs it finds larger cliques than almost all other tested algorithms. We argue that this form of backtracking is actually local search in a space of consistent partial assignments, offering a generic way of combining standard pruning techniques with local search.

Abstract. Arc consistency plays a central role in solving Constraint Satisfaction Problems. This is the reason why many algorithms have been proposed to establish it. Recently, an algorithm called AC2001 and AC3.1 has been independently presented by their authors. This algorithm which is considered as a refinement of the basic algorithm AC3 has the advantage of being simple and competitive. However, it does not take into account constraint bidirectionality as AC7 does. In this paper, we address this issue, and, in particular, introduce two new algorithms called AC3.2 and AC3.3 which benefit from good properties of both AC3 and AC7. Indeed, AC3.2 and AC3.3 are as easy to implement as AC3 and take advantage of bidirectionality as AC7 does. More precisely, AC3.2 is a general algorithm which partially exploits bidirectionality whereas AC3.3 is a binary algorithm which fully exploits bidirectionality. It turns out that, when Maintaining Arc Consistency during search, MAC3.2, due to a memorization effect, is more efficient than MAC3.3 both in terms of constraint checks and cpu time. Compared to MAC2001/3.1, our experimental results show that MAC3.2 saves about 50% of constraint checks and, on average, 15 % of cpu time. 1

Constraint programs with one or more matrices of decision variables are commonly and naturally used to model real-world problems.

. We describe how proof planning is being extended to generate implied algebraic constraints. This inference problem introduces a number of challenging problems like deciding a termination condition and evaluating constraint utility. We have implemented a number of methods for reasoning about algebraic constraints. For example, the eliminate method performs Gaussian-like elimination of variables and terms. We are also re-using proof methods from the PRESS equation solving system like (variable) isolation. 1

Since the origins of the constraint satisfaction paradigm, its restriction to binary constraints has concentrated a significant part of the work. This is understandable because new ideas/techniques are usually much simpler to present/elaborate by first restricting them to the binary case. (See for example the arc consistency algorithms, such as AC-3 or AC-4, which have been presented first in their binary version [10, 12], before being extended to non-binary constraints [11, 13].) But this inclination has highly increased in the early nineties. Authors indeed justied this restriction by the fact that any non-binary constraint network can polyniomally be converted into an equivalent binary one [6, 8, 5, 19]. And, in most cases, they never extended their work to non-binary constraints. Up to now, constraint reasoning has generated robust formal definitions (local consistencies, etc.), original resolution methods (filtering, look-back schemes, decomposition techniques, heuristics, etc.), and theo...

Symmetry breaking and implied constraints can speed up both exhaustive search and the search for a single solution. We experiment with both types of constraint, using three search algorithms (backtracking, local and hybrid) to find single solutions for SAT encodings of three combinatorial problems (clique, set cover and balanced incomplete block design generation). Both show strong positive and negative effects, depending on the problem class and algorithm. However, symmetry breaking constraints consistently have a negative effect on the incomplete search algorithms. This suggests an opposite strategy when applying stochastic search: maximising symmetry in the constraint model.

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Local search has been successfully applied to many combinatorial optimisation problems, and is often more efficient on large instances than systematic backtracking. However, local search usually permits constraint violation and is less suited than backtracking to highly structured problems. To solve large structured problems, hybrid approaches are a promising research direction. A recent hybrid is a randomised form of backtracking, which sacrifices completeness in order to boost scalability to that of local search. This approach is evaluated on two hard combinatorial optimisation problems: maximum cliques and Golomb rulers. In both cases very competitive algorithms are obtained. Some problem modeling issues are also discussed.