We study the computational complexity of reasoning with global constraints. We show that reasoning with such constraints is intractable in general. We then demonstrate how the same tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when decomposing constraints will lose pruning, and when combining constraints is tractable. We also show how the same tools can be used to study symmetry breaking, meta-constraints like the cardinality constraint, and learning nogoods.

. In some recent works, a general framework for finite domains constraint satisfaction has been defined, where classical CSPs, fuzzy CSPs, weighted CSPs, partial CSPs and others can be easily cast. This framework, based on a semiring structure, allows, under certain conditions, to compute arc-consistency. Restricting to that case and integrating semiring-based constraint solving in the Constraint Logic Programming paradigm, we have implemented a generic language, clp(FD,S), for semiring-based constraint satisfaction. In this paper, we describe the kernel of the language: the SFD system and our implementation of clp(FD,S). We also give some performance results on various examples. 1 Introduction In [1, 2], a general framework for finite domains constraint satisfaction and optimization has been defined, where classical CSPs [15, 13, 14], fuzzy CSPs [16, 9, 17], partial CSPs [10] and others can be easily cast. This framework is based on a semiring structure. Moreover, the authors show t...

Optimization and constraint satisfaction methods are complementary to a large extent, and there has been much recent interest in combining them. Yet no generally accepted principle or scheme for their merger has evolved. We propose a scheme based on two fundamental dualities, the duality of search and inference and the duality of strengthening and relaxation. Optimization as well as constraint satisfaction methods can be seen as exploiting these dualities in their respective ways. Our proposal is that rather than employ either type of method exclusively, one can focus on how these dualities can be exploited in a given problem class. The resulting algorithm is likely to contain elements from both optimization and constraint satisfaction, and perhaps new methods that belong to neither.

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.

In recent years, it has been increasingly recognized that constraint and integer programming have orthogonal and complementary strengths in stating and solving combinatorial optimization applications. In addition, their integration has become an active research topic. The optimization programming language opl was a first attempt at integrating these technologies both at the language and at the solver levels. In particular, opl is a modeling language integrating the rich language of constraint programming and the ability to specify search procedures at a high level of abstraction. Its implementation includes both constraint and mathematical programming solvers, as well as some cooperation schemes to make them collaborate on a given problem. The purpose of this paper is to illustrate, using opl, the constraint-programming approach to combinatorial optimization and the complementary strengths of constraint and integer programming. (Artificial Intelligence; Computer Science; Integer Programming) 1.

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.

This paper presents an algorithm that achieves hyper-arc consistency for the soft alldifferent constraint. To this end, we prove and exploit the equivalence with a minimum-cost flow problem. Consistency of the constraint can be checked in O(nm) time, and hyper-arc consistency is achieved in O(m) time, where n is the number of variables involved and m is the sum of the cardinalities of the domains. It improves a previous method that did not ensure hyper-arc consistency.

. The industrial and commercial worlds are increasingly competitive, requiring companies to be more productiveand more responsive to market changes (e.g. globalisation and privatisation). As a consequence, there is a strong need for solutions to large scale optimization problems, in domains such as production scheduling, transport, finance and network management. This means that more experts in constraint programming and optimization technology are required to develop adequate software. Given the computational complexity of Large Scale Combinatorial Optimization problems, a key question is how to help/guide in the tackling of LSCO problems in industry. Optimization technology is certainly reaching a level of maturity. Having emerged in the 50s within the Operational Research community, it has evolved and comprises new paradigms such as constraint programming and stochastic search techniques. There is a practical need, i.e. efficiency, scalability and tractability, to integrate techniques from the different paradigms. This adds complexity to the design of LSCO models and solutions. Various forms of guidance are available in the literature in terms of 1) case studies that map powerful algorithms to problem instances, and 2) visualization and programming tools that ease the modelling and solving of LSCOs. However, there is little guidance to address the process of building applications for new LSCO problems (independently of any language). This article gives an overview of the CHIC-2 methodology which aims at filling a gap in this direction. In particular, we describe some management issues specific to LSCOs such as risk management and team structures, and focus on the technical development guidance for scoping, designing and implementing LSCO appl...

: Large Scale Combinatorial Optimization problems (LSCO) appear in numerous types of industrial applications (e.g. production scheduling, routing problems, financial applications). They are NP-complete problems characterized by large sets of data, constraints and variables, and often have an impure structure. Tackling such problems successfully requires both experience and skill. The current trend in the Constraint Programming (CP) community is to enhance the features of CP languages to ease the modelling and solving of LSCO problems by providing 1) natural modelling, 2) built-in constraints facilities (global constraints), and 3) integration of different constraint solvers (from mathematical programming, constraint programming, stochastic search methods). The common aspect is the growing awareness that we need to hybridize different models and methods and go beyond the constraint programming framework, essentially for efficiency and scaling reasons. This sets strong requirements upstr...

Constraint propagation is one of the techniques central to the success of constraint programming. To reduce search, fast algorithms associated with each constraint prune the domains of variables. With global (or non-binary) constraints, the cost of such propagation may be much greater than the quadratic cost for binary constraints. We therefore study the computational complexity of reasoning with global constraints. We first characterise a number of important questions related to constraint propagation. We show that such questions are intractable in general, and identify dependencies between the tractability and intractability of the different questions. We then demonstrate how the tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when constraints can be safely generalized, when decomposing constraints will reduce the amount of pruning, and when combining constraints is tractable.