Abstract. In an increasing number of domains such as bioinformatics, combinatorial graph problems arise. We propose a novel way to solve these problems, mainly those that can be translated to constrained subgraph finding. Our approach extends constraint programming by introducing CP(Graph), a new computation domain focused on graphs including a new type of variable: graph domain variables as well as constraints over these variables and their propagators. These constraints are subdivided into kernel constraints and additional constraints formulated as networks of kernel constraints. For some of these constraints a dedicated global constraint and its associated propagator are sketched. CP(Graph) is integrated with finite domain and finite sets computation domains, allowing the combining of constraints of these domains with graph constraints. A prototype of CP(Graph) built over finite domains and finite sets in Oz is presented. And we show that a problem of biochemical network analysis can be very simply described and solved within CP(Graph). 1

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Abstract. The constraint NValue counts the number of different values assigned to a vector of variables. Propagating generalized arc consistency on this constraint is NP-hard. We show that computing even the lower bound on the number of values is NP-hard. We therefore study different approximation heuristics for this problem. We introduce three new methods for computing a lower bound on the number of values. The first two are based on the maximum independent set problem and are incomparable to a previous approach based on intervals. The last method is a linear relaxation of the problem. This gives a tighter lower bound than all other methods, but at a greater asymptotic cost. 1 Introduction The NValue constraint counts the number of distinct values used by a vectorof variables. It is a generalization of the widely used AllDifferent constraint[12]. It was introduced in [4] to model a musical play-list configuration problem so

Abstract. We present some general constraints for breaking symmetries in constraint satisfaction problems. These constraints can be used to break symmetries acting on variables, values, or both. We also consider symmetry breaking constraints to deal with conditional symmetries, and symmetries acting on set and other types of variables. 1

We have started a systematic study of global constraints on set and multiset variables. We consider here disjoint, partition, and intersection constraints.

We propose a simple declarative language for specifying a wide range of counting and occurrence constraints. This specification language is executable since it immediately provides a polynomial propagation algorithm. To illustrate the capabilities of this language, we specify a dozen global constraints taken from the literature. We observe one of three outcomes: we achieve generalized arc-consistency; we do not achieve generalized arc-consistency, but achieving generalized arcconsistency is NP-hard; we do not achieve generalized arc-consistency, but specialized propagation algorithms can do so in polynomial time. Experiments demonstrate that this specification language is both efficient and effective in practice.

Abstract. We present a comprehensive study of the use of valueprecedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is bothefficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreathvalue and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we con-sider the interaction between value precedence and symmetry breaking constraints for variable symmetries. 1 INTRODUCTION Symmetry is an important aspect of many search problems. Symme-try occurs naturally in many problems (e.g. if we have two identical

Many constraint toolkits provide logical connectives like disjunction, negation and implication. These permit complex constraint expressions to be built from primitive constraints. However, the propagation of such complex constraint expressions is typically limited. We therefore present a simple and light weight method for propagating complex constraint expressions. We provide a precise characterization of when this method enforces generalized arc-consistency. In addition, we demonstrate that with our method many different global constraints can be easily implemented. 1

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