Abstract. Efficient constraint propagation is crucial to any constraint solver. We show that watched literals, already a great success in the propositional satisfiability community, can also be used to provide highly efficient implementations of constraint propagators. We describe in detail three important aspects of watched literals as we apply them to constraints, and we describe how they are implemented in the Minion constraint solver. We show three successful applications of watched literals to constraint propagators: the sum of boolean variables; generalised arc consistency for the ‘element ’ constraint; and generalised arc consistency for the ‘table ’ constraint. 1

Abstract. In order to apply constraint programming to a particular domain, the problem must first be modelled as a constraint satisfaction problem. There are typically many alternative models of a given problem, and formulating an effective model requires a great deal of expertise. To reduce this bottleneck, the Essence language allows the specification of a problem abstractly, i.e. without making modelling decisions. This specification is refined automatically by the Conjure system to a solverindependent constraint modelling language Essence ′. However, there is still significant work involved in translating an Essence ′ model for use with a particular constraint solver. This paper discusses this ‘tailoring’ process with reference to the constraint solver Minion. 1

There is no standard modelling language for constraint programming (CP) problems. Most solvers have their own modelling language. This makes it difficult for modellers to experiment with different solvers for a problem. In this paper we present MiniZinc, a simple but expressive CP modelling language which is suitable for modelling problems for a range of solvers and provides a reasonable compromise between many design possibilities. Equally importantly, we also propose a low-level solver-input language called FlatZinc, and a straightforward translation from MiniZinc to FlatZinc that preserves all solver-supported global constraints. This lets a solver writer support MiniZinc with a minimum of effort— they only need to provide a simple FlatZinc front-end to their solver, and then combine it with an existing MiniZinc-to-FlatZinc translator. Such a front-end may then serve as a stepping stone towards a full MiniZinc implementation that is more tailored to the particular solver. A standard language for modelling CP problems will encourage experimentation with and comparisons between different solvers. Although MiniZinc is not perfect—no standard modelling language will be—we believe its simplicity, expressiveness, and ease of implementation make it a practical choice for a standard language.

Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised arc consistency on extensional constraints. The first, the Next-Difference list, is novel and has been developed with this application in mind. The second, the trie, is well known but its use in this context is novel. Empirical analyses demonstrate the efficiency of the resulting approaches, both in GACschema, and in the watched-literal table constraint in Minion.

ABSTRACT. The AllDifferent constraint is a crucial component of any constraint toolkit, language or solver, since it is very widely used in a variety of constraint models. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, we focus on the highest strength of inference, enforcing a property known as generalised arc consistency (GAC). This work is an analytical survey of optimizations of the main algorithm for GAC for the AllDifferent constraint. We evaluate empirically a number of key techniques from the literature. We also report important implementation details of those techniques, which have often not been described in published papers. We pay particular attention to improving incrementality by exploiting the strongly-connected components discovered during the standard propagation process, since this has not been detailed before. Our empirical work represents by far the most extensive set of experiments on variants of GAC algorithms for AllDifferent. Overall, the best combination of optimizations gives a mean speedup of 168 times over the same implementation without the optimizations. 1.

Abstract. Typically, there are many alternative models of a given problem as a constraint satisfaction problem, and formulating an effective model requires a great deal of expertise. To reduce this bottleneck, specification of a problem, which can then be refined automatically to a solver-independent modelling language. The final step is to tailor the model to a particular constraint solver. We show that we can eliminate common subexpressions in the tailoring step, as compilers do when compiling source code. We show that common subexpression elimination has two key benefits. First, it can lead to a dramatic reduction in the size of a constraint problem, to the extent that solving time is reduced by an order of magnitude when the number of nodes searched is the same. Second, it can lead to enhanced propagation and reduced search. The effect of this can be even more dramatic, leading to reductions in nodes searched and time taken by several orders of magnitude. Where the technique does not lead to improved search, we have not seen it cause a significant overhead. Therefore, we propose that common subexpression elimination is an important technique for constraint programming. 1

A central feature of relational database management systems is the ability to define multiple different views over an underlying database schema. Views provide a method of defining access control to the underlying database, since a view exposes a part of the

This paper addresses the problem of explaining missing answers in queries that include selection, projection, join, union, aggregation and grouping (SPJUA). Explaining missing answers of queries is useful in various scenarios, including query understanding and debugging. We present a general framework for the generation of these explanations based on source data. We describe the algorithms used to generate a correct, finite, and, when possible, minimal set of explanations. These algorithms are part of Artemis, a system that assists query developers in analyzing queries by, for instance, allowing them to ask why certain tuples are not in the query results. Experimental results demonstrate that Artemis generates explanations of missing tuples at a pace that allows developers to effectively use them for query analysis. 1.

While incremental propagation for global constraints is recognized to be important, little research has been devoted to how propagator-centered constraint programming systems should support incremental propagation. This paper introduces advisors as a simple and efficient, yet widely applicable method for supporting incremental propagation in a propagator-centered setting. The paper presents how advisors can be used for achieving different forms of incrementality and evaluates cost and benefit for several global constraints.

Abstract Equidistant Frequency Permutation Arrays are combinatorial objects of interest in coding theory. A frequency permutation array is a type of constant composition code in which each symbol occurs the same number of times in each codeword. The problem is to find a set of codewords such that any pair of codewords are a given uniform Hamming distance apart. The equidistant case is of special interest given the result that any optimal constant composition code is equidistant. This paper presents, compares and combines a number of different constraint formulations of this problem class, including a new method of representing permutations with constraints. Using these constraint models, we are able to establish several new results, which are contributing directly to mathematical research in this area. 3 1