Justin Pearson

Breaking Row and Column Symmetries in Matrix Models

We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetry-breaking constraints.

Efficient Timed Reachability Analysis Using Clock Difference Diagrams

One of the major problems in applying automatic verification tools to industrial-size systems is the excessive amount of memory required during the state-space exploration of a model. In the setting of real-time, this problem of state-explosion requires extra attention as information must be kept not only on the discrete control structure but also on the values of continuous clock variables. In this paper, we exploit Clock Difference Diagrams, CDDs, a BDD-like data-structure for representing and effectively manipulating certain nonconvex subsets of the Euclidean space, notably those encountered during verification of timed automata. A version of the real-time verification tool Uppaal using CDDs as a compact data-structure for storing explored symbolic states has been implemented. Our experimental results demonstrate significant spacesavings: for eight industrial examples, the savings are in average 42% with moderate increase in runtime. We further report on how the symbolic state-space exploration itself may be carried out using CDDs.

Introducing ESRA, a relational language for modelling combinatorial problems

Current-generation constraint programming languages are considered by many, especially in industry, to be too low-level, difficult, and large. We argue that solver-independent, high-level relational constraint modelling leads to a simpler and smaller language, to more concise, intuitive, and analysable models, as well as to more efficient and effective model formulation, maintenance, reformulation, and verification. All this can be achieved without sacrificing the possibility of efficient solving, so that even time-pressed or less competent modellers can be well assisted. Towards this, we propose the esra relational constraint modelling language, showcase its elegance on some well-known problems, and outline a compilation philosophy for such languages.

Closure Functions and Width 1 Problems

Local Consistency has proven to be an important notion in the study of constraint satisfaction problems. We give an algebraic condition that characterizes all the constraint types for which generalized arc-consistency is sufficient to ensure the existence of a solution. We give some examples to illustrate the application of this result.

Constraints and universal algebra

In this paper we explore the links between constraint satisfaction problems and universal algebra. We show that a constraint satisfaction problem instance can be viewed as a pair of relational structures, and the solutions to the problem are then the structure preserving mappings between these two relational structures. We give a number of examples to illustrate how this framework can be used to express a wide variety of combinatorial problems, many of which are not generally considered as constraint satisfaction problems. We also show that certain key aspects of the mathematical structure of constraint satisfaction problems can be precisely described in terms of the notion of a Galois connection, which is a standard notion of universal algebra. Using this result, we obtain an algebraic characterisation of the property of minimality in a constraint satisfaction problem. We also obtain a similar algebraic criterion for determining whether or not a given set of solutions can be expressed by a constraint satisfaction problem with a given structure, or a given set of allowed constraint types.

Compositional derivation of symmetries for constraint satisfaction

This paper reconsiders the problems of discovering symmetries in constraint satisfaction problems (CSPs). It proposes a compositional approach which derives symmetries of the applications from primitive constraints. The key insight is the recognition of the special role of global constraints in symmetry detection. Once the symmetries of global constraints are available, it often becomes much easier to derive symmetries compositionally and efficiently. The paper demonstrates the potential of this approach by studying several classes of value and variable symmetries and applying the resulting techniques to two non-trivial applications. The paper also discusses the potential of reformulations and high-level modeling abstractions to strengthen symmetry discovery.

On the reification of global constraints

We introduce a simple idea for deriving reified global constraints in a systematic way. It is based on the observation that most global constraints can be reformulated as a conjunction of total function constraints together with a constraint that can be easily reified.

Towards solver-independent propagators

We present an extension to indexicals to describe propagators for global constraints. The resulting language is compiled into actual propagators for different solvers, and is solver-independent. In addition, we show how this high-level description eases the proof of propagator properties, such as correctness and monotonicity. Experimental results show that propagators compiled from their indexical descriptions are sometimes not significantly slower than built-in propagators of Gecode. Therefore, our language can be used for the rapid prototyping of new global constraints.

Dynamic structural symmetry breaking for constraint satisfaction problems

In recent years, symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention. Various general schemes have been proposed to eliminate symmetries. In general, these schemes may take exponential space or time to eliminate all the symmetries. We identify several classes of CSPs that encompass many practical problems and for which symmetry breaking for various forms of value or variable interchangeability is tractable using dedicated search procedures. We also show the limits of efficient symmetry breaking for such dominance-detection schemes by proving intractability results for some classes of CSPs.

A constraint-based local search backend for MiniZinc

MiniZinc is a modelling language for combinatorial problems, which can then be solved by a solver provided in a backend. There are many backends, based on technologies such as constraint programming, integer programming, or Boolean satisfiability solving. However, to the best of our knowledge, there is currently no constraint-based local search (CBLS) backend. We discuss the challenges to develop such a backend and give an overview of the design of a CBLS backend for MiniZinc. Experimental results show that for some MiniZinc models, our CBLS backend, based on the OscaR/CBLS solver, is able to give good-quality results in competitive time.