## 1 Introduction

Currently, applying constraint technology to a large, complex problem
requires significant manual tuning by an expert. Such experts are rare. The
central aim of this project is to improve the scalability of constraint
technology, while simultaneously removing its reliance on manual tuning by an
expert. We propose a novel, elegant means to achieve this – a *constraint
solver synthesiser*, which generates a constraint solver specialised to a given
problem. Constraints research has mostly focused on the incremental improvement
of general-purpose solvers so far. The closest point of comparison is currently
the G12 project [1], which aims to combine existing general constraint
solvers and solvers from related fields into a hybrid. There are previous
efforts at generating specialised constraint solvers in the literature,
e.g. [2]; we aim to use state-of-the-art constraint solver technology
employing a broad range of different techniques. Synthesising a constraint
solver has two key benefits. First, it will enable a fine-grained optimisation
not possible for a general solver, allowing the solving of much larger, more
difficult problems. Second, it will open up many new research possibilities.
There are many techniques in the literature that, although effective in a
limited number of cases, are not suitable for general use. Hence, they are
omitted from current general solvers and remain relatively undeveloped. Among
these are for example conflict recording [3], backjumping [4],
singleton arc consistency [5], and neighbourhood inverse
consistency [6]. The synthesiser will select such techniques as
they are appropriate for an input problem. Additionally, it can also vary basic
design decisions, which can have a significant impact on
performance [7].

The system we are proposing in this paper, Dominion, implements a design that is capable of achieving said goals effectively and efficiently. The design decisions we have made are based on our experience with Minion [9] and other constraint programming systems.

The remainder of this paper is structured as follows. In the next section, we describe the design of Dominion and which challenges it addresses in particular. We then present the current partial implementation of the proposed system and give experimental results obtained with it. We conclude by proposing directions for future work.

## 2 Design of a synthesiser for specialised constraint solvers

The design of Dominion distinguishes two main parts. The
*analyser* analyses the problem model and produces a solver specification
that describes what components the specialised solver needs to have and which
algorithms and data structures to use. The *generator* takes the solver
specification and generates a solver that conforms to it. The flow of
information is illustrated in Figure 1.

Both the analyser and the generator optimise the solver. While the analyser performs the high-level optimisations that depend on the structure of the problem model, the generator performs low-level optimisations which depend on the implementation of the solver. Those two parts are independent and linked by the solver specification, which is completely agnostic of the format of the problem model and the implementation of the specialised solver. There can be different front ends for both the analyser and the generator to handle problems specified in a variety of formats and specialise solvers in a number of different ways, e.g. based on existing building blocks or synthesised from scratch.

### 2.1 The analyser

The analyser operates on the model of a constraint problem class or instance. It determines the constraints, variables, and associated domains required to solve the problem and reasons about the algorithms and data structures the specialised solver should use. It makes high-level design decisions, such as whether to use trailing or copying for backtracking memory. It also decides what propagation algorithms to use for specific constraints and what level of consistency to enforce.

The output of the analyser is a solver specification that describes all the design decisions made. It does not necessarily fix all design decisions – it may use default values – if the analyser is unable to specialise a particular part of the solver for a particular problem model.

In general terms, the requirements for the solver specification are that it (a) describes a solver which is able to find solutions to the analysed problem model and (b) describes optimisations which will make this solver perform better than a general solver.

The notion of better performance includes run time as well as other resources such as memory. It is furthermore possible to optimise with respect to a particular resource; for example a solver which uses less memory at the expense of run time for embedded systems with little memory can be specified.

The solver specification may include a representation of the original problem model such that a specialised solver which encodes the problem can be produced – the generated solver does not require any input when run or only values for the parameters of a problem class. It may furthermore modify the original model in a limited way; for example split variables which were defined as one type into several new types. It does not, however, optimise it like for example Tailor [8].

The analyser may read a partial solver specification along with the model of the problem to be analysed to still allow fine-tuning by human experts while not requiring it. This also allows for running the analyser incrementally, refining the solver specification based on analysis and decisions made in earlier steps.

The analyser creates a constraint optimisation model of the problem of specialising a constraint solver. The decision variables are the design decisions to be made and the values in their domains are the options which are available for their implementation. The constraints encode which parts are required to solve the problem and how they interact. For example, the constraints could require the presence of an integer variable type and an equals constraint which is able to handle integer variables. A solution to this constraint problem is a solver specification that describes a solver which is able to solve the problem described in the original model. The weight attached to each solution describes the performance of the specialised solver and could be based on static measures of performance as well as dynamic ones; e.g. predefined numbers describing the performance of a specific algorithm and experimental results from probing a specific implementation.

This metamodel enables the use of constraint programming techniques for generating the specialised solver and ensures that a solver specification can be created efficiently even for large metamodels.

The result of running the analyser phase of the system is a solver specification which specifies a solver tailored to the analysed problem model.

### 2.2 The generator

The generator reads the solver specification produced by the analyser and constructs a specialised constraint solver accordingly. It may modify an existing solver, or synthesise one from scratch. The generated solver has to conform to the solver specification, but beyond that, no restrictions are imposed. In particular, the generator does not guarantee that the generated specialised solver will have better performance than a general solver, or indeed be able to solve constraint problems at all – this is encoded in the solver specification.

In addition to the high-level design decisions fixed in the solver specification, the generator can perform low-level optimisations which are specific to the implementation of the specialised solver. It could for example decide to represent domains with a data type of smaller range than the default one to save space.

The scope of the generator is not limited to generating the source code which implements the specialised solver, but also includes the system to build it.

The result of running the generator phase of the system is a specialised solver which conforms to the solver specification.

## 3 Preliminary implementation and experimental results

We have started implementing the design proposed above in a system which operates on top of Minion [9]. The analyser reads Minion input files and writes a solver specification which describes the constraints and the variable types which are required to solve the problem. It does not currently create a metamodel of the problem. The generator modifies Minion to support only those constraints and variable types. It furthermore does some additional low-level optimisations by removing infrastructure code which is not required for the specialised solver. The current implementation of Dominion sits between the existing Tailor and Minion projects – it takes Minion problem files, which may have been generated by Tailor, as input, and generates a specialised Minion solver.

The generated solver is specialised for models of problem instances from the problem class the analysed instance belongs to. The models have to be the same with respect to the constraints and variable types used.

Experimental results for models from four different problem classes are shown in Figure 2. The graph only compares the CPU time Minion and the specialised solver took to solve the problem; it does not take into account the overhead of running Dominion – analysing the problem model, generating the solver, and compiling it, which was in the order of a few minutes for all of the benchmarks.

The problem classes Balanced Incomplete Block Design, Golomb Ruler, -Queens, and Social Golfers were chosen because they use a range of different constraints and variable types. Hence the optimisations Dominion can perform are different for each of these problem classes. This is reflected in the experimental results by different performance improvements for different classes.

Figure 2 illustrates two key points. The first point is that even a quite basic implementation of Dominion which does only a few optimisations can yield significant performance improvements over standard Minion. The second point is that the performance improvement does not only depend on the problem class, but also on the instance, even if no additional optimisations beyond the class level were performed. For both the Balanced Incomplete Block Design and the Social Golfers problem classes the largest instances yield significantly higher improvements than smaller ones.

At this stage of the implementation, our aim is to show that a specialised solver can perform better than a general one. We believe that Figure 2 conclusively shows that. As the problem models become larger and take longer to solve, the improvement in terms of absolute run time difference becomes larger as well. Hence the more or less constant overhead of running Dominion is amortised for larger and more difficult problem models, which are our main focus. Generating a specialised solver for problem classes and instances is always going to entail a certain overhead, making the approach infeasible for small and quick-to-solve problems.

## 4 Conclusion and future work

We have described the design of Dominion, a solver generator, and demonstrated its feasibility by providing a preliminary implementation. We have furthermore demonstrated the feasibility and effectiveness of the general approach of generating specialised constraint solvers for problem models by running experiments with Minion and Dominion-generated solvers and obtaining results which show significant performance improvements. These results do not take the overhead of running Dominion into account, but we are confident that for large problem models there will be an overall performance improvement despite the overhead.

Based on our experiences with Dominion, we propose that the next step should be the generation of specialised variable types for the model of a problem instance. Dominion will extend Minion and create variable types of the sort “Integer domain ranging from 10 to 22”. This not only allows us to choose different representations for variables based on the domain, but also to simplify and speed up services provided by the variable, such as checking the bounds of the domain or checking whether a particular value is in the domain.

The implementation of specialised variable types requires generating solvers for models of problem instances because the analysed problem model is essentially rewritten. The instance the solver was specialised for will be encoded in it and no further input will be required to solve the problem. We expect this optimisation to provide an additional improvement in performance which is more consistent across different problem classes, i.e. we expect significant improvements for all problem models and not just some.

We are also planning on continuing to specify the details of Dominion and implementing it.

## 5 Acknowledgements

The authors thank Chris Jefferson for extensive help with the internals of Minion and the anonymous reviewers for their feedback. Lars Kotthoff is supported by a SICSA studentship.

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