Christopher Mears

On implementing symmetry detection

Automatic symmetry detection has received a significant amount of interest, which has resulted in a large number of proposed methods. This paper reports on our experiences while implementing the approach of Puget (CP2005, LNCS, vol. 3709, pp. 475–489. Springer, 2005). In particular, it proposes a modification to the approach to deal with general expressions, discusses the insights gained, and gives the results of an experimental evaluation of the accuracy and efficiency of the approach.

Symmetries and lazy clause generation

Lazy clause generation is a powerful approach to reducing search in constraint programming. This is achieved by recording sets of domain restrictions that previously led to failure as new clausal propagators. Symmetry breaking approaches are also powerful methods for reducing search by recognizing that parts of the search tree are symmetric and do not need to be explored. In this paper we show how we can successfully combine symmetry breaking methods with lazy clause generation. Further, we show that the more precise nogoods generated by a lazy clause solver allow our combined approach to exploit redundancies that cannot be exploited via any previous symmetry breaking method, be it static or dynamic.

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.

Proving symmetries by model transformation

The presence of symmetries in a constraint satisfaction problem gives an opportunity for more efficient search. Within the class of matrix models, we show that the problem of deciding whether some well known permutations are model symmetries (solution symmetries on every instance) is undecidable. We then provide a new approach to proving the model symmetries by way of model transformations. Given a model M and a candidate symmetry σ, the approach first syntactically applies σ to M and then shows that the resulting model σ(M) is semantically equivalent to M. We demonstrate this approach with an implementation that reduces equivalence to a sentence in Presburger arithmetic, using the modelling language MiniZinc and the term re-writing language Cadmium, and show that it is capable of proving common symmetries in models.

Symmetry Propagation: Improved Dynamic Symmetry Breaking in SAT

For constraint programming, many well performing dynamic symmetry breaking techniques have been devised. For propositional satisfiability solving, dynamic symmetry breaking is still either slower or less general than static symmetry breaking. This paper presents Symmetry Propagation, which is an improvement to Lightweight Dynamic Symmetry Breaking, a dynamic symmetry breaking approach from CP. Symmetry Propagation uses any given symmetry as a propagator, and as a result is a general symmetry breaking technique. Experiments with an implementation in the SAT solver Minisat show that on many benchmarks, Symmetry Propagation outperforms the state-of-the-art static symmetry breaking method Shatter.

What do Constraint Programming Users Want to See? Exploring the Role of Visualisation in Profiling of Models and Search

Constraint programming allows difficult combinatorial problems to be modelled declaratively and solved automatically. Advances in solver technologies over recent years have allowed the successful use of constraint programming in many application areas. However, when a particular solvers search for a solution takes too long, the complexity of the constraint program execution hinders the programmers ability to profile that search and understand how it relates to their model. Therefore, effective tools to support such profiling and allow users of constraint programming technologies to refine their model or experiment with different search parameters are essential. This paper details the first user-centred design process for visual profiling tools in this domain. We report on: our insights and opportunities identified through an on-line questionnaire and a creativity workshop with domain experts carried out to elicit requirements for analytical and visual profiling techniques; our designs and functional prototypes realising such techniques; and case studies demonstrating how these techniques shed light on the behaviour of the solvers in practice.

Globalizing Constraint Models

We present a method that, given a constraint model, suggests global constraints to replace parts of it. This helps non-expert users to write higher-level models that are easier to reason about and may result in better solving performance. Our method exploits the structure of the model by considering combinations of the constraints, collections of variables, parameters and loops already present in the model, as well as parameter data from several data files. We assign a score to a candidate global constraint by comparing a sample of its solution space with that of the part of the model it is intended to replace. The top-scoring global constraints are presented to the user through an interactive display, which shows how they could be incorporated into the model. The MiniZinc Globalizer, our implementation of the method for the MiniZinc modelling language, is available on the web. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council. This research was partly sponsored by the Australian Research Council grant DP110102258.

Visual search tree profiling

Understanding how the search space is explored for a given constraint problem – and how it changes for different models, solvers or search strategies – is crucial for efficient solving. Yet programmers often have to rely on the crude aggregate measures of the search that are provided by solvers, or on visualisation tools that can show the search tree, but do not offer sophisticated ways to navigate and analyse it, particularly for large trees. We present an architecture for profiling a constraint programming search that is based on a lightweight instrumentation of the solver. The architecture combines a visualisation of the search tree with various tools for convenient navigation and analysis of the search. These include identifying repeated subtrees, high-level abstraction and navigation of the tree, and the comparison of two search trees. The resulting system is akin to a traditional program profiler, which helps the user to focus on the parts of the execution where an improvement to their program would have the greatest effect.

Learning from Learning Solvers

Modern constraint programming solvers incorporate SAT-style clause learning, where sets of domain restrictions that lead to failure are recorded as new clausal propagators. While this can yield dramatic reductions in search, there are also cases where clause learning does not improve or even hinders performance. Unfortunately, the reasons for these differences in behaviour are not well understood in practice. We aim to cast some light on the practical behaviour of learning solvers by profiling their execution. In particular, we instrument the learning solver Chuffed to produce a detailed record of its execution and extend a graphical profiling tool to appropriately display this information. Further, this profiler enables users to measure the impact of the learnt clauses by comparing Chuffed’s execution with that of a non-learning solver, and examining the points at which their behaviours diverge. We show that analysing a solver’s execution in this way can be useful not only to better understand its behaviour — opening what is typically a black box — but also to infer modifications to the original constraint model that can improve the performance of both learning and non-learning solvers.

Optimised robust treatment plans for prostate cancer focal brachytherapy

Abstract Focal brachytherapy is a clinical procedure that can be used to treat low-risk prostate cancer with reduced side-effects compared to conventional brachytherapy. Current practice is to man- ually plan the placement of radioactive seeds inside the prostate to achieve a desired treatment dose. Problems with the current practice are that the manual planning is time-consuming and high doses to the urethra and rectum cause undesirable side-effects. To address this problem, we have designed an optimisation algorithm that constructs treatment plans which achieve the desired dose while minimizing dose to organs at risk. We also show that these seed plans are robust to post-operative movement of the seeds within the prostate.