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Symmetries of Symmetry Breaking Constraints
"... Symmetry is an important feature of many constraint programs. We show that any symmetry acting on a set of symmetry breaking constraints can be used to break symmetry. Different symmetries pick out different solutions in each symmetry class. We use these observations in two methods for eliminating s ..."
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Symmetry is an important feature of many constraint programs. We show that any symmetry acting on a set of symmetry breaking constraints can be used to break symmetry. Different symmetries pick out different solutions in each symmetry class. We use these observations in two methods for eliminating symmetry from a problem. These methods are designed to have many of the advantages of symmetry breaking methods that post static symmetry breaking constraint without some of the disadvantages. In particular, the two methods prune the search space using fast and efficient propagation of posted constraints, whilst reducing the conflict between symmetry breaking and branching heuristics. Experimental results show that the two methods perform well on some standard benchmarks. 1
On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry ⋆
"... Abstract. We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of pos ..."
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Abstract. We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NPhard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems. 1
Parameterized Complexity Results in Symmetry Breaking
, 2010
"... Symmetry is a common feature of many combinatorial problems. Unfortunately eliminating all symmetry from a problem is often computationally intractable. This paper argues that recent parameterized complexity results provide insight into that intractability and help identify special cases in which sy ..."
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Symmetry is a common feature of many combinatorial problems. Unfortunately eliminating all symmetry from a problem is often computationally intractable. This paper argues that recent parameterized complexity results provide insight into that intractability and help identify special cases in which symmetry can be dealt with more tractably.
The Extended Global Cardinality Constraint: An Empirical Survey
, 2010
"... The Extended Global Cardinality Constraint (EGCC) is a vital component of constraint solving systems, since it is very widely used to model diverse problems. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this pape ..."
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The Extended Global Cardinality Constraint (EGCC) is a vital component of constraint solving systems, since it is very widely used to model diverse problems. The literature contains many different versions of this constraint, which trade strength of inference against computational cost. In this paper, I focus on the highest strength of inference usually considered, enforcing generalized arc consistency (GAC) on the target variables. This work is an extensive empirical survey of algorithms and optimizations, considering both GAC on the target variables, and tightening the bounds of the cardinality variables. I evaluate a number of key techniques from the literature, and report important implementation details of those techniques, which have often not been described in published papers. Two new optimizations are proposed for EGCC. One of the novel optimizations (dynamic partitioning, generalized from AllDifferent) was found to speed up search by 5.6 times in the best case and 1.56 times on average, while exploring the same search tree. The empirical work represents by far the most extensive set of experiments on variants of GAC algorithms for EGCC. Overall, the best combination of optimizations gives a mean speedup of over 50 times compared to the same implementation without the optimizations.
Symmetry Breaking Constraints: Recent Results
, 2012
"... Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symm ..."
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Symmetry is an important problem in many combinatorial problems. One way of dealing with symmetry is to add constraints that eliminate symmetric solutions. We survey recent results in this area, focusing especially on two common and useful cases: symmetry breaking constraints for row and column symmetry, and symmetry breaking constraints for eliminating value symmetry.
Automated symmetry breaking and model selection in Conjure
 IN: 19TH INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING
, 2013
"... Constraint modelling is widely recognised as a key bottleneck in applying constraint solving to a problem of interest. The CONJURE automated constraint modelling system addresses this problem by automatically refining constraint models from problem specifications written in the ESSENCE language. ESS ..."
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Constraint modelling is widely recognised as a key bottleneck in applying constraint solving to a problem of interest. The CONJURE automated constraint modelling system addresses this problem by automatically refining constraint models from problem specifications written in the ESSENCE language. ESSENCE provides familiar mathematical concepts like sets, functions and relations nested to any depth. To date, CONJURE has been able to produce a set of alternative model kernels (i.e. without advanced features such as symmetry breaking or implied constraints) for a given specification. The first contribution of this paper is a method by which CONJURE can break symmetry in a model as it is introduced by the modelling process. This works at the problem class level, rather than just individual instances, and does not require an expensive detection step after the model has been formulated. This allows CONJURE to produce a higher quality set of models. A further limitation of CONJURE has been the lack of a mechanism to select among the models it produces. The second contribution of this paper is to present two such mechanisms, allowing effective models to be chosen automatically.
Partial Symmetry Breaking by Local Search in the Group ⋆
"... Abstract. The presence of symmetry in constraint satisfaction problems can cause a great deal of wasted search effort, and several methods for breaking symmetries have been reported. In this paper we describe a new method called Symmetry Breaking by Nonstationary Optimisation, which interleaves loca ..."
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Abstract. The presence of symmetry in constraint satisfaction problems can cause a great deal of wasted search effort, and several methods for breaking symmetries have been reported. In this paper we describe a new method called Symmetry Breaking by Nonstationary Optimisation, which interleaves local search in the symmetry group with backtrack search on the constraint problem. It can be tuned to break arbitrarily many symmetries with high runtime overhead, or as a lightweight but powerful method with low runtime overhead. It has negligible memory requirement, it combines well with lexleader constraints, and its benefit increases with problem hardness. 1