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31
Symmetry breaking using value precedence
 In: Proceedings of the 17th ECAI, European Conference on Artificial Intelligence, IOS
, 2006
"... Abstract. We present a comprehensive study of the use of valueprecedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is bothefficient and effective at breaking symmetry. We then extend value precedence to deal with a number o ..."
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Cited by 24 (22 self)
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Abstract. We present a comprehensive study of the use of valueprecedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is bothefficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreathvalue and partial interchangeability. We also show that value precedence is closely related to lexicographical ordering. Finally, we consider the interaction between value precedence and symmetry breaking constraints for variable symmetries. 1 INTRODUCTION Symmetry is an important aspect of many search problems. Symmetry occurs naturally in many problems (e.g. if we have two identical
Breaking value symmetry.
 In 13th Int.
, 2007
"... Abstract Symmetry is an important factor in solving many constraint satisfaction problems. One common type of symmetry is when we have symmetric values. In a recent series of papers, we have studied methods to break value symmetries ..."
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Cited by 22 (18 self)
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Abstract Symmetry is an important factor in solving many constraint satisfaction problems. One common type of symmetry is when we have symmetric values. In a recent series of papers, we have studied methods to break value symmetries
Breaking instanceindependent symmetries in exact graph coloring
 JOURNAL OF ARTIFICIAL INTELLIGENCE RESEARCH
, 2004
"... Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Naturallyoccurring instances of such problems are often small and can be solved optimally. A recent wave of improvements in algorithms for ..."
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Cited by 18 (6 self)
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Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Naturallyoccurring instances of such problems are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 01 ILP suggests generic problemreduction methods, rather than problemspecific heuristics, because (1) heuristics are easily upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. The NPspec project offers a language to specify NPproblems and automatic reductions to SAT. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetrybreaking, in particular when certain kinds of symmetry are present in all generated instances. Our surprising conclusion is that instanceindependent symmetries should often be processed together with instancespecific symmetries rather than earlier, at the specification level.
M.: Detecting and breaking symmetries by reasoning on problem specifications
 In: Abstraction, Reformulation and Approximation. Lecture Notes in Computer Science
, 2005
"... Abstract. In this paper we address the problem of detecting and breaking symmetries in combinatorial problems, following the approach of imposing additional symmetrybreaking constraints. Differently from other works in the literature, we attack the problem at the specification level. In fact, man ..."
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Cited by 18 (9 self)
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Abstract. In this paper we address the problem of detecting and breaking symmetries in combinatorial problems, following the approach of imposing additional symmetrybreaking constraints. Differently from other works in the literature, we attack the problem at the specification level. In fact, many symmetries depend on the structure of the problem, and not on the particular input instance. Hence, they can be easily detected by reasoning on the specification, and appropriate symmetrybreaking formulae generated. We give formal definitions of symmetries and symmetrybreaking formulae on specifications written in existential secondorder logic, clarifying the new definitions on some specifications: Graph 3coloring, Social golfer, and Protein folding problems. Finally, we show experimentally that, applying this technique, even if in a naive way, to specifications written in stateoftheart languages, e.g., opl, may greatly improve search efficiency. 1
Symmetry Breaking Constraints for Value Symmetries in Constraint Satisfaction
 Constraints
, 2006
"... Constraint satisfaction problems (CSPs) sometimes contain both variable symmetries and value symmetries, causing adverse effects on CSP solvers based on tree search. As a remedy, symmetry breaking constraints are commonly used. While variable symmetry breaking constraints can be expressed easily and ..."
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Cited by 17 (2 self)
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Constraint satisfaction problems (CSPs) sometimes contain both variable symmetries and value symmetries, causing adverse effects on CSP solvers based on tree search. As a remedy, symmetry breaking constraints are commonly used. While variable symmetry breaking constraints can be expressed easily and propagated efficiently using lexicographic ordering, value symmetry breaking constraints are often difficult to formulate. In this paper, we propose two methods of using symmetry breaking constraints to tackle value symmetries. First, we show theoretically when value symmetries in one CSP correspond to variable symmetries in another CSP of the same problem. We also show when variable symmetry breaking constraints in the two CSPs, combined using channeling constraints, are consistent. Such results allow us to tackle value symmetries efficiently using additional CSP variables and channeling constraints. Second, we introduce value precedence, a notion which can be used to break a common class of value symmetries, namely symmetries of indistinguishable values. While value precedence can be expressed using inefficient ifthen constraints in existing CSP solvers, we propose efficient propagation algorithms for implementing global value precedence constraints. We also characterize several theoretical properties of the value precedence constraints. Extensive experiments are conducted to verify the feasibility and efficiency of the two proposals. 1.
Static and dynamic structural symmetry breaking
 PROCEEDINGS OF CP’06, VOLUME 4204 OF LNCS
, 2006
"... We reconsider the idea of structural symmetry breaking for constraint satisfaction problems (CSPs). We show that the dynamic dominance checks used in symmetry breaking by dominancedetection search for CSPs with piecewise variable and value symmetries have a static counterpart: there exists a set o ..."
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Cited by 16 (2 self)
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We reconsider the idea of structural symmetry breaking for constraint satisfaction problems (CSPs). We show that the dynamic dominance checks used in symmetry breaking by dominancedetection search for CSPs with piecewise variable and value symmetries have a static counterpart: there exists a set of constraints that can be posted at the root node and that breaks all the compositions of these (unconditional) symmetries. The amount of these symmetrybreaking constraints is linear in the size of the problem, and yet they are able to remove a superexponential number of symmetries on both values and variables. Moreover, we compare the search trees under static and dynamic structural symmetry breaking when using fixed variable and value orderings. These results are then generalized to wreathsymmetric CSPs with both variable and value symmetries. We show that there also exists a polynomialtime dominancedetection algorithm for this class of CSPs, as well as a linearsized set of constraints that breaks these symmetries statically.
Compositional derivation of symmetries for constraint satisfaction
 Proceedings of SARA’05, volume 3607 of LNAI
"... Abstract. 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. Its key insight is the recognition of the special role of global constrain ..."
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Cited by 12 (3 self)
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Abstract. 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. Its 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 nontrivial applications. The paper also discusses the potential of reformulations and highlevel modelling abstractions to strengthen symmetry discovery. 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. Its 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 nontrivial applications. The paper also discusses the potential of reformulations and highlevel modeling abstractions to strengthen symmetry discovery.
Model restarts for structural symmetry breaking
 in 14th Int. Conf. on the Principles and Practice of Constraint Programming
, 2008
"... There exist various methods to break symmetries. The two that concern us in this paper are static symmetry breaking where we add static constraints to the problem (see e.g. [1, 3]) and symmetry breaking by dominance detection (SBDD) where we filter values based on a symmetric dominance analysis when ..."
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Cited by 5 (1 self)
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There exist various methods to break symmetries. The two that concern us in this paper are static symmetry breaking where we add static constraints to the problem (see e.g. [1, 3]) and symmetry breaking by dominance detection (SBDD) where we filter values based on a symmetric dominance analysis when comparing the current searchnode with those that were previously expanded [2, 5]. The core task of SBDD is dominance detection. The first provably polynomialtime dominance checkers for value symmetry were devised in [18] and [14]. For problems exhibiting both “piecewise ” symmetric values and variables, [15] devised structural symmetry breaking (SSB). SSB is a polynomialtime dominance checker for piecewise symmetries which, used within SBDD, eliminates symmetric subproblems from the searchtree. Piecewise symmetries are of particular interest as they result naturally from symmetry detection based on a static analysis of a given CSP that exploits the knowledge about problem substructures as captured in global constraints [17]. Static SSB was developed in [4] and is based on the structural abstractions that were introduced in [17]. Compared with other symmetrybreaking techniques, the big advantage of dynamic
The steel mill slab design problem revisited
 Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, 5th International Conference, CPAIOR 2008, volume 5015 of Lecture Notes in Computer Science
, 2008
"... Abstract. Recently, Gargani and Refalo (G&R) presented an elegant model for the Steel Mill Slab Design Problem (Problem 38 in the CSP LIB). Contrary to earlier approaches, their model does not use 0/1 variables but exploits the traditional expressiveness of constraint programming. G&R indica ..."
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Cited by 5 (2 self)
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Abstract. Recently, Gargani and Refalo (G&R) presented an elegant model for the Steel Mill Slab Design Problem (Problem 38 in the CSP LIB). Contrary to earlier approaches, their model does not use 0/1 variables but exploits the traditional expressiveness of constraint programming. G&R indicated that static symmetrybreaking constraints proposed earlier are not effective on this model, as these interact with their heuristic. Instead they use large neighborhood search to obtain solutions quickly. This paper shows that a simple search procedure breaking symmetries dynamically leads to a constraint program solving the problem in a few seconds, while maintaining the completeness of the approach and removing the need for large neighborhood search. 1
SubOptimality Approximation
 In CP05
, 2005
"... Abstract. The suboptimality approximation problem considers an optimization problem O, its optimal solution σ ∗ , and a variable x with domain {d1,..., dm} and returns approximations to O[x ← d1],..., O[x ← dm], where O[x ← d1] denotes the problem O with x assigned to di. The suboptimality approxi ..."
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Cited by 5 (3 self)
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Abstract. The suboptimality approximation problem considers an optimization problem O, its optimal solution σ ∗ , and a variable x with domain {d1,..., dm} and returns approximations to O[x ← d1],..., O[x ← dm], where O[x ← d1] denotes the problem O with x assigned to di. The suboptimality approximation problem is at the core of online stochastic optimization algorithms and it can also be used for solution repair and approximate filtering of optimization constraints. This paper formalizes the problem and presents suboptimality approximation algorithms for metric TSPs, packet scheduling, and metric kmedians that run faster than the optimal or approximation algorithms. It also presents results on the hardness/easiness of suboptimality approximations. 1