Efficient structural symmetry breaking for constraint satisfaction problems

Symmetry breaking for constraint satisfaction problems (CSPs) has attracted considerable attention in recent years. 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 and variable interchangeability is tractable using dedicated search procedures or symmetry-breaking constraints that allow nogoods and their symmetrically equivalent solutions to be stored and checked efficiently

A Partial Taxonomy of Substitutability and Interchangeability

Substitutability, interchangeability and related concepts in Constraint Programming were introduced approximately twenty years ago and have given rise to considerable subsequent research. We survey this work, classify, and relate the different concepts, and indicate directions for future work, in particular with respect to making connections with research into symmetry breaking. This paper is a condensed version of a larger work in progress.Comment: 18 pages, The 10th International Workshop on Symmetry in Constraint Satisfaction Problems (SymCon'10

Symmetry Breaking Using Value Precedence

We present a comprehensive study of the use of value precedence constraints to break value symmetry. We first give a simple encoding of value precedence into ternary constraints that is both efficient and effective at breaking symmetry. We then extend value precedence to deal with a number of generalizations like wreath value 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.Comment: 17th European Conference on Artificial Intelligenc

Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. 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 symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Abstraction-based action ordering in planning

Many planning problems contain collections of symmetric objects, actions and structures which render them difficult to solve efficiently. It has been shown that the detection and exploitation of symmetric structure in planning problems can dramatically reduce the size of the search space and the time taken to find a solution. We present the idea of using an abstraction of the problem domain to reveal symmetric structure and guide the navigation of the search space. We show that this is effective even in domains in which there is little accessible symmetric structure available for pruning. Proactive exploitation represents a flexible and powerfulalternative to the symmetry-breaking strategies exploited in earlier work in planning and CSPs. The notion of almost symmetry is defined and results are presented showing that proactive exploitation of almost symmetry can improve the performance of a heuristic forward search planner

Increasing symmetry breaking by preserving target symmetries and eliminating eliminated symmetries in constraint satisfaction.

在約束滿足問題中，破壞指數量級數量的所有對稱通常過於昂貴。在實踐中，我們通常只有效地破壞對稱的一個子集。我們稱之為目標對稱。在靜態對稱破壞中，我們的目標是發佈一套約束去破壞這些目標對稱，以達到減少解集以及搜索空間的效果。一個問題中的所有對稱之間是互相交織的。一個旨在特定對稱的破壞對稱約束几乎總會產生副作用，而不僅僅破壞了預期的對稱。破壞相同目標對稱的不同約束可以有不同的副作用。傳統智慧告訴我們應該選擇一個破壞更多對稱從而有更多副作用的破壞對稱約束。雖然這樣的說法在許多方面上都是有效的，我們應該更加注意副作用發生的地方。給與一個約束滿足問題，一個對稱被一個約束保留當且僅當該對稱仍然是新的約束滿足問題的對稱。這個新的約束滿足問題是有原問題加上該約束組成的。我們給出定律和例子，以表明發佈儘量保留目標對稱以及限制它的副作用發生在非目標對稱上的破壞約束是有利的。這些好處來自于被破壞的對稱數目以及一個對稱被破壞（或消除）的程度，并導致一個較小的解集和搜索空間。但是，對稱不一定會被保留。我們顯示，旨在一個已經被消除的目標對稱的破壞對稱約束仍然可以被發佈。我們建議根據問題的約束以及其他破壞對稱約束來選擇破壞對稱約束，以繼續消除更多的對稱。我們進行了廣泛的實驗來確認我們的建議的可行性與效率。Breaking the exponential number of all symmetries of a constraint satisfaction problem is often too costly. In practice, we often aim at breaking a subset of the symmetries efficiently, which we call target symmetries. In static sym-metry breaking, the goal is to post a set of constraints to break these target symmetries in order to reduce the solution set and thus also the search space. Symmetries of a problem are all intertwined. A symmetry breaking constraint intended for a particular symmetry almost always breaks more than just the intended symmetry as a side-effect. Different constraints for breaking the same target symmetry can have different side-effects. Conventional wisdom suggests that we should select a symmetry breaking constraint that has more side-effects by breaking more symmetries. While this wisdom is valid in many ways, we should be careful where the side-effects take place.A symmetry σ of a CSP P =(V, D, C) is preserved by a set of symmetry breaking constraints C{U+02E2}{U+1D47} i σ is a symmetry of P¹ =(V, D, CU C{U+02E2}{U+1D47}). We give theorems and examples to demonstrate that it is beneficial to post symmetry breaking constraints that preserve the target symmetries and restrict the side-effects to only non-target symmetries as much as possible. The benefits are in terms of the number of symmetries broken and the extent to which a symmetry is broken (or eliminated), resulting in a smaller solution set and search space. However, symmetry preservation may not always hold. We illustrate that symmetry breaking constraints, which aim at a target symmetry that is already eliminated, can still be posted. To continue eliminating more symmetries, we suggest to select symmetry breaking constraints based on problem constraints and other symmetry breaking constraints. Extensive experiments are also conducted to confirm the feasibility and efficiency of our proposal empirically.Detailed summary in vernacular field only.Detailed summary in vernacular field only.Li, Jingying.Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.Includes bibliographical references (leaves 101-112).Abstracts also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Motivation and Goals --- p.3Chapter 1.3 --- Outline of Thesis --- p.5Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Search --- p.9Chapter 2.1.2 --- Consistency Techniques --- p.12Chapter 2.1.3 --- Local Consistencies with Backtracking Search --- p.15Chapter 2.2 --- Symmetry Breaking in CSPs --- p.16Chapter 2.2.1 --- Symmetry Classes --- p.18Chapter 2.2.2 --- Breaking Symmetries --- p.22Chapter 2.2.3 --- Variable and Value Symmetries --- p.23Chapter 2.2.4 --- Symmetry Breaking Constraints --- p.26Chapter 3 --- Effects of Symmetry Breaking Constraints --- p.29Chapter 3.1 --- Removing Symmetric Search Space --- p.29Chapter 3.1.1 --- Properties --- p.30Chapter 3.1.2 --- Canonical Variable Orderings --- p.31Chapter 3.1.3 --- Regenerating All Solutions --- p.33Chapter 3.1.4 --- Remaining Solution Set Sizes --- p.36Chapter 3.2 --- Constraint Interactions in Propagation --- p.43Chapter 4 --- Choices of Symmetry Breaking Constraints --- p.45Chapter 4.1 --- Side-Effects --- p.45Chapter 4.2 --- Symmetry Preservation --- p.50Chapter 4.2.1 --- De nition and Properties --- p.50Chapter 4.2.2 --- Solution Reduction --- p.54Chapter 4.2.3 --- Preservation Examples --- p.55Chapter 4.2.4 --- Preserving Order --- p.64Chapter 4.3 --- Eliminating Eliminated Symmetries --- p.65Chapter 4.3.1 --- Further Elimination --- p.65Chapter 4.3.2 --- Aggressive Elimination --- p.71Chapter 4.4 --- Interactions with Problem Constraints --- p.72Chapter 4.4.1 --- Further Simplification --- p.72Chapter 4.4.2 --- Increasing Constraint Propagation --- p.73Chapter 5 --- Experiments --- p.75Chapter 5.1 --- Symmetry Preservation --- p.75Chapter 5.1.1 --- Diagonal Latin Square Problem --- p.76Chapter 5.1.2 --- NN-Queen Problem --- p.77Chapter 5.1.3 --- Error Correcting Code - Lee Distance (ECCLD) --- p.78Chapter 5.2 --- Eliminating Eliminated Symmetries --- p.80Chapter 5.2.1 --- Equidistance Frequency Permutation Array Problem --- p.80Chapter 5.2.2 --- Cover Array Problem --- p.82Chapter 5.2.3 --- Sports League Scheduling Problem --- p.83Chapter 6 --- Related Work --- p.86Chapter 6.1 --- Symmetry Breaking Approaches --- p.86Chapter 6.2 --- Reducing Overhead and Increasing Propagation --- p.90Chapter 6.3 --- Selecting and Generating Choices --- p.91Chapter 6.3.1 --- Reducing Conflict with Search Heuristic --- p.92Chapter 6.3.2 --- Choosing the Subset of Symmetries --- p.93Chapter 6.4 --- Detecting Symmetries --- p.93Chapter 7 --- Conclusion and Remarks --- p.95Chapter 7.1 --- Conclusion --- p.95Chapter 7.2 --- Discussions --- p.97Chapter 7.3 --- Future Work --- p.99Bibliography --- p.10

Symmetry breaking in numeric constraint problems

Symmetry-breaking constraints in the form of inequalities between variables have been proposed for a few kind of solution symmetries in numeric CSPs. We show that, for the variable symmetries among those, the proposed inequalities are but a specific case of a relaxation of the well-known LEX constraints extensively used for discrete CSPs. We discuss the merits of this relaxation and present experimental evidences of its practical interest.Postprint (author’s final draft

Symmetry in constraint programming

Constraint programming is an invaluable tool for solving many of the complex NP-complete problems that we need solutions to. These problems can be easily described as Constraint Satisfaction Problems (CSPs) and then passed to constraint solvers: complex pieces of software written to solve general CSPs efficiently. Many of the problems we need solutions to are real world problems: planning (e.g. vehicle routing), scheduling (e.g. job shop schedules) and timetabling problems (e.g. staff rotas) to name but a few. In the real world, we place structure on objects to make them easier to deal with. This manifests itself as symmetry. The symmetry in these real world problems make them easier to deal with for humans. However, they lead to a great deal of redundancy when using computational methods of problem solving. Thus, this thesis examines some of the many aspects of utilising the symmetry of CSPs to reduce the amount of computation needed by constraint solvers. In this thesis we look at the ease of use of previous symmetry breaking methods. We introduce a new and novel method of describing the symmetries of CSPs. We look at previous methods of symmetry breaking and show how we can drastically reduce their computation while still breaking all symmetry. We give the first detailed investigation into the behaviour of breaking only subsets of all symmetry. We look at how this affects the performance of constraint solvers before discovering the properties of a good symmetry. We then present an original method for choosing the best symmetries to use. Finally, we look at areas of redundant computation in constraint solvers that no other research has examined. New ways of dealing with this redundancy are proposed with results of an example implementation which improves efficiency by several orders of magnitude

Techniques for Bundling the Solution Space of Finite Constraint Satisfaction Problems

We study the backtrack-search procedure with forward checking (FCBT) for finding all solutions to a finite Constraint Satisfaction Problem (CSP). We describe how to use dynamic interchangeability to enhance the performance of search and represent the solution space in a compact manner. We evaluate this strategy (FC-DNPI) in terms of the numbers of nodes visited, constraints checked, and solution bundles generated by comparing it, theoretically and empirically, to other search strategies. We show that FC-DNPI is equivalent to search with the Cross Product Representation (FC-CPR) of [Hubbe and Freuder 1992] in terms of the numbers of solution bundles and constraint checks, while it reduces the number of nodes visited. We establish that both strategies are always superior to FC-BT in terms of all three criteria and dynamic bundling is always beneficial. Further, we compare FC-DNPI to the search procedure of [Haselböck 1993], which exploits static, pre-computed interchangeability relations. We show that the former never generates more solution bundles nor expands more nodes than the latter, and often reduces the number of constraint checks. We also propose, without evaluating them, amendments to the strategy of [Haselböck 1993] to improve its performance and reduce the number of constraint checks