Symmetry within Solutions

We define the concept of an internal symmetry. This is a symmety within a solution of a constraint satisfaction problem. We compare this to solution symmetry, which is a mapping between different solutions of the same problem. We argue that we may be able to exploit both types of symmetry when finding solutions. We illustrate the potential of exploiting internal symmetries on two benchmark domains: Van der Waerden numbers and graceful graphs. By identifying internal symmetries we are able to extend the state of the art in both cases.Comment: AAAI 2010, Proceedings of Twenty-Fourth AAAI Conference on Artificial Intelligenc

Symmetry Breaking Constraints: Recent Results

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 symmetryComment: To appear in Proceedings of Twenty-Sixth Conference on Artificial Intelligence (AAAI-12

Bus driver rostering by hybrid methods based on column generation

Tese de doutoramento, Informática (Engenharia Informática), Universidade de Lisboa, Faculdade de Ciências, 2018Rostering problems arise in a diversity of areas where, according to the business and labor rules, distinct variants of the problem are obtained with different constraints and objectives considered. The diversity of existing rostering problems, allied with their complexity, justifies the activity of the research community addressing them. The current research on rostering problems is mainly devoted to achieving near-optimal solutions since, most of the times, the time needed to obtain optimal solutions is very high. In this thesis, a Bus Driver Rostering Problem is addressed, to which an integer programming model is adapted from the literature, and a new decomposition model with three distinct subproblems representations is proposed. The main objective of this research is to develop and evaluate a new approach to obtain solutions to the problem in study. The new approach follows the concept of search based on column generation, which consists in using the column generation method to solve problems represented by decomposition models and, after, applying metaheuristics to search for the best combination of subproblem solutions that, when combined, result in a feasible integer solution to the complete problem. Besides the new decomposition models proposed for the Bus Driver Rostering Problem, this thesis proposes the extension of the concept of search by column generation to allow using population-based metaheuristics and presents the implementation of the first metaheuristic using populations, based on the extension, which is an evolutionary algorithm. There are two additional contributions of this thesis. The first is an heuristic allowing to obtain solutions for the subproblems in an individual or aggregated way and the second is a repair operator which can be used by the metaheuristics to repair infeasible solutions and, eventually, generate missing subproblem solutions needed. The thesis includes the description and results from an extensive set of computational tests. Multiple configurations of the column generation with three decomposition models are tested to assess the best configuration to use in the generation of the search space for the metaheuristic. Additional tests compare distinct single-solution metaheuristics and our basic evolutionary algorithm in the search for integer solutions in the search space obtained by the column generation. A final set of tests compares the results of our final algorithm (with the best column generation configuration and the evolutionary algorithm using the repair operator) and the solutions obtained by solving the problem represented by the integer programming model with a commercial solver.Programa de Apoio à Formação Avançada de Docentes do Ensino Superior Politécnico (PROTEC), SFRH/PROTEC/67405/201

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

Proceedings of CSCLP 2007: Annual ERCIM Workshop on Constraint Solving and Constraint Logic Programming

Ce fichier regroupe en un seul document l'ensemble des articles acceptés pour la conférence CSCLP 2007Constraints are a natural way to represent knowledge, and constraint programming is a declarative programming paradigm that has been successfully used to express and solve many practical combinatorial optimization problems. Examples of application domains are scheduling, production planning, resource allocation, communication networks, robotics, and bioinformatics. These proceedings contain the research papers presented at the 12th International Workshop on Constraint Solving and Constraint Logic Programming (CSCLP'07), held on June 7th and 8th 2007, at INRIA Rocquencourt, France. This workshop, open to all, is organized as the twelfth meeting of the working group on Constraints of the European Research Consortium for Informatics and Mathematics (ERCIM). It continues a series of workshops organized since the creation of the working group in 1997, that have led since 2002 to the publication of a series of books entitled ”Recent Advances in Constraints” in the Lecture Notes in Artificial Intelligence, edited by Springer-Verlag. In addition to the contributed papers collected in this volume, two invited talks were given at CSCLP'07, one by Gilles Pesant, Ecole Polytechnique de Montreal, Canada, and one by Jean-Charles R égin, ILOG, France. The editors would like to take the opportunity to thank all the authors who submitted a paper, as well as the reviewers for their helpful work. CSCLP'07 has been made possible thanks to the support of the European Research Consortium for Informatics and Mathematics (ERCIM), the Institut National de la Recherche en Informatique et Automatique (INRIA) and the Association for Constraint programming (ACP)

Breaking Symmetry of Interchangeable Variables and Values

A common type of symmetry is when both variables and values partition into interchangeable sets. Polynomial methods have been introduced to eliminate all symmetric solutions introduced by such interchangeability. Unfortunately, whilst eliminating all symmetric solutions is tractable in this case, pruning all symmetric values is NP-hard. We introduce a new global constraint called SIGLEX and its GAC propagator for pruning some (but not necessarily all) symmetric values. We also investigate how different postings of the SIGLEX constraints affect the pruning performance during constraint solving. Finally, we test these static symmetry breaking constraints experimentally for the first time