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

On The Complexity and Completeness of Static Constraints for Breaking Row and Column Symmetry

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 NP-hard. 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.Comment: To appear in the Proceedings of the 16th International Conference on Principles and Practice of Constraint Programming (CP 2010

SLIDE: A Useful Special Case of the CARDPATH Constraint

We study the CardPath constraint. This ensures a given constraint holds a number of times down a sequence of variables. We show that SLIDE, a special case of CardPath where the slid constraint must hold always, can be used to encode a wide range of sliding sequence constraints including CardPath itself. We consider how to propagate SLIDE and provide a complete propagator for CardPath. Since propagation is NP-hard in general, we identify special cases where propagation takes polynomial time. Our experiments demonstrate that using SLIDE to encode global constraints can be as efficient and effective as specialised propagators.Comment: 18th European Conference on Artificial Intelligenc

Constraint Programming for Multi-criteria Conceptual Clustering

International audienceA conceptual clustering is a set of formal concepts (i.e., closed itemsets) that defines a partition of a set of transactions. Finding a conceptual clustering is an N P-complete problem for which Constraint Programming (CP) and Integer Linear Programming (ILP) approaches have been recently proposed. We introduce new CP models to solve this problem: a pure CP model that uses set constraints, and an hybrid model that uses a data mining tool to extract formal concepts in a preprocessing step and then uses CP to select a subset of formal concepts that defines a partition. We compare our new models with recent CP and ILP approaches on classical machine learning instances. We also introduce a new set of instances coming from a real application case, which aims at extracting setting concepts from an Enterprise Resource Planning (ERP) software. We consider two classic criteria to optimize, i.e., the frequency and the size. We show that these criteria lead to extreme solutions with either very few small formal concepts or many large formal concepts, and that compromise clusterings may be obtained by computing the Pareto front of non dominated clusterings

Symmetry Breaking for Answer Set Programming

In the context of answer set programming, this work investigates symmetry detection and symmetry breaking to eliminate symmetric parts of the search space and, thereby, simplify the solution process. We contribute a reduction of symmetry detection to a graph automorphism problem which allows to extract symmetries of a logic program from the symmetries of the constructed coloured graph. We also propose an encoding of symmetry-breaking constraints in terms of permutation cycles and use only generators in this process which implicitly represent symmetries and always with exponential compression. These ideas are formulated as preprocessing and implemented in a completely automated flow that first detects symmetries from a given answer set program, adds symmetry-breaking constraints, and can be applied to any existing answer set solver. We demonstrate computational impact on benchmarks versus direct application of the solver. Furthermore, we explore symmetry breaking for answer set programming in two domains: first, constraint answer set programming as a novel approach to represent and solve constraint satisfaction problems, and second, distributed nonmonotonic multi-context systems. In particular, we formulate a translation-based approach to constraint answer set solving which allows for the application of our symmetry detection and symmetry breaking methods. To compare their performance with a-priori symmetry breaking techniques, we also contribute a decomposition of the global value precedence constraint that enforces domain consistency on the original constraint via the unit-propagation of an answer set solver. We evaluate both options in an empirical analysis. In the context of distributed nonmonotonic multi-context system, we develop an algorithm for distributed symmetry detection and also carry over symmetry-breaking constraints for distributed answer set programming.Comment: Diploma thesis. Vienna University of Technology, August 201

Breakout group allocation schedules and the social golfer problem with adjacent group sizes

The current pandemic has led schools and universities to turn to online meeting software solutions such as Zoom and Microsoft Teams. The teaching experience can be enhanced via the use of breakout rooms for small group interaction. Over the course of a class (or over several classes), the class will be allocated to breakout groups multiple times over several rounds. It is desirable to mix the groups as much as possible, the ideal being that no two students appear in the same group in more than one round. In this paper, we discuss how the problem of scheduling balanced allocations of students to sequential breakout rooms directly corresponds to a novel variation of a well-known problem in combinatorics (the social golfer problem), which we call the social golfer problem with adjacent group sizes. We explain how solutions to this problem can be obtained using constructions from combinatorial design theory and how they can be used to obtain good, balanced breakout room allocation schedules. We present our solutions for up to 50 students and introduce an online resource that educators can access to immediately generate suitable allocation schedules

Résolution exacte du problème de partitionnement de données avec minimisation de variance sous contraintes de cardinalité par programmation par contraintes

Le partitionnement de données représente une procédure destinée à regrouper un ensemble d’observations dans plusieurs sous ensembles homogènes et/ou bien séparés. L’idée derrière une telle activité est de simplifier l’extraction d’information utile en étudiant les groupes résultants plutôt que les observations elles-mêmes. Cela dit, plusieurs situations appellent à ce que la solution générée respecte un ensemble de contraintes données. En particulier, on exige parfois que les groupes résultants comportent un nombre prédéfini d’éléments. On parle de partitionnement avec contraintes de cardinalité. On présente alors, dans ce travail, une approche de résolution exacte pour le partitionnement de données avec minimisation de la variance sous contraintes de cardinalité. En utilisant le paradigme de la Programmation par Contraintes, on propose d’abord un modèle adéquat du problème selon celui-ci. Ensuite, on suggère à la fois une stratégie de recherche rehaussée ainsi que deux algorithmes de filtrage. Ces outils ainsi développés tirent avantage de la structure particulière du problème afin de naviguer l’espace de recherche de façon efficace, à la recherche d’une solution globalement optimale. Des expérimentations pratiques montrent que notre approche procure un avantage important par rapport aux autres méthodes exactes existantes lors de la résolution de plusieurs exemplaires du problème.----------ABSTRACT: Data clustering is a procedure designed to group a set of observations into subsets that are homogeneous and/or well separated. The idea behind such an endeavor is to simplify extraction of useful information by studying the resulting groups instead of directly dealing with the observations themselves. However, many situations mandate that the answer conform to a set of constraints. Particularly one that involves the target number of elements each group must possess. This is known as cardinality constrained clustering. In this work we present an exact approach to solve the cardinality constrained Euclidian minimum sum-of-squares clustering. Based on the Constraint Programming paradigm, we first present an adequate model for this problem in the aforementioned framework. We then suggest both an upgraded search heuristic as well as two filtering algorithms. We take advantage of the structure of the problem in designing these tools to efficiently navigate the search space, looking for a globally optimal solution. Computational experiments show that our approach provides a substantial boost to the resolution of several instances of the problem in comparison to existing exact methods