Weighted constraint satisfaction with set variables.

Siu Fai Keung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 79-83).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- (Weighted) Constraint Satisfaction --- p.1Chapter 1.2 --- Set Variables --- p.2Chapter 1.3 --- Motivations and Goals --- p.3Chapter 1.4 --- Overview of the Thesis --- p.4Chapter 2 --- Background --- p.6Chapter 2.1 --- Constraint Satisfaction Problems --- p.6Chapter 2.1.1 --- Backtracking Tree Search --- p.8Chapter 2.1.2 --- Consistency Notions --- p.10Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.14Chapter 2.2.1 --- Branch and Bound Search --- p.17Chapter 2.2.2 --- Consistency Notions --- p.19Chapter 2.3 --- Classical CSPs with Set Variables --- p.23Chapter 2.3.1 --- Set Variables and Set Domains --- p.24Chapter 2.3.2 --- Set Constraints --- p.24Chapter 2.3.3 --- Searching with Set Variables --- p.26Chapter 2.3.4 --- Set Bounds Consistency --- p.27Chapter 3 --- Weighted Constraint Satisfaction with Set Variables --- p.30Chapter 3.1 --- Set Variables --- p.30Chapter 3.2 --- Set Domains --- p.31Chapter 3.3 --- Set Constraints --- p.31Chapter 3.3.1 --- Zero-arity Constraint --- p.33Chapter 3.3.2 --- Unary Constraints --- p.33Chapter 3.3.3 --- Binary Constraints --- p.36Chapter 3.3.4 --- Ternary Constraints --- p.36Chapter 3.3.5 --- Cardinality Constraints --- p.37Chapter 3.4 --- Characteristics --- p.37Chapter 3.4.1 --- Space Complexity --- p.37Chapter 3.4.2 --- Generalization --- p.38Chapter 4 --- Consistency Notions and Algorithms for Set Variables --- p.41Chapter 4.1 --- Consistency Notions --- p.41Chapter 4.1.1 --- Element Node Consistency --- p.41Chapter 4.1.2 --- Element Arc Consistency --- p.43Chapter 4.1.3 --- Element Hyper-arc Consistency --- p.43Chapter 4.1.4 --- Weighted Cardinality Consistency --- p.45Chapter 4.1.5 --- Weighted Set Bounds Consistency --- p.46Chapter 4.2 --- Consistency Enforcing Algorithms --- p.47Chapter 4.2.1 --- "Enforcing Element, Node Consistency" --- p.48Chapter 4.2.2 --- Enforcing Element Arc Consistency --- p.51Chapter 4.2.3 --- Enforcing Element Hyper-arc Consistency --- p.52Chapter 4.2.4 --- Enforcing Weighted Cardinality Consistency --- p.54Chapter 4.2.5 --- Enforcing Weighted Set Bounds Consistency --- p.56Chapter 5 --- Experiments --- p.59Chapter 5.1 --- Modeling Set Variables Using 0-1 Variables --- p.60Chapter 5.2 --- Softening the Problems --- p.61Chapter 5.3 --- Steiner Triple System --- p.62Chapter 5.4 --- Social Golfer Problem --- p.63Chapter 5.5 --- Discussions --- p.66Chapter 6 --- Related Work --- p.68Chapter 6.1 --- Other Consistency Notions in WCSPs --- p.68Chapter 6.1.1 --- Full Directional Arc Consistency --- p.68Chapter 6.1.2 --- Existential Directional Arc Consistency --- p.69Chapter 6.2 --- Classical CSPs with Set Variables --- p.70Chapter 6.2.1 --- Bounds Reasoning --- p.70Chapter 6.2.2 --- Cardinality Reasoning --- p.70Chapter 7 --- Concluding Remarks --- p.72Chapter 7.1 --- Contributions --- p.72Chapter 7.2 --- Future Work --- p.74List of Symbols --- p.76Bibliography --- p.7

Experimental Evaluation of Branching Schemes for the CSP

The search strategy of a CP solver is determined by the variable and value ordering heuristics it employs and by the branching scheme it follows. Although the effects of variable and value ordering heuristics on search effort have been widely studied, the effects of different branching schemes have received less attention. In this paper we study this effect through an experimental evaluation that includes standard branching schemes such as 2-way, d-way, and dichotomic domain splitting, as well as variations of set branching where branching is performed on sets of values. We also propose and evaluate a generic approach to set branching where the partition of a domain into sets is created using the scores assigned to values by a value ordering heuristic, and a clustering algorithm from machine learning. Experimental results demonstrate that although exponential differences between branching schemes, as predicted in theory between 2-way and d-way branching, are not very common, still the choice of branching scheme can make quite a difference on certain classes of problems. Set branching methods are very competitive with 2-way branching and outperform it on some problem classes. A statistical analysis of the results reveals that our generic clustering-based set branching method is the best among the methods compared.Comment: To appear in the 3rd workshop on techniques for implementing constraint programming systems (TRICS workshop at the 16th CP Conference), St. Andrews, Scotland 201

Robustness and stability in dynamic constraint satisfaction problems

Constraint programming is a paradigm wherein relations between variables are stated in the form of constraints. It is well-known that many real life problems can be modeled as Constraint Satisfaction Problems (CSPs). Much effort has been spent to increase the efficiency of algorithms for solving CSPs. However, many of these techniques assume that the set of variables, domains and constraints involved in the CSP are known and fixed when the problem is modeled. This is a strong limitation because many problems come from uncertain and dynamic environments, where both the original problem may evolve because of the environment, the user or other agents. In such situations, a solution that holds for the original problem can become invalid after changes. There are two main approaches for dealing with these situations: reactive and proactive approaches. Using reactive approaches entails re-solving the CSP after each solution loss, which is a time consuming. That is a clear disadvantage, especially when we deal with short-term changes, where solution loss is frequent. In addition, in many applications, such as on-line planning and scheduling, the delivery time of a new solution may be too long for actions to be taken on time, so a solution loss can produce several negative effects in the modeled problem. For a task assignment production system with several machines, it could cause the shutdown of the production system, the breakage of machines, the loss of the material/object in production, etc. In a transport timetabling problem, the solution loss, due to some disruption at a point, may produce a delay that propagates through the entire schedule. In addition, all the negative effects stated above will probably entail an economic loss. In this thesis we develop several proactive approaches. Proactive approaches use knowledge about possible future changes in order to avoid or minimize their effects. These approaches are applied before the changes occur. Thus, our approaches search for robust solutions, which have a high probability to remain valid after changes. Furthermore, some of our approaches also consider that the solutions can be easily adapted when they did not resist the changes in the original problem. Thus, these approaches search for stable solutions, which have an alternative solution that is similar to the previous one and therefore can be used in case of a value breakage. In this context, sometimes there exists knowledge about the uncertain and dynamic environment. However in many cases, this information is unknown or hard to obtain. For this reason, for the majority of our approaches (specifically 3 of the 4 developed approaches), the only assumptions made about changes are those inherent in the structure of problems with ordered domains. Given this framework and therefore the existence of a significant order over domain values, it is reasonable to assume that the original bounds of the solution space may undergo restrictive or relaxed modifications. Note that the possibility of solution loss only exists when changes over the original bounds of the solution space are restrictive. Therefore, the main objective for searching robust solutions in this framework is to find solutions located as far away as possible from the bounds of the solution space. In order to meet this criterion, we propose several approaches that can be divided in enumeration-based techniques and a search algorithm.Climent Aunés, LI. (2013). Robustness and stability in dynamic constraint satisfaction problems [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/34785TESI

Adaptive and Opportunistic Exploitation of Tree-decompositions for Weighted CSPs

International audienceWhen solving weighted constraint satisfaction problems , methods based on tree-decompositions constitute an interesting approach depending on the nature of the considered instances. The exploited decompositions often aim to reduce the maximal size of the clusters, which is known as the width of the decomposition. Indeed, the interest of this parameter is related to its importance with respect to the theoretical complexity of these methods. However, its practical interest for the solving of instances remains limited if we consider its multiple drawbacks, notably due to the restrictions imposed on the freedom of the variable ordering heuristic. So, we first propose to exploit new decompositions for solving the constraint optimization problem. These decompositions aim to take into account criteria allowing to increase the solving efficiency. Secondly, we propose to use these decompositions in a more dynamic manner in the sense that the solving of a subprob-lem would be based on the decomposition, totally or locally, only when it seems to be useful. The performed experiments show the practical interest of these new decompositions and the benefit of their dynamic exploitation