Solving Set Constraint Satisfaction Problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems

Constraint Reasoning and Kernel Clustering for Pattern Decomposition with Scaling

Abstract. Motivated by an important and challenging task encountered in material discovery, we consider the problem of finding K basis patterns of numbers that jointly compose N observed patterns while enforcing additional spatial and scaling constraints. We propose a Constraint Pro-gramming (CP) model which captures the exact problem structure yet fails to scale in the presence of noisy data about the patterns. We allevi-ate this issue by employing Machine Learning (ML) techniques, namely kernel methods and clustering, to decompose the problem into smaller ones based on a global data-driven view, and then stitch the partial solu-tions together using a global CP model. Combining the complementary strengths of CP and ML techniques yields a more accurate and scalable method than the few found in the literature for this complex problem.

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

Solving set constraint satisfaction problems using ROBDDs

In this paper we present a new approach to modeling finite set domain constraint problems using Reduced Ordered Binary Decision Diagrams (ROBDDs). We show that it is possible to construct an efficient set domain propagator which compactly represents many set domains and set constraints using ROBDDs. We demonstrate that the ROBDD-based approach provides unprecedented flexibility in modeling constraint satisfaction problems, leading to performance improvements. We also show that the ROBDD-based modeling approach can be extended to the modeling of integer and multiset constraint problems in a straightforward manner. Since domain propagation is not always practical, we also show how to incorporate less strict consistency notions into the ROBDD framework, such as set bounds, cardinality bounds and lexicographic bounds consistency. Finally, we present experimental results that demonstrate the ROBDD-based solver performs better than various more conventional constraint solvers on several standard set constraint problems. 1