Quantified weighted constraint satisfaction problems.

Mak, Wai Keung Terrence.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 100-104).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.2Chapter 1.3 --- Quantified Constraint Satisfaction Problems --- p.3Chapter 1.4 --- Motivation and Goal --- p.4Chapter 1.5 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.7Chapter 2.1 --- Constraint Satisfaction Problems --- p.7Chapter 2.1.1 --- Backtracking Tree Search --- p.9Chapter 2.1.2 --- Local Consistencies for solving CSPs --- p.11Node Consistency (NC) --- p.13Arc Consistency (AC) --- p.14Searching by Maintaining Arc Consistency --- p.16Chapter 2.1.3 --- Constraint Optimization Problems --- p.17Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.19Chapter 2.2.1 --- Branch and Bound Search (B&B) --- p.23Chapter 2.2.2 --- Local Consistencies for WCSPs --- p.25Node Consistency --- p.26Arc Consistency --- p.28Chapter 2.3 --- Quantified Constraint Satisfaction Problems --- p.32Chapter 2.3.1 --- Backtracking Free search --- p.37Chapter 2.3.2 --- Consistencies for QCSPs --- p.38Chapter 2.3.3 --- Look Ahead for QCSPs --- p.45Chapter 3 --- Quantified Weighted CSPs --- p.48Chapter 4 --- Branch & Bound with Consistency Techniques --- p.54Chapter 4.1 --- Alpha-Beta Pruning --- p.54Chapter 4.2 --- Consistency Techniques --- p.57Chapter 4.2.1 --- Node Consistency --- p.62Overview --- p.62Lower Bound of A-Cost --- p.62Upper Bound of A-Cost --- p.66Projecting Unary Costs to Cθ --- p.67Chapter 4.2.2 --- Enforcing Algorithm for NC --- p.68Projection Phase --- p.69Pruning Phase --- p.69Time Complexity --- p.71Chapter 4.2.3 --- Arc Consistency --- p.73Overview --- p.73Lower Bound of A-Cost --- p.73Upper Bound of A-Cost --- p.75Projecting Binary Costs to Unary Constraint --- p.75Chapter 4.2.4 --- Enforcing Algorithm for AC --- p.76Projection Phase --- p.77Pruning Phase --- p.77Time complexity --- p.79Chapter 5 --- Performance Evaluation --- p.83Chapter 5.1 --- Definitions of QCOP/QCOP+ --- p.83Chapter 5.2 --- Transforming QWCSPs into QCOPs --- p.90Chapter 5.3 --- Empirical Evaluation --- p.91Chapter 5.3.1 --- Random Generated Problems --- p.92Chapter 5.3.2 --- Graph Coloring Game --- p.92Chapter 5.3.3 --- Min-Max Resource Allocation Problem --- p.93Chapter 5.3.4 --- Value Ordering Heuristics --- p.94Chapter 6 --- Concluding Remarks --- p.96Chapter 6.1 --- Contributions --- p.96Chapter 6.2 --- Limitations and Related Works --- p.97Chapter 6.3 --- Future Works --- p.99Bibliography --- p.10

Tractable projection-safe soft global constraints in weighted constraint satisfaction.

Wu, Yi.Thesis (M.Phil.)--Chinese University of Hong Kong, 2011.Includes bibliographical references (p. 74-80).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.3Chapter 1.3 --- Motivation and Goal --- p.4Chapter 1.4 --- Outline of the Thesis --- p.5Chapter 2 --- Background --- p.7Chapter 2.1 --- Constraint Satisfaction Problems --- p.7Chapter 2.1.1 --- Backtracking Tree search --- p.8Chapter 2.1.2 --- Local consistencies in CSP --- p.11Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.18Chapter 2.2.1 --- Branch and Bound Search --- p.20Chapter 2.2.2 --- Local Consistencies in WCSP --- p.21Chapter 2.3 --- Global Constraints --- p.31Chapter 3 --- Tractable Projection-Safety --- p.36Chapter 3.1 --- Tractable Projection-Safety: Definition and Analysis --- p.37Chapter 3.2 --- Polynomially Decomposable Soft Constraints --- p.42Chapter 4 --- Examples of Polynomially Decomposable Soft Global Constraints --- p.48Chapter 4.1 --- Soft Among Constraint --- p.49Chapter 4.2 --- Soft Regular Constraint --- p.51Chapter 4.3 --- Soft Grammar Constraint --- p.54Chapter 4.4 --- Max_Weight/Min Weight Constraint --- p.57Chapter 5 --- Experiments --- p.61Chapter 5.1 --- The car Sequencing Problem --- p.61Chapter 5.2 --- The nonogram problem --- p.62Chapter 5.3 --- Well-Formed Parenthesis --- p.64Chapter 5.4 --- Minimum Energy Broadcasting Problem --- p.64Chapter 6 --- Related Work --- p.67Chapter 6.1 --- WCSP Consistencies --- p.67Chapter 6.2 --- Global Constraints . --- p.68Chapter 7 --- Conclusion --- p.71Chapter 7.1 --- Contributions --- p.71Chapter 7.2 --- Future Work --- p.72Bibliography --- p.7

Speeding up weighted constraint satisfaction using redundant modeling.

Woo Hiu Chun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 91-99).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction Problems --- p.1Chapter 1.2 --- Weighted Constraint Satisfaction Problems --- p.3Chapter 1.3 --- Redundant Modeling --- p.4Chapter 1.4 --- Motivations and Goals --- p.5Chapter 1.5 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.9Chapter 2.1.2 --- Local Consistencies --- p.12Chapter 2.1.3 --- Local Consistencies in Backtracking Search --- p.17Chapter 2.1.4 --- Permutation CSPs --- p.19Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.20Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistencies --- p.26Chapter 2.2.3 --- Local Consistencies in Branch and Bound Search --- p.32Chapter 2.3 --- Redundant Modeling --- p.34Chapter 3 --- Generating Redundant WCSP Models --- p.37Chapter 3.1 --- Model Induction for CSPs --- p.38Chapter 3.1.1 --- Stated Constraints --- p.39Chapter 3.1.2 --- No-Double-Assignment Constraints --- p.39Chapter 3.1.3 --- At-Least-One-Assignment Constraints --- p.40Chapter 3.2 --- Generalized Model Induction for WCSPs --- p.43Chapter 4 --- Combining Mutually Redundant WCSPs --- p.47Chapter 4.1 --- Naive Approach --- p.47Chapter 4.2 --- Node Consistency Revisited --- p.51Chapter 4.2.1 --- Refining Node Consistency Definition --- p.52Chapter 4.2.2 --- Enforcing m-NC* c Algorithm --- p.55Chapter 4.3 --- Arc Consistency Revisited --- p.58Chapter 4.3.1 --- Refining Arc Consistency Definition --- p.60Chapter 4.3.2 --- Enforcing m-AC*c Algorithm --- p.62Chapter 5 --- Experiments --- p.67Chapter 5.1 --- Langford's Problem --- p.68Chapter 5.2 --- Latin Square Problem --- p.72Chapter 5.3 --- Discussion --- p.75Chapter 6 --- Related Work --- p.77Chapter 6.1 --- Soft Constraint Satisfaction Problems --- p.77Chapter 6.2 --- Other Local Consistencies in WCSPs --- p.79Chapter 6.2.1 --- Full Arc Consistency --- p.79Chapter 6.2.2 --- Pull Directional Arc Consistency --- p.81Chapter 6.2.3 --- Existential Directional Arc Consistency --- p.82Chapter 6.3 --- Redundant Modeling and Channeling Constraints --- p.83Chapter 7 --- Concluding Remarks --- p.85Chapter 7.1 --- Contributions --- p.85Chapter 7.2 --- Future Work --- p.87List of Symbols --- p.88Bibliograph

Finding robust solutions for constraint satisfaction problems with discrete and ordered domains by coverings

Constraint programming is a paradigm wherein relations between variables are stated in the form of constraints. Many real life problems come from uncertain and dynamic environments, where the initial constraints and domains may change during its execution. Thus, the solution found for the problem may become invalid. The search forrobustsolutions for constraint satisfaction problems (CSPs) has become an important issue in the ¿eld of constraint programming. In some cases, there exists knowledge about the uncertain and dynamic environment. In other cases, this information is unknown or hard to obtain. In this paper, we consider CSPs with discrete and ordered domains where changes only involve restrictions or expansions of domains or constraints. To this end, we model CSPs as weighted CSPs (WCSPs) by assigning weights to each valid tuple of the problem constraints and domains. The weight of each valid tuple is based on its distance from the borders of the space of valid tuples in the corresponding constraint/domain. This distance is estimated by a new concept introduced in this paper: coverings. Thus, the best solution for the modeled WCSP can be considered as a most robust solution for the original CSP according to these assumptionsThis work has been partially supported by the research projects TIN2010-20976-C02-01 (Min. de Ciencia e Innovacion, Spain) and P19/08 (Min. de Fomento, Spain-FEDER), and the fellowship program FPU.Climent Aunés, LI.; Wallace, RJ.; Salido Gregorio, MA.; Barber Sanchís, F. (2013). Finding robust solutions for constraint satisfaction problems with discrete and ordered domains by coverings. Artificial Intelligence Review. 1-26. https://doi.org/10.1007/s10462-013-9420-0S126Climent L, Salido M, Barber F (2011) Reformulating dynamic linear constraint satisfaction problems as weighted csps for searching robust solutions. In: Ninth symposium of abstraction, reformulation, and approximation (SARA-11), pp 34–41Dechter R, Dechter A (1988) Belief maintenance in dynamic constraint networks. In: Proceedings of the 7th national conference on, artificial intelligence (AAAI-88), pp 37–42Dechter R, Meiri I, Pearl J (1991) Temporal constraint networks. Artif Intell 49(1):61–95Fargier H, Lang J (1993) Uncertainty in constraint satisfaction problems: a probabilistic approach. In: Proceedings of the symbolic and quantitative approaches to reasoning and uncertainty (EC-SQARU-93), pp 97–104Fargier H, Lang J, Schiex T (1996) Mixed constraint satisfaction: a framework for decision problems under incomplete knowledge. In: Proceedings of the 13th national conference on, artificial intelligence, pp 175–180Fowler D, Brown K (2000) Branching constraint satisfaction problems for solutions robust under likely changes. In: Proceedings of the international conference on principles and practice of constraint programming (CP-2000), pp 500–504Goles E, Martínez S (1990) Neural and automata networks: dynamical behavior and applications. Kluwer Academic Publishers, DordrechtHays W (1973) Statistics for the social sciences, vol 410, 2nd edn. Holt, Rinehart and Winston, New YorkHebrard E (2006) Robust solutions for constraint satisfaction and optimisation under uncertainty. PhD thesis, University of New South WalesHerrmann H, Schneider C, Moreira A, Andrade Jr J, Havlin S (2011) Onion-like network topology enhances robustness against malicious attacks. J Stat Mech Theory Exp 2011(1):P01,027Larrosa J, Schiex T (2004) Solving weighted CSP by maintaining arc consistency. Artif Intell 159:1–26Larrosa J, Meseguer P, Schiex T (1999) Maintaining reversible DAC for Max-CSP. J Artif Intell 107(1):149–163Mackworth A (1977) On reading sketch maps. In: Proceedings of IJCAI’77, pp 598–606Sam J (1995) Constraint consistency techniques for continuous domains. These de doctorat, École polytechnique fédérale de LausanneSchiex T, Fargier H, Verfaillie G (1995) Valued constraint satisfaction problems: hard and easy problems. In: Proceedings of the 14th international joint conference on, artificial intelligence (IJCAI-95), pp 631–637Taillard E (1993) Benchmarks for basic scheduling problems. Eur J Oper Res 64(2):278–285Verfaillie G, Jussien N (2005) Constraint solving in uncertain and dynamic environments: a survey. Constraints 10(3):253–281Wallace R, Freuder E (1998) Stable solutions for dynamic constraint satisfaction problems. In: Proceedings of the 4th international conference on principles and practice of constraint programming (CP-98), pp 447–461Wallace RJ, Grimes D (2010) Problem-structure versus solution-based methods for solving dynamic constraint satisfaction problems. In: Proceedings of the 22nd international conference on tools with artificial intelligence (ICTAI-10), IEEEWalsh T (2002) Stochastic constraint programming. In: Proceedings of the 15th European conference on, artificial intelligence (ECAI-02), pp 111–115William F (2006) Topology and its applications. Wiley, New YorkWiner B (1971) Statistical principles in experimental design, 2nd edn. McGraw-Hill, New YorkYorke-Smith N, Gervet C (2009) Certainty closure: reliable constraint reasoning with incomplete or erroneous data. J ACM Trans Comput Log (TOCL) 10(1):

Conflict History Based Branching Heuristic for CSP Solving

International audienceAn important feature in designing algorithms to solve Constraint Satisfaction Problems (CSP) is the definition of a branching heuristic to explore efficiently the search space and exploit the problem structure. We propose Conflict-History Search (CHS), a new dynamic and adaptive branching heuristic for CSP solving. It is based on the search history by considering the temporality of search failures. To achieve that, we use the exponential recency weighted average to estimate the evolution of the hardness of constraints throughout the search. The experimental evaluation on XCSP3 instances shows that integrating CHS to solvers based on MAC obtains competitive results and can improve those obtained through other heuristics of the state of the art

Identifying sources of global contention in constraint satisfaction search

Much work has been done on learning from failure in search to boost solving of combinatorial problems, such as clause-learning and clause-weighting in boolean satisfiability (SAT), nogood and explanation-based learning, and constraint weighting in constraint satisfaction problems (CSPs). Many of the top solvers in SAT use clause learning to good effect. A similar approach (nogood learning) has not had as large an impact in CSPs. Constraint weighting is a less fine-grained approach where the information learnt gives an approximation as to which variables may be the sources of greatest contention. In this work we present two methods for learning from search using restarts, in order to identify these critical variables prior to solving. Both methods are based on the conflict-directed heuristic (weighted-degree heuristic) introduced by Boussemart et al. and are aimed at producing a better-informed version of the heuristic by gathering information through restarting and probing of the search space prior to solving, while minimizing the overhead of these restarts. We further examine the impact of different sampling strategies and different measurements of contention, and assess different restarting strategies for the heuristic. Finally, two applications for constraint weighting are considered in detail: dynamic constraint satisfaction problems and unary resource scheduling problems

An algebraic theory of complexity for valued constraints: Establishing a Galois connection

Abstract. The complexity of any optimisation problem depends critically on the form of the objective function. Valued constraint satisfaction problems are discrete optimisation problems where the function to be minimised is given as a sum of cost functions defined on specified subsets of variables. These cost functions are chosen from some fixed set of available cost functions, known as a valued constraint language. We show in this paper that when the costs are non-negative rational numbers or infinite, then the complexity of a valued constraint problem is determined by certain algebraic properties of this valued constraint language, which we call weighted polymorphisms. We define a Galois connection between valued constraint languages and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterised. These results provide a new approach in the search for tractable valued constraint languages

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

Soft global constraints in constraint optimization and weighted constraint satisfaction.

Leung, Ka Lun.Thesis (M.Phil.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 118-126).Abstract also in Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- Constraint Satisfaction and Global Constraints --- p.3Chapter 1.2 --- Soft Constraints --- p.4Chapter 1.3 --- Motivation and Goal --- p.5Chapter 1.4 --- Outline of the Thesis --- p.6Chapter 2 --- Background --- p.8Chapter 2.1 --- Constraint Satisfaction Problems --- p.8Chapter 2.1.1 --- Backtracking Tree Search --- p.10Chapter 2.1.2 --- Local Consistency in CSP --- p.11Chapter 2.1.3 --- Constraint Optimization Problem --- p.16Chapter 2.2 --- Weighted Constraint Satisfaction --- p.21Chapter 2.2.1 --- Branch and Bound Search --- p.23Chapter 2.2.2 --- Local Consistency in WCSP --- p.26Chapter 2.3 --- Global Constraints --- p.35Chapter 2.4 --- Flow Theory --- p.37Chapter 3 --- Related Work --- p.39Chapter 3.1 --- Handling Soft Constraints in COPs --- p.39Chapter 3.2 --- Global Constraints --- p.40Chapter 3.2.1 --- Hard Global Constraints --- p.40Chapter 3.2.2 --- Soft Global Constraints --- p.41Chapter 3.3 --- Local Consistency in Weighted CSP --- p.42Chapter 4 --- “Soft as Hard´ح Approach --- p.44Chapter 4.1 --- The General “Soft as Hard´ح Approach --- p.44Chapter 4.2 --- Cost-based GAC --- p.49Chapter 4.3 --- Empirical Results --- p.53Chapter 5 --- Weighted CSP Approach --- p.55Chapter 5.1 --- Strong 0-Inverse Consistency --- p.55Chapter 5.1.1 --- 0-Inverse Consistency and Strong 0-Inverse Consistency --- p.56Chapter 5.1.2 --- Comparison with Other Consistencies --- p.62Chapter 5.2 --- Generalized Arc Consistency Star --- p.65Chapter 5.3 --- Full Directional Generalized Arc Consistency Star --- p.72Chapter 5.4 --- Generalizing EDAC* --- p.78Chapter 5.5 --- Implementation Issues --- p.87Chapter 6 --- Towards A Library of Efficient Soft Global Constraints --- p.90Chapter 6.1 --- The allDifferent Constraint --- p.91Chapter 6.1.1 --- All Interval Series --- p.93Chapter 6.1.2 --- Latin Square --- p.95Chapter 6.2 --- The GCC Constraint --- p.97Chapter 6.2.1 --- Latin Square --- p.100Chapter 6.2.2 --- Round Robin Tournament --- p.100Chapter 6.3 --- The Same Constraint --- p.102Chapter 6.3.1 --- Fair Scheduling --- p.104Chapter 6.3.2 --- People-Mission Scheduling --- p.105Chapter 6.4 --- The Regular Constraint --- p.106Chapter 6.4.1 --- Nurse Rostering Problem --- p.110Chapter 6.4.2 --- Modelling Stretch() Constraint --- p.111Chapter 6.5 --- Discussion --- p.113Chapter 7 --- Conclusion and Remarks --- p.115Chapter 7.1 --- Contributions --- p.115Chapter 7.2 --- Future Work --- p.117Bibliography --- p.11