Practical Tractability of CSPS by Higher Level Consistency and Tree Decomposition (original) (raw)
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The tractability of a Constraint Satisfaction Problem (CSP) is guaranteed by a direct relationship between its consistency level and a structural parameter of its constraint network such as the treewidth. This result is not widely exploited in practice because enforcing higher-level consistencies can be costly and can change the structure of the constraint network and increase its width. Recently, R( * ,m)C was proposed as a relational consistency property that does not modify the structure of the graph and, thus, does not affect its width. In this paper, we explore two main strategies, based on a tree decomposition of the CSP, for improving the performance of enforcing R( * ,m)C and getting closer to the above tractability condition. Those strategies are: a) localizing the application of the consistency algorithm to the clusters of the tree decomposition, and b) bolstering constraint propagation between clusters by adding redundant constraints at their separators, for which we propose three new schemes. We characterize the resulting consistency properties by comparing them, theoretically and empirically, to the original R( * ,m)C and the popular GAC and maxRPWC, and establish the benefits of our approach for solving difficult problems.
Beyond NP: Arc-Consistency for Quantified Constraints
Lecture Notes in Computer Science, 2002
The generalization of the satisfiability problem with arbitrary quantifiers is a challenging problem of both theoretical and practical relevance. Being PSPACE-complete, it provides a canonical model for solving other PSPACE tasks which naturally arise in AI. Effective SAT-based solvers have been designed very recently for the special case of boolean constraints. We propose to consider the more general problem where constraints are arbitrary relations over finite domains. Adopting the viewpoint of constraint-propagation techniques so successful for CSPs, we provide a theoretical study of this problem. Our main result is to propose quantified arc-consistency as a natural extension of the classical CSP notion.
Exploiting Structure in Constraint Propagation
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Local consistency properties and algorithms for enforcing them are central to the success of Constraint Processing. In this paper, we explore how to exploit the structure of the problem on the performance of the algorithm for enforcing consistency. We propose various strategies for managing the propagation queue of an algorithm for enforcing consistency, and empirically compare their effectiveness for solving CSPs with backtrack search and full lookahead. We focus our investigations on consistency algorithms that operate on the dual graph of a CSP and demonstrate the importance of exploiting a tree decomposition of the dual graph. Further, we note that exploiting structure is particularly striking on unsatisfiable instances. We conjecture that the approach for queue-management strategies benefits virtually all other propagation algorithms.
A Framework for Decision-Based Consistencies
Lecture Notes in Computer Science, 2011
Consistencies are properties of constraint networks that can be enforced by appropriate algorithms to reduce the size of the search space to be explored. Recently, many consistencies built upon taking decisions (most often, variable assignments) and stronger than (generalized) arc consistency have been introduced. In this paper, our ambition is to present a clear picture of decision-based consistencies. We identify four general classes (or levels) of decision-based consistencies, denoted by S φ ∆ , E φ ∆ , B φ ∆ and D φ ∆ , study their relationships, and show that known consistencies are particular cases of these classes. Interestingly, this general framework provides us with a better insight into decision-based consistencies, and allows us to derive many new consistencies that can be directly integrated and compared with other ones.
Constraint propagation as a proof system
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Refutation proofs can be viewed as a special case of constraint propagation, which is a fundamental technique in solving constraint-satisfaction problems. The generalization lifts, in a uniform way, the concept of refutation from Boolean satisfiability problems to general constraint-satisfaction problems. On the one hand, this enables us to study and characterize basic concepts, such as refutation width, using tools from finite-model theory.
Inference rules for high-order consistency in weighted CSP
Recently defined resolution calculi for Max-SAT and signed Max-SAT have provided a logical characterization of the solving techniques applied by Max-SAT and WCSP solvers. In this paper we first define a new resolution rule, called signed Max-SAT parallel resolution, and prove that it is sound and complete for signed Max-SAT. Second, we define a restriction and a generalization of the previous rule called, respectively, signed Max-SAT i-consistency resolution and signed Max-SAT (i, j)-consistency resolution. These rules have the following property: if a WCSP signed encoding is closed under signed Max-SAT i-consistency, then the WCSP is i-consistent, and if it is closed under signed Max-SAT (i, j)-consistency, then the WCSP is (i, j)-consistent. A new and practical insight derived from the definition of these new rules is that algorithms for enforcing high order consistency should incorporate an efficient and effective component for detecting minimal unsatisfiable cores. Finally, we describe an algorithm that applies directional soft consistency with the previous rules. *
A filtering technique to achieve 2-consistency in constraint satisfaction problems
2012
Arc-Consistency algorithms are the most commonly used filtering techniques to prune the search space in Constraint Satisfaction Problems (CSPs). 2-consistency is a similar technique that guarantees that any instantiation of a value to a variable can be consistently extended to any second variable. Thus, 2-consistency can be stronger than arc-consistency in binary CSPs. In this work we present a new algorithm to achieve 2consistency called 2-C4. This algorithm is a reformulation of AC4 algorithm that is able to reduce unnecessary checking and prune more search space than AC4. The experimental results show that 2-C4 was able to prune more search space than arc-consistency algorithms in non-normalized instances. Furthermore, 2-C4 was more efficient than other 2-consistency algorithms presented in the literature.