The collapse of the bounded width hierarchy

Among the local consistency techniques used for solving constraint networks, path-consistency (PC) has received a great deal of attention. However, enforcing PC is computationally expensive and sometimes even unnecessary. Directional path-consistency (DPC) is a weaker notion of PC that considers a given variable ordering and can thus be enforced more efficiently than PC. This paper shows that DPC (the DPC enforcing algorithm of Dechter and Pearl) decides the constraint satisfaction problem (CSP) of a constraint language Γ if it is complete and has the variable elimination property (VEP). However, we also show that no complete VEP constraint language can have a domain with more than 2 values.We then present a simple variant of the DPC algorithm, called DPC * , and show that the CSP of a constraint language can be decided by DPC * if it is closed under a majority operation. In fact, DPC * is sufficient for guaranteeing backtrack-free search for such constraint networks. Examples of majority-closed constraint classes include the classes of connected row-convex (CRC) constraints and tree-preserving constraints, which have found applications in various domains, such as scene labeling, temporal reasoning, geometric reasoning, and logical filtering. Our experimental evaluations show that DPC * significantly outperforms the state-of-the-art algorithms for solving majority-closed constraints.

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“…× D n . For any tuple t ∈ R and any 1 ≤ i ≤ n, we denote by t[i] the value in the i-th coordinate position of t and write t as t [1], . .…”

Section: Preliminariesmentioning

confidence: 99%

“…[27,29,32,6,10]). Recently, it was shown that PC can be used to decide the satisfiability of a problem if and only if the problem does not have the ability to count [2,1]; however, it remains unclear whether backtrack-free search can be used to extract a solution for such a problem after enforcing PC.…”

Section: Introductionmentioning

confidence: 99%

Exploring Directional Path-Consistency for Solving Constraint Networks

Kong

Sioutis

2017

The Computer Journal

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“…Any language preserved by WNU polymorphisms of all arities greater than or equal to 3 has bounded width [9]; this class is extremely large and encompasses all tractable classes discussed above. Finally, it has been shown that every language with bounded width has width (2, 3) [8,24], which allows for fairly efficient solving.…”

Section: Language Classes Solved By Local Consistencymentioning

confidence: 99%

“…In the case of rigid cores, the class of languages with WNU polymorphisms of all arities greater than or equal to 3 can be recognized in polynomial time [8] by combining an alternative characterization involving only two WNU polymorphisms of fixed arity [124] and a variant of the recognition algorithm used for near-unanimity polymorphisms. However, deciding whether an arbitrary language has bounded width is NP-hard [34].…”

Section: Recognition Of Different Families Of Polymorphismsmentioning

confidence: 99%

Tractability in constraint satisfaction problems: a survey

Carbonnel

Cooper

2015

Constraints

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Even though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP. What is tractability?The idea that an algorithm is efficient if its time complexity is a polynomial function of the size of its input can be traced back to pioneering work of Cobham [45] and Edmonds [83], but the foundations of complexity theory are based on the seminal work of Cook [52] and Karp [118]. A computational decision problem (such as the constraint satisfaction problem or CSP) consists of a generic instance (in the case of the CSP, a set of variables, their

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“…(1) The relational product of two binary relations R and S on a set A is the relation R • S defined to be{(a, b) ∈ A 2 : (a, c) ∈ R and (c, b) ∈ S for some c ∈ A}.…”

mentioning

confidence: 99%

Characterizations of several Maltsev conditions

et al. 2015

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Tame congruence theory identifies six Maltsev conditions associated with locally finite varieties omitting certain types of local behaviour. Extending a result of Siggers, we show that of these six Maltsev conditions only two of them are equivalent to strong Maltsev conditions for locally finite varieties. Besides omitting the unary type, the only other of these conditions that is strong is that of omitting the unary and affine types.We also provide novel presentations of some of the above Maltsev conditions.

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The collapse of the bounded width hierarchy

Cited by 49 publications

References 21 publications

Exploring Directional Path-Consistency for Solving Constraint Networks

Exploring Directional Path-Consistency for Solving Constraint Networks

Tractability in constraint satisfaction problems: a survey

Characterizations of several Maltsev conditions

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