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The Tractability of CSP Classes Defined by Forbidden Patterns

D. A. Cohen, M. C. Cooper, P. Creed, D. Marx, A. Z. Salamon
2012 The Journal of Artificial Intelligence Research  
Our key contribution is the novel concept of a CSP pattern and classes of problems defined by forbidden patterns (which can be viewed as forbidding generic sub-problems).  ...  We describe the theoretical framework which can be used to reason about classes of problems defined by forbidden patterns.  ...  Acknowledgments The authors acknowledge support from ANR Project ANR-10-BLAN-0210, EPSRC grants EP/F011776/1 and EP/I011935/1, ERC Starting Grant PARAMTIGHT (No. 280152), and EPSRC platform grant EP/F028288  ... 
doi:10.1613/jair.3651 fatcat:6tfaqcsoeffu7oxfwrkj426wz4

A Characterisation of the Complexity of Forbidding Subproblems in Binary Max-CSP [chapter]

Martin C. Cooper, Guillaume Escamocher, Stanislav Živný
2012 Lecture Notes in Computer Science  
Tractable classes of binary CSP and binary Max-CSP have recently been discovered by studying classes of instances defined by excluding subproblems.  ...  In the resulting dichotomy, the only non-trivial tractable class defined by a forbidden subproblem corresponds to the set of instances satisfying the so-called joint-winner property.  ...  out that the tractable classes we have already identified, defined by forbidden subproblems on two or three variables, are the only possible tractable classes defined by forbidding a single subproblem  ... 
doi:10.1007/978-3-642-33558-7_21 fatcat:vhrbm7jvvzhc5ofkm3imzv7ree

Binary Constraint Satisfaction Problems Defined by Excluded Topological Minors [article]

David A. Cohen, Martin C. Cooper, Peter G. Jeavons, Stanislav Zivny
2018 arXiv   pre-print
This captures some tractable classes of the CSP, but does not capture classes defined by language restrictions, or the well-known structural property of acyclicity.  ...  Finally, we introduce the idea of augmented patterns that allows for the identification of more tractable classes, including all language restrictions of the binary CSP.  ...  To go beyond the earlier results for forbidden sub-patterns [7, 8, 12, 23] , and define a wider range of restricted classes, we use Definition 3.4 to define restricted classes of binary CSP instances  ... 
arXiv:1608.05358v2 fatcat:sgaphwoh6rh4xle6xvvmjcclua

Galois Connections for Patterns: An Algebra of Labelled Graphs [chapter]

David A. Cohen, Martin C. Cooper, Peter G. Jeavons, Stanislav Živný
2021 Lecture Notes in Computer Science  
The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP.  ...  ).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes.  ...  Defining sets of binary CSP instances by forbidding patterns has led to the discovery of novel tractable classes [9, 18] .  ... 
doi:10.1007/978-3-030-72308-8_9 fatcat:oz44glrizjgoza2jvvwmohti7e

Characterising the complexity of constraint satisfaction problems defined by 2-constraint forbidden patterns

Martin C. Cooper, Guillaume Escamocher
2015 Discrete Applied Mathematics  
We study classes of binary CSP instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes.  ...  We characterise all tractable non-mergeable two-constraint patterns. It is worth observing that, in a class of CSP instances defined by forbidding a pattern, there is no bound on the size of domains.  ...  The fact that forbidding patterns only on one point per variable has allowed us to find a strict generalisation of the tractable class defined by the pattern T 5 indicates the validity of this approach  ... 
doi:10.1016/j.dam.2014.10.035 fatcat:vbursdr47ngepatajiudv5ggcy

A Dichotomy for 2-Constraint Forbidden CSP Patterns [article]

Martin C. Cooper, Guillaume Escamocher
2012 arXiv   pre-print
The complete characterisation of all tractable classes defined by forbidding patterns (where a pattern is simply a compact representation of a set of subproblems) is a challenging problem.  ...  We study classes of instances defined by excluding subproblems. This approach has recently led to the discovery of novel tractable classes.  ...  As we will show, each T i defines a tractable class of binary CSP instances.  ... 
arXiv:1201.3868v1 fatcat:3itmaekgw5e4zfu4j2txpintp4

Discovering Archipelagos of Tractability for Constraint Satisfaction and Counting [article]

Robert Ganian, M. S. Ramanujan, Stefan Szeider
2015 arXiv   pre-print
For this purpose we utilize the notion of a strong backdoor of a CSP instance, as introduced by Williams et al.  ...  A prominent way of defining islands of tractability for CSP is to restrict the relations that may occur in the constraints to a fixed set, called a constraint language, whereas a constraint language is  ...  from a list of tractable classes H 1 , . . . , H d .  ... 
arXiv:1507.02479v2 fatcat:pbu6tmcwxzck7pwacyes6vddvu

Tractability in constraint satisfaction problems: a survey

Clément Carbonnel, Martin C. Cooper
2015 Constraints  
Acknowledgments We are grateful to Peter Jeavons and StanislavŽivný for their detailed comments on a first draft of this paper, and to the reviewers for their constructive comments.  ...  Any class of instances defined by a forbidden pattern is necessarily recognisable in polynomial time by a simple exhaustive search for the pattern.  ...  A direct generalisation of the broken-triangle class CSP(BTP) to general-arity CSPs has also recently been given [60] , but the definition of tractable classes by forbidden patterns in general-arity CSPs  ... 
doi:10.1007/s10601-015-9198-6 fatcat:fl7kxmceh5bqzpyd2c37rtohii

The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns

Martin C. Cooper, Stanislav Živný
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
We note that focusing on classes of CSPs defined by forbidden patterns is very natural as AC cannot introduce forbidden patterns because pattern occurrence is defined by the presence of some compatibility  ...  One approach to hybrid CSPs that has been rather successful studies the classes of CSPs defined by forbidden patterns; that is, by forbidding certain generic subinstances.  ...  We would like to thank the reviewers of both the extended abstract of this paper [19] and this full version for their very detailed comments and pertinent suggestions.  ... 
doi:10.1145/2933575.2933587 dblp:conf/lics/CooperZ16 fatcat:wlde6fsfcje3jgudptddj55334

Transformations of representation in constraint satisfaction

András Z. Salamon
2015 Constraints  
Such tractable hereditary classes are defined by using the technique of excluded induced subgraphs, such as classes of graphs that contain neither odd cycles with five or more vertices, nor their complements  ...  For the common case where all constraints involve only two variables, I show how the microstructure can be used to define CSPs that are tractable because their clause structures fall within classes of  ...  A class defined by a single forbidden induced substructure S is equivalent to the intersection of several larger classes defined by coloured versions of S.  ... 
doi:10.1007/s10601-015-9224-8 fatcat:lollakqzx5h7xeajtfrmaz6u4q

On the Containment of Forbidden Patterns Problems [chapter]

Florent Madelaine
2010 Lecture Notes in Computer Science  
In the presence of a recolouring from a forbidden patterns problem Ω1 to another forbidden patterns problem Ω2, containment of Ω1 in Ω2 follows.  ...  Forbidden patterns problems are a generalisation of (finite) constraint satisfaction problems which are definable in Feder and Vardi's logic mmsnp [1].  ...  We denote by fpp the class of forbidden patterns problems.  ... 
doi:10.1007/978-3-642-15396-9_29 fatcat:5wwjwvhaxnan3kxjk5poho2qly

On Broken Triangles [chapter]

Martin C. Cooper, Achref El Mouelhi, Cyril Terrioux, Bruno Zanuttini
2014 Lecture Notes in Computer Science  
A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP.  ...  We show that a local version of the BTP allows the merging of domain values in binary CSPs, thus providing a novel polynomial-time reduction operation.  ...  Inspired by the BTP, investigation of other forbidden patterns in the coloured microstructure has led to the discovery of new tractable classes [Cohen et al., 2012; Cooper and Escamocher, 2015; Cooper  ... 
doi:10.1007/978-3-319-10428-7_5 fatcat:dofu6lctjvd4nnquf3owzmhrcy

On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

Clément Carbonnel, David A. Cohen, Martin C. Cooper, Stanislav Živný
2018 Algorithmica  
We aim to characterise all classes of CSPs defined by a forbidden pattern that are solved by singleton arc consistency and closed under removing constraints.  ...  We identify five new patterns whose absence ensures solvability by singleton arc consistency, four of which are provably maximal and three of which generalise 2-SAT.  ...  Conclusion We have established SAC-solvability of five novel classes of binary CSPs defined by a forbidden pattern, three of which are generalisations of 2-SAT.  ... 
doi:10.1007/s00453-018-0498-2 pmid:31007327 pmcid:PMC6449328 fatcat:hexfkxyuh5awngv76xkzamhqze

A Microstructure-Based Family of Tractable Classes for CSPs [chapter]

Martin C. Cooper, Philippe Jégou, Cyril Terrioux
2015 Lecture Notes in Computer Science  
In this context, the Broken Triangle Property (BTP) is a state-of-the-art microstructure-based tractable class which generalizes well-known and previously-defined tractable classes, notably the set of  ...  Moreover, we show that if k-TW is the class of instances having tree-width bounded by a constant k, then k-TW (k + 1)-BTP.  ...  In this paper, we return to this type of approach by generalizing the tractable class BTP which is defined by a property excluding certain patterns (called Broken Triangles) in the microstructure graph  ... 
doi:10.1007/978-3-319-23219-5_6 fatcat:uris5oeuiffr7afzrz4evmk3ka

Hybrid tractability of valued constraint problems

Martin C. Cooper, Stanislav Živný
2011 Artificial Intelligence  
An important tool in our investigation will be the notion of forbidden substructures.  ...  Soft constraint problems are a generalisation of the CSP which allow the user to model optimisation problems.  ...  We have demonstrated 3-maximality of the tractable class defined by the JWP.  ... 
doi:10.1016/j.artint.2011.02.003 fatcat:otqplqno5fah3if7z2f4gc2zaq
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