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The Tractability of CSP Classes Defined by Forbidden Patterns

2012
*
The Journal of Artificial Intelligence Research
*

Our key contribution is

doi:10.1613/jair.3651
fatcat:6tfaqcsoeffu7oxfwrkj426wz4
*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 ...##
###
A Characterisation of the Complexity of Forbidding Subproblems in Binary Max-CSP
[chapter]

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 ...

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

2018
*
arXiv
*
pre-print

This captures some

arXiv:1608.05358v2
fatcat:sgaphwoh6rh4xle6xvvmjcclua
*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 ...##
###
Galois Connections for Patterns: An Algebra of Labelled Graphs
[chapter]

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] . ...

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

2015
*
Discrete Applied Mathematics
*

We study

doi:10.1016/j.dam.2014.10.035
fatcat:vbursdr47ngepatajiudv5ggcy
*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 ...##
###
A Dichotomy for 2-Constraint Forbidden CSP Patterns
[article]

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. ...

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

2015
*
arXiv
*
pre-print

For this purpose we utilize

arXiv:1507.02479v2
fatcat:pbu6tmcwxzck7pwacyes6vddvu
*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 . ...##
###
Tractability in constraint satisfaction problems: a survey

2015
*
Constraints
*

Acknowledgments We are grateful to Peter Jeavons and StanislavŽivný for their detailed comments on a first draft

doi:10.1007/s10601-015-9198-6
fatcat:fl7kxmceh5bqzpyd2c37rtohii
*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*...##
###
The Power of Arc Consistency for CSPs Defined by Partially-Ordered Forbidden Patterns

2016
*
Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16
*

We note that focusing on

doi:10.1145/2933575.2933587
dblp:conf/lics/CooperZ16
fatcat:wlde6fsfcje3jgudptddj55334
*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. ...##
###
Transformations of representation in constraint satisfaction

2015
*
Constraints
*

Such

doi:10.1007/s10601-015-9224-8
fatcat:lollakqzx5h7xeajtfrmaz6u4q
*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. ...##
###
On the Containment of Forbidden Patterns Problems
[chapter]

2010
*
Lecture Notes in Computer Science
*

In

doi:10.1007/978-3-642-15396-9_29
fatcat:5wwjwvhaxnan3kxjk5poho2qly
*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. ...##
###
On Broken Triangles
[chapter]

2014
*
Lecture Notes in Computer Science
*

A directional version

doi:10.1007/978-3-319-10428-7_5
fatcat:dofu6lctjvd4nnquf3owzmhrcy
*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 ...##
###
On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

2018
*
Algorithmica
*

We aim to characterise all

doi:10.1007/s00453-018-0498-2
pmid:31007327
pmcid:PMC6449328
fatcat:hexfkxyuh5awngv76xkzamhqze
*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. ...##
###
A Microstructure-Based Family of Tractable Classes for CSPs
[chapter]

2015
*
Lecture Notes in Computer Science
*

In this context,

doi:10.1007/978-3-319-23219-5_6
fatcat:uris5oeuiffr7afzrz4evmk3ka
*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 ...##
###
Hybrid tractability of valued constraint problems

2011
*
Artificial Intelligence
*

An important tool in our investigation will be

doi:10.1016/j.artint.2011.02.003
fatcat:otqplqno5fah3if7z2f4gc2zaq
*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. ...
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