Unifying and extending hybrid tractable classes of CSPs

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15358The contribution was presented at CP 2015 :http://booleconferences.ucc.ie/cp2015 Abstract. The study of tractable classes is an important issue in Artificial Intelligence, especially in Constraint Satisfaction Problems. 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 instances whose constraint graph is a tree. In this paper, we propose to extend and to generalize this class using a more general approach based on a parameter k which is a given constant. To this end, we introduce the k-BTP property (and the class of instances satisfying this property) such that we have 2-BTP = BTP, and for k > 2, k-BTP is a relaxation of BTP in the sense that k-BTP (k + 1)-BTP. 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. Concerning tractability, we show that instances satisfying k-BTP and which are strong k-consistent are tractable, that is, can be recognized and solved in polynomial time. We also study the relationship between k-BTP and the approach of Naanaa who proposed a set-theoretical tool, known as the directional rank, to extend tractable classes in a parameterized way. Finally we propose an experimental study of 3-BTP which shows the practical interest of this class, particularly w.r.t. the practical solving of instances satisfying 3-BTP and for other instances, w.r.t. to backdoors based on this tractable class.

Section: Introductionmentioning

confidence: 99%

A Microstructure-Based Family of Tractable Classes for CSPs

Cooper

Jégou²,

Terrioux³

2015

“…Naanaa has given two other generalisations of BTP which define tractable classes [18,19]. It has been shown [8] that the notion of directional rank k −1 [18] strictly generalises k-BTP.…”

Section: Definition 7 (K-btp [8])mentioning

confidence: 99%

“…It has been shown [8] that the notion of directional rank k −1 [18] strictly generalises k-BTP. We can deduce that the example of Fig.…”

Section: Definition 7 (K-btp [8])mentioning

confidence: 99%

“…However, by imposing some restrictions on the constraint scopes and/or relations, we can define tractable classes of instances which can be solved in polynomial time. The BTP (Broken Triangle Property) tractable class, is an important tractable class since it generalises certain previously known classes based exclusively on properties of the constraint scopes or the constraint relations and has been the inspiration for a new branch of research on tractable classes of CSPs based on forbidden patterns [1,4,9,11,18]. The Broken Triangle Property imposes the absence of so-called broken triangles.…”

Section: Preliminariesmentioning

confidence: 99%

“…This class has some interesting characteristics, both from a theoretical and practical viewpoint: it generalises existing language-based and structural classes and is solved in polynomial time by the algorithm MAC which is omnipresent in constraint solvers [20]. Besides, many extensions of the broken-triangle property have led to the definition of new tractable classes [8,10,11,14,18,19]. Local versions of the BTP have also given rise to novel reduction operations for CSP instances.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Extending Broken Triangles and Enhanced Value-Merging

Cooper

Mouelhi

Terrioux

2016

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 17155The contribution was presented at CP 2016 :http://cp2016.a4cp.org/ Abstract. Broken triangles constitute an important concept not only for solving constraint satisfaction problems in polynomial time, but also for variable elimination or domain reduction by merging domain values. Specifically, for a given variable in a binary arc-consistent CSP, if no broken triangle occurs on any pair of values, then this variable can be eliminated while preserving satisfiability. More recently, it has been shown that even when this rule cannot be applied, it could be possible that for a given pair of values no broken triangle occurs. In this case, we can apply a domain-reduction operation which consists in merging these values while preserving satisfiability.In this paper we show that under certain conditions, and even if there are some broken triangles on a pair of values, these values can be merged without changing the satisfiability of the instance. This allows us to define a stronger merging operation and a new tractable class of binary CSP instances. We report experimental trials on benchmark instances.

A Join-Based Hybrid Parameter for Constraint Satisfaction

Ganian

Ordyniak

Szeider

2019

We propose joinwidth, a new complexity parameter for the Constraint Satisfaction Problem (CSP). The definition of joinwidth is based on the arrangement of basic operations on relations (joins, projections, and pruning), which inherently reflects the steps required to solve the instance. We use joinwidth to obtain polynomial-time algorithms (if a corresponding decomposition is provided in the input) as well as fixed-parameter algorithms (if no such decomposition is provided) for solving the CSP. Joinwidth is a hybrid parameter, as it takes both the graphical structure as well as the constraint relations that appear in the instance into account. It has, therefore, the potential to capture larger classes of tractable instances than purely structural parameters like hypertree width and the more general fractional hypertree width (fhtw). Indeed, we show that any class of instances of bounded fhtw also has bounded joinwidth, and that there exist classes of instances of bounded joinwidth and unbounded fhtw, so bounded joinwidth properly generalizes bounded fhtw. We further show that bounded joinwidth also properly generalizes several other known hybrid restrictions, such as fhtw with degree constraints and functional dependencies. In this sense, bounded joinwidth can be seen as a unifying principle that explains the tractability of several seemingly unrelated classes of CSP instances.⋆ Robert Ganian acknowledges support by the Austrian Science Fund (FWF, Project P31336) and is also affiliated with FI MUNI, Czech Republic.