A Microstructure-Based Family of Tractable Classes for CSPs

Abstract

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.

Variable Elimination in Binary CSPs

“…Unfortunately, the search for lack-of-support variables for each broken triangle may render such techniques prohibitively expensive in terms of time complexity (since, in the worst case, the number of broken triangles is Θ(n 3 d 4 )). Other generalisations of BTP require levels of consistency, such as strong path consistency, which may change positive edges into negative edges; this has the disadvantage of possibly introducing new broken triangles besides the extra memory required to store new binary constraints (Cooper, Jégou, & Terrioux, 2015;Naanaa, 2013).…”

Extending Broken Triangles and Enhanced Value-Merging

“…5). A lighter version of BTP, called k-BTP, which allows certain broken triangles, has recently been defined [8]. Binary CSP instances which satisfy both strong k-consistency and k-BTP constitute a tractable class.…”

“…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.…”

“…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.…”

“…6 we show that it does allow us to define a tractable class. We also compare m-wBTP with certain other generalisations of BTP, such as k-BTP [8] and WBTP [19]. In Sect.…”