A BTP-Based Family of Variable Elimination Rules for Binary CSPs

International audienceThe study of broken-triangles is becoming increasingly ambitious , by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through value merging or variable elimination. Considerable progress has been made in extending this important concept, such as dual broken-triangle and weakly broken-triangle, in order to maximize the number of captured tractable CSP instances and/or the number of merged values. Specifically, m-wBTP allows to merge more values than BTP. k-BTP, WBTP and m-BTP permit to capture more tractable instances than BTP. Here, we introduce a new weaker form of BTP, which will be called m-fBTP for flexible broken-triangle property. m-fBTP allows on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but less than m-wBTP

On Broken Triangles (IJCAI 2016)

International audienceA binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. 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. Experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP

Sur la complexité des algorithmes de backtracking et quelques nouvelles classes polynomiales pour CSP

National audienceThe question of tractable classes of constraint satisfaction problems (CSPs) has been studied for a long time, and is now a very active research domain. However, studies of tractable classes are typically very theoretical. They usually introduce classes of instances together with polynomial time algorithms for recognizing and solving them, and the algorithms can be used only for the new class. In this paper, we address the issue of tractable classes of CSPs from a di erent perspective. We investigate the complexity of classical, generic algorithms for solving CSPs (such as Forward Checking). We introduce a new parameter for measuring their complexity and derive new complexity bounds. By relating the com- plexity of CSP algorithms to graph-theoretic parameters, our analysis allows us to point at new tractable classes, which can be solved directly by the usual CSP algorithms in polynomial time, and without the need to recognize the classes in advance.L'étude des classes polynomiales, pour les problèmes de satisfaction de contraintes (CSP), constitue depuis longtemps un domaine de recherche important qui s'avère aujourd'hui très actif. Cependant, les travaux réalisés jusqu'à présent se sont révélés pour l'essentiel théoriques. En effet, ils se cantonnent en général à la définition de classes d'instances pour lesquelles des algorithmes polynomiaux ad hoc, à la fois pour la reconnaissance et pour la résolution, sont proposes. Ces algorithmes ne peuvent être, en fait, utilisés que pour le traitement d'une classe d'instances donnée. Ils s'avèrent ainsi difficilement exploitables en pratique, et ne sont donc pas exploités au sein de solveurs généraux. L'intérêt pratique des classes polynomiales est ainsi très limitée. Dans cet article, nous abordons la question des classes polynomiales CSP d'un point de vue différent de l'approche classique, en nous intéressant aux algorithmes que l'on peut retrouver dans les systèmes de résolution opérationnels. Pour cela, nous _étudions d'abord la complexité d'algorithmes génériques de résolution de CSP tels que le Forward-Checking par exemple. Cette étude s'appuie sur l'exploitation d'un paramètre issu de la théorie des graphes, et qui permet de proposer de nouvelles bornes de complexité. La mise en relation de ces nouvelles bornes avec certains résultats issus de la théorie des graphes nous permet d'exhiber de nouvelles classes polynomiales. De cette façon, nous montrons comment des algorithmes classiques de résolution de CSP peuvent traiter efficacement en pratique ainsi qu'en théorie, des instances de CSP, sans devoir reconnaître au préalable leur appartenance à d'éventuelles classes polynomiales

Autour des Triangles Cassés

National audienceUne instance CSP binaire qui satisfait la propriété des triangles cassés (BTP) peut etre résolue en temps polynomial. Malheureusement, en pratique, peu d'ins-tances satisfont cette propriété. Nous montrons qu'une version locale de BTP permet de fusionner des valeurs dans les domaines d'instances binaires quelconques. Des expérimentations démontrent la diminution significative de la taille de l'instance pour certaines classes de pro-bì emes. Ensuite, nous proposons une généralisation de cette fusion a des contraintes d'arité quelconque. En-fin, une version orientée nous permet d'´ etendre la classe polynomiale BTP. Ce papier est un résumé de l'article M. C. Cooper, A. El Mouelhi, C. Terrioux et B. Zanuttini. On Broken Triangles In Proceedings of CP,LNCS 8656, 9–24, 2014

On the Efficiency of Backtracking Algorithms for Binary Constraint Satisfaction Problems

International audienceThe question of tractable classes of constraint satisfaction problems (CSPs) has been studied for a long time, and is now a very active research domain. However, studies of tractable classes are typically very theoretical. They usually introduce classes of instances together with polynomial time algorithms for recognizing and solving them, and the algorithms can be used only for the new class. In this paper, we address the issue of tractable classes of CSPs from a different perspective. We investigate the complexity of classical, generic algorithms for solving CSPs (such as Forward Checking). We introduce a new parameter for measuring their complexity and derive new complexity bounds. By relating the complexity of CSP algorithms to graph-theoretic parameters, our analysis allows us to point at new tractable classes, which can be solved directly by the usual CSP algorithms in polynomial time, and without the need to recognize the classes in advance

On Broken Triangles

A binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. 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. Experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP

Broken triangles: From value merging to a tractable class of general-arity constraint satisfaction problems

International audienceA binary CSP instance satisfying the broken-triangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in arbitrary instances of binary CSP, thus providing a novel polynomial-time reduction operation. Extensive experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity and we investigate the theoretical relationship with resolution in SAT. A directional version of general-arity BTP-merging then allows us to extend the BTP tractable class previously defined only for binary CSP. We investigate the complexity of several related problems including the recognition problem for the general-arity BTP class when the variable order is unknown, finding an optimal order in which to apply BTP merges and detecting BTP-merges in the presence of global constraints such as AllDifferent

Tractable Classes for CSPs of Arbitrary Arity: From Theory to Practice

International audienceThe research of this thesis focuses on the analysis of polynomial classes and their practical exploitation for solving constraint satisfaction problems (CSPs) with finite domains. In particular, I worked on bridging the gap between theoretical works and practical results in constraint solvers. Specifically, the goal of this thesis is to find explanation for the effectiveness of solvers, and also to show that studied tractable classes are not artificial since several real-problems among the ones used in the CSP 2008 Competition belong to them. Our work is organized into three main parts. In the first part, we proposed several types of microstructures for CSPs of arbitrary arity which are based on some knwon binary encoding of non-binary CSPs like, dual encoding, hidden-variable transformation and mixed (or double) encoding. These theoretical tools are designed to facilitate the study of tractable classes, sets of CSP instances which can be solved in polytime, when the constraints are non-binary. After that, we propose a new tractable classes of CSPs whose the highlighting should allow on the one hand to explain the effectiveness of solvers of the state of the art namely FC, MAC, RFL and on the second hand to provide the opportunities for easy integration in these solvers. These would include the definition of new tractable classes without using of an ad hoc algorithms as in the traditional case. These new tractable classes are related to the number of maximal cliques in the microstructure of binary or non-binary CSP. In the last part, we focus on the presence of instances belonging to polynomial classes in classical benchmarks used by the CP community. We study in particular the Broken-Triangle Property (BTP) and its extension DBTP to CSP of arbitrary arity. Next, we prove that BTP can also be used to reduce the size of the search space by merging pairs of values on which no broken triangle exists. Finally, we introduce a formal framework, called transformation, and we develop the concept of hidden tractable class that we exploit from an experimental point of view

Tractable Classes for CSPs of Arbitrary Arity: From Theory to Practice

International audienceThe research of this thesis focuses on the analysis of polynomial classes and their practical exploitation for solving constraint satisfaction problems (CSPs) with finite domains. In particular, I worked on bridging the gap between theoretical works and practical results in constraint solvers. Specifically, the goal of this thesis is to find explanation for the effectiveness of solvers, and also to show that studied tractable classes are not artificial since several real-problems among the ones used in the CSP 2008 Competition belong to them. Our work is organized into three main parts. In the first part, we proposed several types of microstructures for CSPs of arbitrary arity which are based on some knwon binary encoding of non-binary CSPs like, dual encoding, hidden-variable transformation and mixed (or double) encoding. These theoretical tools are designed to facilitate the study of tractable classes, sets of CSP instances which can be solved in polytime, when the constraints are non-binary. After that, we propose a new tractable classes of CSPs whose the highlighting should allow on the one hand to explain the effectiveness of solvers of the state of the art namely FC, MAC, RFL and on the second hand to provide the opportunities for easy integration in these solvers. These would include the definition of new tractable classes without using of an ad hoc algorithms as in the traditional case. These new tractable classes are related to the number of maximal cliques in the microstructure of binary or non-binary CSP. In the last part, we focus on the presence of instances belonging to polynomial classes in classical benchmarks used by the CP community. We study in particular the Broken-Triangle Property (BTP) and its extension DBTP to CSP of arbitrary arity. Next, we prove that BTP can also be used to reduce the size of the search space by merging pairs of values on which no broken triangle exists. Finally, we introduce a formal framework, called transformation, and we develop the concept of hidden tractable class that we exploit from an experimental point of view

Une famille de règles d'élimination de variables pour les CSP binaires basées sur BTP

International audienceL'´ etude des triangles cassés devient de plus en plus ambitieuse, par la résolution desprobì emes de satisfaction de contraintes (CSP) en temps polynomial d'un coté, et par la réduction de l'espace de recherchè a tra-vers l'´ elimination de variables et la fusion de valeurs de l'autre. Pour cela, plusieurs extensions de ce concept ontétéétudiées ontétéontétéétudiées dans le passé récent, tel que les triangles cassés duaux et les triangles légèrement cassés. Ces extensions ontétéontété introduites dans le but de maximiser soit le nombre de valeurs fusionnées et/ou le nombre d'ins-tances traitables capturées. Mais, aucune d'entre elles n'a préservé toutes les caractéristiques de BTP. Ici, nous introduisons une nouvelle version légère de BTP, que nous appelons m-fBTP (pour flexible broken-triangle property). m-fBTP permet la fusion de valeurs, l'´ elimination de variables et définit une plus grande classe polynomiale pour laquelle la cohérence d'arc est une pro-cédure de décision. Une version plus détaillée en langue anglaise a ´ eté publiéè a AAAI'17 [4]