Achref El Mouelhi

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.

A Hybrid Tractable Class for Non-binary CSPs

Find new islands of tractability, that is classes of CSPs for which polytime algorithms exist, is a fundamental task in the study of constraint satisfaction problems. The concept of hybrid tractable class, which allows to deal simultaneously with the restrictions of languages and, for example, the satisfaction of structural properties, is an approach which has already shown its interest in this domain. Here we study a hybrid class for non-binary CSPs. With this aim in view, we consider the tractable class BTP introduced in [1].While this class has been defined for binary CSPs, the authors have suggested to extend it to CSPs with constraints of arbitrary arities, using the dual representation of such CSPs. We develop this idea by proposing a new definition without exploiting the dual representation, but using a semantic property associated to the compatibility relations of the constraints. This class, called DBTP for Dual BTP, is firstly shown to be tractable. Then it is compared to some known classes. In particular, we prove that DBTP is incomparable with BTP and that it includes some well known classes of CSPs such as beta-acyclic CSPs.

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 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.

Some New Tractable Classes of CSPs and Their Relations with Backtracking Algorithms

In this paper, we investigate the complexity of algorithms for solving CSPs which are classically implemented in real practical solvers, such as Forward Checking or Bactracking with Arc Consistency (RFL or MAC).. We introduce a new parameter for measuring their complexity and then we derive new complexity bounds. By relating the complexity of CSP algorithms to graph-theoretical parameters, our analysis allows us to define 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. So, our approach allows us to propose new tractable classes of CSPs that are naturally exploited by solvers, which indicates new ways to explain in some cases the practical efficiency of classical search algorithms.

A hybrid tractable class for non-binary CSPs

Find new islands of tractability, that is classes of CSP instances for which polytime algorithms exist, is a fundamental task in the study of constraint satisfaction problems. The concept of hybrid tractable class, which allows to deal simultaneously with the restrictions of languages and, for example, the satisfaction of structural properties, is an approach which has already shown its interest in this domain. Here we study a hybrid class for non-binary CSP instances. With this aim in view, we consider the Broken Triangle Property (BTP) introduced in Cooper et al. (Artificial Intelligence, 174, 570–584 2010). While this tractable class has been defined for binary instances, the authors have suggested to extend it to instances with constraints of arbitrary arities, using the dual representation of such CSPs. We develop this idea by proposing a new definition without exploiting the dual representation, but using a semantic property associated to the compatibility relations of the constraints. This class is called DBTP for Dual Broken Triangle Property. We study it in depth, firstly to show that it is tractable. Then we compare it to some known classes. In particular, we prove that DBTP is incomparable with BTP and that it includes some well known tractable classes of CSPs such as β-acyclic CSPs. Then, we compare it with the Hyper-k-Consistency, which allows us to also present new results for BTP. Finally, we analyse DBTP from a practical viewpoint, by first highlighting that some benchmarks which are classically used to compare the solvers are included in DBTP and then by explaining the efficiency of solvers of the state of the art on such instances thanks to their membership of the DBTP class.

Extending Broken Triangles and Enhanced Value-Merging

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.

On the Decomposition of Non-binary Constraint into Equivalent Binary Constraints

Considerable research effort has been focused on the translation of non-binary CSP into an equivalent binary CSP. Most of this work has been devoted to studying the binary encoding of non-binary CSP. Three encodings have been proposed, namely dual encoding, hidden variable encoding and double encoding. Unfortunately, such encodings do not allow to use some properties and interesting results defined only for the binary case. Another approach consists in the transformation of each non-binary constraint into a set of binary constraints: the CSP obtained is called primal. Unfortunately, this transformation does not preserve satisfiability. In this paper, we propose some conditions under which a non-binary constraint can be decomposed into a set of binary constraints while preserving satisfiability. An experimental study proves that our approach is not artificial since some ternary benchmarks can be transformed into equivalent binary instances and effectively solved by MAC.

Different Classes of Graphs to Represent Microstructures for CSPs

The CSP formalism has shown, for many years, its interest for the representation of numerous kinds of problems, and also often provide effective resolution methods in practice. This formalism has also provided a useful framework for the knowledge representation as well as to implement efficient methods for reasoning about knowledge. The data of a CSP are usually expressed in terms of a constraint network. This network is a (constraints) graph when the arity of the constraints is equal to two (binary constraints), or a (constraint) hypergraph in the case of constraints of arbitrary arity, which is generally the case for problems of real life. The study of the structural properties of these networks has made it possible to highlight certain properties, which led to the definition of new tractable classes, but in most cases, they have been defined for the restricted case of binary constraints. So, several representations by graphs have been proposed for the study of constraint hypergraphs to extend the known results to the binary case. Another approach, finer, is interested in the study of the microstructure of CSP, which is defined by graphs. This helped, offering a new theoretical framework to propose other tractable classes. In this paper, we propose to extend the notion of microstructure to any type of CSP. For this, we propose three kinds of graphs that can take into account the constraints of arbitrary arity. We show how these new theoretical tools can already provide a framework for developing new tractable classes for CSPs. We think that these new representations should be of interest for the community, firstly for the generalization of existing results, but also to obtain original results.

On a new extension of BTP for binary CSPs

The study of broken-triangles is becoming increasingly ambitious, by both solving constraint satisfaction problems (CSPs) in polynomial time and reducing search space size through either 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 us to merge more values than BTP. DBTP, ∀∃-BTP, k-BTP, WBTP and m-wBTP permit us to capture more tractable instances than BTP. However, except BTP, none of these extensions allows variable elimination while preserving satisfiability. Moreover, k-BTP and m-wBTP define bigger tractable classes around BTP but both of them generally need a high level of consistency. Here, we introduce a new weaker form of BTP, called m-fBTP for flexible broken-triangle property, which will represent a compromise between most of these previous tractable properties based on BTP. m-fBTP allows us on the one hand to eliminate more variables than BTP while preserving satisfiability and on the other to define a new bigger tractable class for which arc consistency is a decision procedure. Likewise, m-fBTP permits to merge more values than BTP but fewer than m-wBTP. The binary CSP instances satisfying m-fBTP are solved by algorithms of the state-of-the-art like MAC and RFL in polynomial time. An open question is whether it is possible to compute, in polynomial time, the existence of some variable ordering for which a given instance satisfies m-fBTP.

Tractable classes for CSPs of arbitrary arity: from theory to practice

The 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.The 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.