Cyril Terrioux

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.

Hidden Tractable Classes: From Theory to Practice

Tractable classes constitute an important issue in CP, at least from a theoretical viewpoint. But they are not actually used in practice. Either their treatment is too costly for time complexity or, even if there exist efficient algorithms to manage them, they do not appear in the real problems. We propose here to address this issue thanks to the notion of hidden tractable classes. Such classes are based on a known tractable class C, and a transformation t, and are defined by sets of instances P such that their transformation using t is in C, that is t (P) in C. We propose a general framework to study such notions. After, we focus our study on the tractable class BTP, and several transformations which are the filterings classically used in CP. We show then that the use of filterings allows sometimes to highlight the occurrence of BTP in the benchmarks generally considered for solver comparisons, i.e. That BTP is sometimes hidden in the benchmarks. Thus, this approach allows to extend the set of known tractable classes.

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.

Tree-Decompositions with Connected Clusters for Solving Constraint Networks

From a theoretical viewpoint, the (tree-)decomposition methods offer a good approach for solving Constraint Satisfaction Problems (CSPs) when their (tree)-width is small. In this case, they have often shown their practical interest. So, the literature (coming from Mathematics or AI) has concentrated its efforts on the minimization of a single parameter, the tree-width. Nevertheless, experimental studies have shown that this parameter is not always the most relevant to consider for solving CSPs. In this paper, we experimentally show that the decomposition algorithms of the state of the art produce clusters (a tree-decomposition is a tree of clusters) having several connected components. Then we highlight that such clusters create a real problem for the efficiency of solving methods. To avoid this kind of problem, we consider here a new kind of graph decomposition called Bag-Connected Tree-Decomposition, which considers only tree-decompositions such that each cluster is connected. We propose a first polynomial time algorithm to find such decompositions. Finally, we show experimentally that using these bag-connected tree-decompositions improves significantly the solving of CSPs by decomposition methods.

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 Microstructure-Based Family of Tractable Classes for CSPs

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

Combining restarts, Nogoods and decompositions for solving CSPs

k > 2