Hybrid tractability of soft constraint problems

The constraint satisfaction problem (CSP) is a central generic problem in computer science and artificial intelligence: it provides a common framework for many theoretical problems as well as for many real-life applications. Soft constraint problems are a generalisation of the CSP which allow the user to model optimisation problems. Considerable effort has been made in identifying properties which ensure tractability in such problems. In this work, we initiate the study of hybrid tractability of soft constraint problems; that is, properties which guarantee tractability of the given soft constraint problem, but which do not depend only on the underlying structure of the instance (such as being tree-structured) or only on the types of soft constraints in the instance (such as submodularity). We present several novel hybrid classes of soft constraint problems, which include a machine scheduling problem, constraint problems of arbitrary arities with no overlapping nogoods, and the SoftAllDiff constraint with arbitrary unary soft constraints. An important tool in our investigation will be the notion of forbidden substructures.Comment: A full version of a CP'10 paper, 26 page

New schemes for simplifying binary constraint satisfaction problems

Finding a solution to a Constraint Satisfaction Problem (CSP) is known to be an NP-hard task. This has motivatedthe multitude of works that have been devoted to developing techniques that simplify CSP instances before or duringtheir resolution.The present work proposes rigidly enforced schemes for simplifying binary CSPs that allow the narrowing of valuedomains, either via value merging or via value suppression. The proposed schemes can be viewed as parametrizedgeneralizations of two widely studied CSP simplification techniques, namely, value merging and neighbourhoodsubstitutability. Besides, we show that both schemes may be strengthened in order to allow variable elimination,which may result in more significant simplifications. This work contributes also to the theory of tractable CSPs byidentifying a new tractable class of binary CSP

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

Forbidden patterns in constraint satisfaction problems

Le problème de satisfaction de contraintes (CSP) est NP-complet, même dans le cas où toutes les contraintes sont binaires. Cependant, certaines classes d'instances CSP sont traitables. Récemment, une nouvelle méthode pour définir de telles classes aémergée. Cette approche est centrée autour des motifs interdits, ou l'absence locale de certaines conditions. Elle est l'objet de ma thèse. Nous définissons formellement ce que sont les motifs interdits, présentons les propriétés qu'ils détiennent, et finalement les utilisons afin d'établir plusieurs résultats de complexité importants. En utilisant différentes versions de motifs, toutes basées sur le même concept de base, nous énumérons un nombre important de nouvelles classes traitables, ainsi que certaines NP-completes. Nous combinons ces résultats pour révéler plusieurs dichotomies, chacune englobant une large gamme de classes d'instances CSP. Nous montrons aussi que les motifs interdits représentent un outil intéressant pour la simplification d'instances CSPs. Nous donnons plusieurs nouveaux moyens de réduire la taille des instances CSP, que ce soit en éliminant des variables ou en fusionnant les domaines, et montrons comment ces méthodes sont activées par l'absence locale de certains modèles. Comme les conditions de leurutilisation sont entièrement locales, nos opérations peuvent être utilisés sur un large éventail de problèmes.The Constraint Satisfaction Problem (CSP) is NP-Complete, even in the case where all constraints are binary. However, some classes of CSP instances are tractable. Recently, a new method for defining such classes has emerged. This approach is centered around forbidden patterns, or the local absence of some conditions. It is the focus of my thesis. We formally define what forbidden patterns are, exhibit the properties they hold, and eventually put them to use in order to establish several important tractability results. Using different versions of patterns, all based on the same core concept, we list a significant number of new tractable classes, as well as some NP-Complete ones. We combine these results to reveal several dichotomies, each one encompassing a large range of classes of CSP instances. We also show how useful a tool forbidden patterns can be in the field of CSP instance simplification. We give multiple new ways of decreasing the size of CSP instances, whether by eliminating variables or fusioning domains, and prove how all these methods are enabled by the local absence of some patterns. Since the conditions for their use are entirely local, our operations can be used on a wide array of problems

Datalog and Constraint Satisfaction with Infinite Templates

On finite structures, there is a well-known connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates. If the template Gamma is omega-categorical, we present various equivalent characterizations of those Gamma such that the constraint satisfaction problem (CSP) for Gamma can be solved by a Datalog program. We also show that CSP(Gamma) can be solved in polynomial time for arbitrary omega-categorical structures Gamma if the input is restricted to instances of bounded treewidth. Finally, we characterize those omega-categorical templates whose CSP has Datalog width 1, and those whose CSP has strict Datalog width k.Comment: 28 pages. This is an extended long version of a conference paper that appeared at STACS'06. In the third version in the arxiv we have revised the presentation again and added a section that relates our results to formalizations of CSPs using relation algebra

Domain value mutation and other techniques for constraint satisfaction problems

The term Constraint Satisfaction Problem (CSP) refers to a class of NP-complete problems, a collection of difficult problems for which no fast solution is known. The standard definition of a CSP involves variables, values, and constraints: each variable must be assigned a value from a designated group of possible values (also known as the variable’s domain), while a constraint on a set of variables indicates permissible combinations of values for these variables. Given a CSP, an important objective is to query whether it has a solution — an assignment of each variable to a value such that all constraints are satisfied. Solving a CSP usually requires chronological backtracking search that interleaves variable assignments with various kinds of inferences in order to reduce the search space. This dissertation comprises two parts. The first part deals with a modification of the classical CSP model that allows a value to be broken up and multiple values to be combined. The second part deals with generalized arc consistency algorithms. Both parts share a common theme in that extensional constraints --‐ the most basic expression possible for constraints --- play the central role. Despite being an important class, extensional constraints have received much less attention recently as most efforts have been channelled toward identifying new types of specialized constraints and coming up with corresponding algorithms. Regardless, improvements to algorithms for extensional constraints are more fundamental. This dissertation will attempt to improve existing techniques and algorithms for extensional constraints by examining them critically from the bottom up and approaching them from a novel direction