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

Higher-Level Consistencies: Where, When, and How Much

Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances. We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies. Adviser: B.Y. Choueiry and C. Bessier

Global Inverse Consistency for Interactive Constraint Satisfaction

International audienceSome applications require the interactive resolution of a constraint problem by a human user. In such cases, it is highly desirable that the person who interactively solves the problem is not given the choice to select values that do not lead to solutions. We call this property global inverse consistency. Existing systems simulate this either by maintaining arc consistency after each assignment performed by the user or by compiling offline the problem as a multi-valued decision diagram. In this paper, we define several questions related to global inverse consistency and analyse their complexity. Despite their theoretical intractability, we propose several algorithms for enforcing global inverse consistency and we show that the best version is efficient enough to be used in an interactive setting on several configuration and design problems. We finally extend our contribution to the inverse consistency of tuples

Designing and Optimizing Representations for Non-Binary Constraints

Ph.DDOCTOR OF PHILOSOPH

Higher-Level Consistencies: Where, When, and How Much

Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances. We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies. Adviser: B.Y. Choueiry and C. Bessier

Strong consistencies for weighted constraint satisfaction problems

Cette thèse se focalise sur l'étude de cohérences locales fortes afin de résoudre des problèmes d'optimisation sur des réseaux de fonctions de coûts (ou réseaux de contraintes pondérées). Ces méthodes fournissent le minorant nécessaire pour des approches de type "Séparation-Evaluation". Nous étudions dans un premier temps la cohérence d'Arc virtuelle (VAC), une des plus fortes cohérences d'arcs du domaine, qui est établie via l'établissement de la cohérence d'arc dure dans une séquence de réseaux de contraintes classiques. L'algorithme itératif pour établir VAC est amélioré via l'introduction d'une incrémentalité accrue, exploitant la cohérence d'arc dynamique. La nouvelle méthode est aussi capable de maintenir VAC efficacement pendant la recherche lorsque les réseaux de contraintes pondérées sont dynamiquement modifiés par les opérations de branchement. Dans une seconde partie, nous nous intéressons à des cohérences de domaines plus fortes, inspirées de cohérences similaires dans les réseaux de contraintes classiques (cohérence de chemin inverse, réduite ou Max-réduite). Pour chaque cohérence dure, plusieurs cohérences souples ont été proposées pour les réseaux de contraintes pondérées. Les nouvelles cohérences fournissent un minorant plus fort que celui des cohérences d'arc souples en traitant les triplets de variables connectées deux à deux par des fonctions de coûts binaires. Dans cette thèse, nous étudions les propriétés des nouvelles cohérences, les implémentons et les testons sur une variété de problèmes.This thesis focuses on strong local consistencies for solving optimization problems in cost function networks (or weighted constraint networks). These methods provide the lower bound necessary for Branch-and-Bound search. We first study the Virtual arc consistency, one of the strongest soft arc consistencies, which is enforced by iteratively establishing hard arc consistency in a sequence of classical Constraint Networks. The algorithm enforcing VAC is improved by integrating the dynamic arc consistency to exploit its incremental behavior. The dynamic arc consistency also allows to improve VAC when maintained VAC during search by efficiently exploiting the changes caused by branching operations. Operations. Secondly, we are interested in stronger domain-based soft consistencies, inspired from similar consistencies in hard constraint networks (path inverse consistency, restricted or Max-restricted path consistencies). From each of these hard consistencies, many soft variants have been proposed for weighted constraint networks. The new consistencies provide lower bounds stronger than soft arc consistencies by processing triplets of variables connected two-by-two by binary cost functions. We have studied the properties of these new consistencies, implemented and tested them on a variety of problems

Higher-Level Consistencies: Where, When, and How Much

Determining whether or not a Constraint Satisfaction Problem (CSP) has a solution is NP-complete. CSPs are solved by inference (i.e., enforcing consistency), conditioning (i.e., doing search), or, more commonly, by interleaving the two mechanisms. The most common consistency property enforced during search is Generalized Arc Consistency (GAC). In recent years, new algorithms that enforce consistency properties stronger than GAC have been proposed and shown to be necessary to solve difficult problem instances. We frame the question of balancing the cost and the pruning effectiveness of consistency algorithms as the question of determining where, when, and how much of a higher-level consistency to enforce during search. To answer the `where\u27 question, we exploit the topological structure of a problem instance and target high-level consistency where cycle structures appear. To answer the \u27when\u27 question, we propose a simple, reactive, and effective strategy that monitors the performance of backtrack search and triggers a higher-level consistency as search thrashes. Lastly, for the question of `how much,\u27 we monitor the amount of updates caused by propagation and interrupt the process before it reaches a fixpoint. Empirical evaluations on benchmark problems demonstrate the effectiveness of our strategies. Adviser: B.Y. Choueiry and C. Bessier