A Partial Taxonomy of Substitutability and Interchangeability

Substitutability, interchangeability and related concepts in Constraint Programming were introduced approximately twenty years ago and have given rise to considerable subsequent research. We survey this work, classify, and relate the different concepts, and indicate directions for future work, in particular with respect to making connections with research into symmetry breaking. This paper is a condensed version of a larger work in progress.Comment: 18 pages, The 10th International Workshop on Symmetry in Constraint Satisfaction Problems (SymCon'10

Solution Counting for CSP and SAT with Large Tree-Width

Рассмотрена проблема подсчета количества решений задачи совместимости ограничений (Constraint Satisfaction Problem). Для ее решения был адаптирован метод обратного прослеживания с ацикличным представлением графа ограничений (Backtracking with Tree-Decomposition). Предложен точный алгоритм, сложность которого экспоненциально зависит от ширины дерева, и приближенный алгоритм, экспоненциально зависящий от размера максимальной клики.The problem of counting the number of solutions of a CSP is considered. For solving the problem the Backtracking with a Tree-Decomposition method was adapted. The exact algorithm is suggested which has the worst-time complexity exponential in a tree width, as well as iterative algorithm that has computational complexity exponential in a maximum clique size.Розглянуто проблему підрахунку кількості розв’язків задачі сумісності обмежень (Constraint Satisfaction Problem). Для її розв’язку було адаптовано метод зворотного простеження з ациклічним поданням графа обмежень (Backtracking with Tree-Decomposition). Запропоновано точний алгоритм, складність якого експоненційно залежить від ширини дерева, і наближений алгоритм, експоненційно залежний від розміру максимальної кліки

The Language of Search

This paper is concerned with a class of algorithms that perform exhaustive search on propositional knowledge bases. We show that each of these algorithms defines and generates a propositional language. Specifically, we show that the trace of a search can be interpreted as a combinational circuit, and a search algorithm then defines a propositional language consisting of circuits that are generated across all possible executions of the algorithm. In particular, we show that several versions of exhaustive DPLL search correspond to such well-known languages as FBDD, OBDD, and a precisely-defined subset of d-DNNF. By thus mapping search algorithms to propositional languages, we provide a uniform and practical framework in which successful search techniques can be harnessed for compilation of knowledge into various languages of interest, and a new methodology whereby the power and limitations of search algorithms can be understood by looking up the tractability and succinctness of the corresponding propositional languages

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

Décompositions fonctionnelles et structurelles dans les modèles graphiques probabilistes appliquées à la reconstruction d'haplotypes

Cette thèse s'articule autour de deux thèmes : la décomposition dans les modèles graphiques que sont, entre autres, les réseaux bayésiens et les réseaux de fonctions de coûts (WCSP) et la reconstruction d'haplotypes dans les pedigrees. Nous appliquons les techniques des WCSP pour traiter les réseaux bayésiens, en exploitant les propriétés structurelles et fonctionnelles, de manière exacte et approchée, des instances dans le cadre de l'inférence (ou d'un problème proche, celui de compter le nombre de solutions) et de l'optimisation. Nous définissons en particulier une décomposition de fonctions qui produit des fonctions portant sur un plus petit nombre de variables. Un exemple d'application en optimisation est la reconstruction d'haplotypes. Elle est essentielle pour une meilleure prédiction de la gravité de maladie ou pour comprendre des caractères physiques particuliers. La reconstruction d'haplotypes se modélise sous forme d'un réseau bayésien. La décomposition fonctionnelle permet de réduire ce réseau bayésien en un problème d'optimisation WCSP (Max-2SAT).This thesis is based on two topics : the decomposition in graphical models which are, among others, Bayesian networks and cost function networks (WCSP) and the haplotype reconstruction in pedigrees. We apply techniques of WCSP to treat Bayesian network. We exploit stuctural and fonctional properties, in an exact and approached methods. Particulary, we define a decomposition of function which produces functions with a smaller variable number. An application example in optimization is the haplotype reconstruction. It is essential for a best prediction of seriousness of disease or to understand particular physical characters. Haplotype reconstruction is represented with a Bayesian network. The functionnal decomposition allows to reduce this Bayesian network in an optimization problem WCSP (Max-2SAT)

The Impact of AND/OR Search Spaces on Constraint Satisfaction and Counting

The contribution of this paper is in demonstrating the impact of AND/OR search spaces view on solutions counting. In contrast to the traditional (OR) search space view, the AND/OR search space displays independencies present in the graphical model explicitly and may sometimes reduce the search space exponentially. Empirica