Multi-Dimensional Models Facilitate Automatic Reformulation: The Case Study of the SET Game

In this paper we describe a reformulation strategy for solving multidimensional Constraint Satisfaction Problems (CSPs). This strategy operates by iteratively considering, in isolation, each one of the uni-dimensional constraints in the problem, and exploits the approximate symmetries induced by the selected constraint on the domains in order to enforce this constraint on the simplified problem. We use the game of SET, a combinatorial card game, as a toy problem to motivate our strategy and to explain and illustrate its operation. However, we believe that our approach is applicable to more complex domains of scientific and industrial importance, and deserves more thorough investigations in the future. Our approach sheds a new light on the dynamic reformulation of multidimensional CSPs. Importantly, it advocates that modeling tools for Constraint Programming should allow the user to specify the constraints directly on the attributes of the domain objects (i.e., variables and values) so that their multi-dimensionality can be exploited during problem solving

Tree Projections and Constraint Optimization Problems: Fixed-Parameter Tractability and Parallel Algorithms

Tree projections provide a unifying framework to deal with most structural decomposition methods of constraint satisfaction problems (CSPs). Within this framework, a CSP instance is decomposed into a number of sub-problems, called views, whose solutions are either already available or can be computed efficiently. The goal is to arrange portions of these views in a tree-like structure, called tree projection, which determines an efficiently solvable CSP instance equivalent to the original one. Deciding whether a tree projection exists is NP-hard. Solution methods have therefore been proposed in the literature that do not require a tree projection to be given, and that either correctly decide whether the given CSP instance is satisfiable, or return that a tree projection actually does not exist. These approaches had not been generalized so far on CSP extensions for optimization problems, where the goal is to compute a solution of maximum value/minimum cost. The paper fills the gap, by exhibiting a fixed-parameter polynomial-time algorithm that either disproves the existence of tree projections or computes an optimal solution, with the parameter being the size of the expression of the objective function to be optimized over all possible solutions (and not the size of the whole constraint formula, used in related works). Tractability results are also established for the problem of returning the best K solutions. Finally, parallel algorithms for such optimization problems are proposed and analyzed. Given that the classes of acyclic hypergraphs, hypergraphs of bounded treewidth, and hypergraphs of bounded generalized hypertree width are all covered as special cases of the tree projection framework, the results in this paper directly apply to these classes. These classes are extensively considered in the CSP setting, as well as in conjunctive database query evaluation and optimization

Effectively Enforcing Minimality During Backtrack Search

Constraint Processing is an expressive and powerful framework for modeling and solving combinatorial decision problems. Enforcing consistency during backtrack search is an effective technique for reducing thrashing in a large search tree. The higher the level of the consistency enforced, the stronger the pruning of inconsistent subtrees. Recently, high-level consistencies (HLC) were shown to be instrumental for solving difficult instances. In particular, minimality, which is guaranteed to prune all inconsistent branches, is advantageous even when enforced locally. In this thesis, we study two algorithms for computing minimality and propose three new mechanisms that significantly improve performance. Then, we integrate the resulting algorithms in a portfolio that operates both locally and dynamically during search. Finally, we empirically evaluate the performance of our approach on benchmark problems. Adviser: Berthe Y. Choueir

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

A First Practical Algorithm for High Levels of Relational Consistency

Consistency properties and algorithms for achieving them are at the heart of the success of Constraint Programming. In this paper, we study the relational consistency property R(*,m)C, which is equivalent to m-wise consistency proposed in relational databases. We also define wR(*,m)C, a weaker variant of this property. We propose an algorithm for enforcing these properties on a Constraint Satisfaction Problem by tightening the existing relations and without introducing new ones. We empirically show that wR(*,m)C solves in a backtrackfree manner all the instances of some CSP benchmark classes, thus hinting at the tractability of those classes

Uma condição suficiente para otimização global sem retrocesso

Orientador: Fabiano SilvaTese (doutorado) - Universidade Federal do Paraná, Setor de Ciências Exatas, Programa de Pós-Graduação em Informática. Defesa : Curitiba, 04/05/2018Inclui referências: p.185-193Área de concentração: Ciência da ComputaçãoResumo: Um problema de satisfação de restrições (CSP, do inglês constraint satisfaction problem) consiste em encontrar uma atribuição de valores a um conjunto de variáveis que satisfaça uma rede de restrições. Técnicas de consistência local desempenham um papel central na resolução de CSPs, excluindo valores que certamente não constituem uma solução do problema. Muitos esforços vêm sendo aplicados na identificação de classes de CSPs relacionando a estrutura da rede (representada por um hipergrafo) com o nível de consistência local que garante uma solução livre de retrocesso, isto é, uma busca que encontra uma solução em um número polinomial de passos em relação ao tamanho da instância. Nesta tese, problemas de otimização global são representados por hipergrafos com um vértice raiz que representa a função objetivo a ser minimizada. Uma forma de decomposição de hipergrafos, chamada decomposição Epífita, é apresentada. Através da decomposição Epífita do hipergrafo de restrições, caracteriza-se uma classe de problemas de otimização onde a consistência de arco relacional direcionada garante uma solução livre de retrocesso. Alcançar consistência relacional exige a adição de novas restrições na rede, alterando a sua estrutura; por essa razão, um método de ramificação e poda intervalar para alcançar uma forma relaxada dessa consistência é proposto, encontrando uma aproximação do mínimo global de problemas de otimização. Um otimizador de código-fonte aberto que implementa esse método, chamado OGRe, é apresentado. A fim de generalizar o conceito de decomposição Epífita a todos os problemas de otimização, um parâmetro de largura de hipergrafos chamado largura epífita é introduzido. Como principal contribuição desta tese, mostra-se que problemas de otimização representados por hipergrafos com largura epífita k possuem decomposições k-Epífitas e são resolvidos sem retrocesso se alcançada k-consistência relacional direcionada forte. Palavras-chave: otimização global, consistência relacional, análise intervalar, decomposição epífita, hipergrafo.Abstract: A constraint satisfaction problem (CSP) consists of finding an assignment of values to a set of variables that satisfy a constraint network. Local consistency techniques play a central role in solving CSPs, pruning values that surely do not constitute a solution of the problem. Many efforts have been applied to identify classes of CSPs by linking the constraint network structure (represented by a hypergraph) to the level of local consistency that guarantees a backtrack-free solution, i.e., a search that finds a solution in a polynomial number of steps with relation to the size of the instance. In this thesis, global optimization problems are represented by hypergraphs with a root vertex that represents the objective function to be minimized. A form of hypergraph decomposition is introduced, called Epiphytic decomposition. By the Epiphytic decomposition of constraint hypergraphs a class of optimization problems is characterized, for which directional relational arc-consistency ensures a backtrack-free solution. Achieving relational consistency requires the addition of new constraints on the network, changing its structure; for this reason, an interval branch and bound method to enforce a relaxed form of this consistency is proposed, thus finding an approximation for the global minimum of optimization problems. An open-source optimizer that implements this method, namely OGRe, is introduced. In order to generalize the Epiphytic decomposition concept to cover all optimization problems, a hypergraph width parameter is introduced, called epiphytic width. As the main contribution of this thesis, it is shown that optimization problems represented by hypergraphs with epiphytic width k have k-Epiphytic decompositions and are solved in a backtrack-free manner if achieved strong directional relational k-consistency. Keywords: global optimization, relational consistency, interval analysis, epiphytic decomposition, hypergraph