Set Intersection and Consistency in Constraint Networks

In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This approach not only simplifies the understanding of and unifies many existing consistency results, but also directs the study of consistency to that of set intersection properties in many situations, as demonstrated by the results on the convexity and tightness of constraints in this paper. Specifically, we identify a new class of tree convex constraints where local consistency ensures global consistency. This generalizes row convex constraints. Various consistency results are also obtained on constraint networks where only some, in contrast to all in the existing work,constraints are tight

Set Intersection and Consistency in Constraint Networks

Abstract In this paper, we show that there is a close relation between consistency in a constraint network and set intersection. A proof schema is provided as a generic way to obtain consistency properties from properties on set intersection. This approach not only simplifies the understanding of and unifies many existing consistency results, but also directs the study of consistency to that of set intersection properties in many situations, as demonstrated by the results on the convexity and tightness of constraints in this paper. Specifically, we identify a new class of tree convex constraints where local consistency ensures global consistency. This generalizes row convex constraints. Various consistency results are also obtained on constraint networks where only some, in contrast to all in the existing work, constraints are tight

Constraint tightness and looseness versus local and global consistency

Constraint networks are a simple representation and reasoning framework with diverse applications. In this paper, weidentify two new complementary properties on the restrictiveness of the constraints in a network| constraint tightness and constraint looseness|and we show their usefulness for estimating the level of local consistency needed to ensure global consistency, and for estimating the level of local consistency present ina network. In particular, we present a su cient condition, based on constraint tightness and the level of local consistency, that guarantees that a solution can be found in a backtrack-free manner. The condition can be useful in applications where a knowledge base will be queried over and over and the preprocessing costs can be amortized over many queries. We also present a su cient condition for local consistency, based on constraint looseness, that is straightforward and inexpensive to determine. The condition can be used to estimate the level of local consistency of a network. This in turn can be used in deciding whether it would be useful to preprocess the network before a backtracking search, and in deciding which local consistency conditions, if any, still need to be enforced if we want to ensure that a solution can be found in a backtrack-free manner. Two de-nitions of local consistency are employed in characterizing the conditions: the traditional variable-based notion and a recently introduced de nition of local consistency called relational consistency 1. 1 Some of the results reported in this paper have previously appeared at KR-94 [28] and 1

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