Contraintes de Partitionnement par des Arbres

Nous présentons deux contraintes qui partitionnent les sommets d'un graphe non-orienté G = (V, E), où |V| = n et |E| = m, en un ensemble d'arbres disjoints. La première contrainte, resource-forest, spécifie que chaque arbre dans la forêt doit contenir au moins un sommet ressource. L'ensemble des ressources est un sous-ensemble R ⊆ V. Cette contrainte est la contrepartie non-orienté de la contrainte d'arbre introduite dans [2], qui partitionne un graphe orienté en une forêt d'arbres orientés où seulement certains sommets peuvent être des racines. Nous décrivons un algorithme de consistance-hybride pour la contrainte resource-forest ayant une complexité de O(m + n). Ceci constitue donc une amélioration de la complexité en O(mn) connue pour le cas orienté. La seconde constrainte, proper-forest, est une variante de la première ne nécessitant pas que chaque arbre contienne une ressource. Cependant, tout arbre construit doit être un arbre propre, i.e., un arbre contenant au moins deux sommets. Nous avons développé un algorithme de consistance-hybride ayant une complexité en O(mn) au pire des cas, et en O(m√n) dans la plupart des autres cas

On the Algebraic Complexity of Set Equality and Inclusion

We present a linear-time algorithm in the algebraic computation tree model for checking whether two sets of integers are equal. The significance of this result is in the fact that it shows that set equality testing is computationally easier when the elements of the sets are restricted to be integers. In addition, we show a linear-time algorithm for checking set inclusion in a slightly extended computational model

Constraints and Changes

This thesis consists of four technical chapters. The first two chapters deal with filtering algorithms for global constraints. Namley, we show improved algorithms for the well known Global Cardinality Constraint. Then we define a new constraint, UseBy and its special case Same, and show efficient filtering algorithms for both. All of the filtering algorithms follow the same approach: model the constraint as a flow problem. The next two chapters deal with dynamic algoritms. That is, algorithms that maintain information about a directed acyclic graph (DAG) while the graph changes. The third chapter deals with the problem of maintaining the topological order of the nodes of a DAG upon deals with the problem of maintaining the topological order of the nodes of a DAG upon a sequence of edge insertions. The fourth chapter deals with the problem of maintaining the longest paths in a directed acyclic graph upon edge insertions and deletions

Constraints and Changes

This thesis consists of four technical chapters. The first two chapters deal with filtering algorithms for global constraints. Namley, we show improved algorithms for the well known Global Cardinality Constraint. Then we define a new constraint, UseBy and its special case Same, and show efficient filtering algorithms for both. All of the filtering algorithms follow the same approach: model the constraint as a flow problem. The next two chapters deal with dynamic algoritms. That is, algorithms that maintain information about a directed acyclic graph (DAG) while the graph changes. The third chapter deals with the problem of maintaining the topological order of the nodes of a DAG upon deals with the problem of maintaining the topological order of the nodes of a DAG upon a sequence of edge insertions. The fourth chapter deals with the problem of maintaining the longest paths in a directed acyclic graph upon edge insertions and deletions

On the Algebraic Complexity of Set Equality and Inclusion

We present a linear-time algorithm in the algebraic computation tree model for checking whether two sets of integers are equal. The significance of this result is in the fact that it shows that set equality testing is computationally easier when the elements of the sets are restricted to be integers. In addition, we show a linear-time algorithm for checking set inclusion in a slightly extended computational model

Multiconsistency and robustness with global constraints

Abstract. We propose a natural generalization of arc-consistency, which we call multiconsistency: A value v in the domain of a variable x is kmulticonsistent with respect to a constraint C if there are at least k solutions to C in which x is assigned the value v. We present algorithms that determine which variable-value pairs are k-multiconsistent with respect to several well known global constraints. In addition, we show that finding super solutions is sometimes strictly harder than finding arbitrary solutions for these constraints and suggest multiconsistency as an alternative way to search for robust solutions.

Tree Decompositions With Small Cost

It is shown that the problem of maintaining the topological order of the nodes of a directed acyclic graph while inserting m edges can be solved in O(min{m log n}) time, an improvement over the best known result of O(mn). In addition, we analyze the complexity of the same algorithm with respect to the treewidth k of the underlying undirected graph. We show that the algorithm runs in time O(mk log n) for general k and that it can be implemented to run in O(n log n) time on trees, which is optimal. If the input contains cycles, the algorithm detects this

Online Topological Ordering

It is shown that the problem of maintaining the topological order of the nodes of a directed acyclic graph while inserting m edges can be solved in O(min{m 3/2 log n, m 3/2 + n 2 log n}) time, an improvement over the best known result of O(mn). In addition, we analyze the complexity of the same algorithm with respect to the treewidth k of the underlying undirected graph. We show that the algorithm runs in time O(mk log 2 n) for general k and that it can be implemented to run in O(n log n) time on trees, which is optimal. If the input contains cycles, the algorithm detects this