Vers des réseaux optiques efficaces et tolérants aux pannes : complexité et algorithmes

We study in this thesis optimization problems with application in optical networks. The problems we consider are related to fault-tolerance and efficient resource allocation and the results we obtain are mainly related to the computational complexity of these problems. The first part of this thesis is devoted to finding paths and disjoint paths. Finding a path is crucial in all types of networks in order to set up connections and finding disjoint paths is a common approach used to provide some degree of protection against failures in networks. We study these problems under different settings. We first focus on finding paths and node or link-disjoint paths in networks with asymmetric nodes, which are nodes with restrictions on their internal connectivity. Afterwards, we consider networks with star Shared Risk Link Groups (SRLGs) which are groups of links that might fail simultaneously due to a localized event. In these networks, we investigate the problem of finding SRLG-disjoint paths. The second part of this thesis focuses on the problem of Routing and Spectrum Assignment (RSA) in Elastic Optical Networks (EONs). EONs are proposed as the new generation of optical networks and they aim at an efficient and flexible use of the optical resources. RSA is the key problem in EONs and it deals with allocating resources to requests under multiple constraints. We first study the static version of RSA in tree networks. Afterwards, we examine a dynamic version of RSA in which a non-disruptive spectrum defragmentation technique is used. Finally, we present in the appendix another problem that has been studied during this thesis.Nous étudions dans cette thèse des problèmes d’optimisation avec applications dans les réseaux optiques. Les problèmes étudiés sont liés à la tolérance aux pannes et à l’utilisation efficace des ressources. Les résultats obtenus portent principalement sur la complexité de calcul de ces problèmes. La première partie de cette thèse est consacrée aux problèmes de trouver des chemins et des chemins disjoints. La recherche d’un chemin est essentielle dans tout type de réseaux afin d’y établir des connexions et la recherche de chemins disjoints est souvent utilisée pour garantir un certain niveau de protection contre les pannes dans les réseaux. Nous étudions ces problèmes dans des contextes différents. Nous traitons d’abord les problèmes de trouver un chemin et des chemins lien ou nœud- disjoints dans des réseaux avec nœuds asymétriques, c’est-à-dire des nœuds avec restrictions sur leur connectivité interne. Ensuite, nous considérons les réseaux avec des groupes de liens partageant un risque (SRLG) en étoile : ensembles de liens qui peuvent tomber en panne en même temps suite à un événement local. Dans ce type de réseaux, nous examinons le problème de recherche des chemins SRLG-disjoints. La deuxième partie de cette thèse est consacrée au problème de routage et d’allocation de spectre (RSA) dans les réseaux optiques élastiques (EONs). Les EONs sont proposés comme la nouvelle génération des réseaux optiques et ils visent une utilisation plus efficace et flexible des ressources optiques. Le problème RSA est central dans les EONs. Il concerne l’allocation de ressources aux requêtes sous plusieurs contraintes

On Spectrum Assignment in Elastic Optical Tree-Networks

International audiencePour répondre à la demande croissante du trafic d'Internet, une nouvelle génération de réseaux optiques est en cours de développement ; les réseaux optiques élastiques (EONs). La technologie EON permet d'utiliser le spectre optique de manière efficace et flexible. Cette flexibilité promet de résoudre les difficultés liées à la croissance et l'hétérogénéité du trafic. Toutefois, elle rend le problème d'allocation de ressources plus complexe. Dans ce papier, nous traitons le problème d'allocation de spectre dans les réseaux optiques élastiques en arbre. Dans ce type de réseau , bien que le routage soit fixé, l'allocation de spectre est NP-difficile. Nous présentons des résultats de difficulté et d'approximation pour des cas spéciaux où le réseau est une étoile ou un arbre binaire

On Spectrum Assignment in Elastic Optical Tree-Networks

International audiencePour répondre à la demande croissante du trafic d'Internet, une nouvelle génération de réseaux optiques est en cours de développement ; les réseaux optiques élastiques (EONs). La technologie EON permet d'utiliser le spectre optique de manière efficace et flexible. Cette flexibilité promet de résoudre les difficultés liées à la croissance et l'hétérogénéité du trafic. Toutefois, elle rend le problème d'allocation de ressources plus complexe. Dans ce papier, nous traitons le problème d'allocation de spectre dans les réseaux optiques élastiques en arbre. Dans ce type de réseau , bien que le routage soit fixé, l'allocation de spectre est NP-difficile. Nous présentons des résultats de difficulté et d'approximation pour des cas spéciaux où le réseau est une étoile ou un arbre binaire

Dynamic Routing and Spectrum Assignment with Non-Disruptive Defragmentation

International audienceL'augmentation du trafic Internet motive une évolution des réseaux WDM traditionnels vers les réseaux optiques élastiques (Elastic Optical Networks, EON). Les EONs sont conçus pour optimiser l'utilisation des ressources optiques; ils permettent l'attribution de bandes de largeurs quelconques et offrent ainsi plus de souplesse que les réseaux WDM avec leurs bandes fixes. Cependant, la gestion du spectre dans les EONs devient plus difficile du fait de la fragmentation. Dans ce papier, nous proposons deux algorithmes qui permettent de résoudre le problème de Routage et d'Allocation de Spectre d'une nouvelle requête en utilisant une technique de défragmentation non-perturbatrice

On Spectrum Assignment in Elastic Optical Tree-Networks

To face the explosion of the Internet traffic, a new generation of optical networks is being developed; the Elastic optical Networks (EONs). The aim with EONs is to use the optical spectrum efficiently and flexibly. The benefit of the flexibility is, however, accompanied by more difficulty in the resource allocation problems. In this report, we study the problem of Spectrum Allocation in Elastic Optical Tree-Networks. In trees, even though the routing is fixed, the spectrum allocation is NP-hard. We survey the complexity and approximability results that have been established for the SA in trees and prove new results for stars and binary trees

On spectrum assignment in elastic optical tree-networks

International audienceTo face the explosion of the Internet trac, a new generation of optical networks is being developed; the Elastic Optical Networks (EONs). EONs use the optical spectrum eciently and flexibly, but that gives rise to more diculty in the resource allocation problems. In this article, we study the problem of Spectrum Assignment (SA) in Elastic Optical Tree-Networks. Given a set of trac requests with their routing paths (unique in the case of trees) and their spectrum demand, a spectrum assignment consists in allocating to each request an interval of consecutive slots (spectrum units) such that a slot on a given link can be used by at most one request. The objective of the SA problem is to find an assignment minimizing the total number of spectrum slots to be used. We prove that SA is NP-hard in undirected stars of 3 links and in directed stars of 4 links, and show that it can be approximated within a factor of 4 in general stars. Afterwards, we use the equivalence of SA with a graph coloring problem (interval coloring) to find constant-factor approximation algorithms for SA on binary trees with special demand profiles

Minimum Size Tree-Decompositions

International audienceTree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree- decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic program- ming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree- decompositions. It is then interesting to try to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k >= 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k >= 4 and polynomial for k <= 2. On going work also suggests it is polynomial for k = 3

Minimum Size Tree-Decompositions

International audienceWe study in this paper the problem of computing a tree-decomposition of a graph with width at most k and minimum number of bags. More precisely, we focus on the following problem: given a fixed $k ≥ 1$, what is the complexity of computing a tree-decomposition of width at most k with minimum number of bags in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed $k ≥$4 and polynomial for $k ≤ 2$; for $k = 3$, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs

Size-Constrained Tree Decompositions

Tree-Decompositions are the corner-stone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms based on tree-decompositions is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this report, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. More precisely, we focus on the following problem: given a fixed k ≥ 1, what is the complexity of computing a tree-decomposition of width at most k with minimum size in the class of graphs with treewidth at most k? We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs

Minimum Size Tree-decompositions

International audienceTree-decompositions are the cornerstone of many dynamic programming algorithms for solving graph problems. Since the complexity of such algorithms generally depends exponentially on the width (size of the bags) of the decomposition, much work has been devoted to compute tree-decompositions with small width. However, practical algorithms computing tree-decompositions only exist for graphs with treewidth less than 4. In such graphs, the time-complexity of dynamic programming algorithms is dominated by the size (number of bags) of the tree-decompositions. It is then interesting to minimize the size of the tree-decompositions. In this extended abstract, we consider the problem of computing a tree-decomposition of a graph with width at most k and minimum size. We prove that the problem is NP-complete for any fixed k ≥ 4 and polynomial for k ≤ 2; for k = 3, we show that it is polynomial in the class of trees and 2-connected outerplanar graphs