Dans cette thèse, nous étudions les décompositions arborescentes qui satisfont certaines contraintes supplémentaires et nous proposons des algorithmes pour les calculer dans certaines classes de graphes. Finalement, nous résolvons des problèmes liés au routage en utilisant ces décompositions ainsi que des propriétés structurelles des graphes. Cette thèse est divisée en deux parties. Dans la première partie, nous étudions les décompositions arborescentes satisfaisant des propriétés spécifiques. Dans le Chapitre 2, nous étudions les décompositions de taille minimum, c’est-À-Dire avec un nombre minimum de sacs. Etant donné une entier k 4 fixé, nous prouvons que le problème de calculer une décomposition arborescente de largeur au plus k et de taille minimum est NP-Complet dans les graphes de largeur arborescente au plus 4. Nous décrivons ensuite des algorithmes qui calculent des décompositions de taille minimum dans certaines classes de graphes de largeur arborescente au plus 3. Ces résultats ont été présentés au workshop international ICGT 2014. Dans le Chapitre 3, nous étudions la cordalité des graphes et nous introduisons la notion de k-Good décomposition arborescente. Nous étudions tout d’abord les jeux de Gendarmes et Voleur dans les graphes sans long cycle induit. Notre résultat principal est un algorithme polynomial qui, étant donné un graphe G, soit trouve un cycle induit de longueur au moins k+1, ou calcule une k-Good décomposition de G. Ces résultats ont été publiés à la conférence internationale ICALP’12 et dans la revue internationale Algorithmica. Dans la seconde partie de la thèse, nous nous concentrons sur des problèmes de routage. / A tree decomposition of a graph is a way to represent it as a tree by preserving some connectivity properties of the initial graph. Tree decompositions have been widely studied for their algorithmic applications, in particular using dynamic programming approach. In this thesis, we study tree decompositions satisfying various constraints and design algorithms to compute them in some graph classes. We then use tree decompositions or specific graph properties to solve several problems related to routing. The thesis is divided into two parts. In the first part, we study tree decompositions satisfying some properties. In Chapter 2, we investigate minimum size tree decompositions, i.e., with minimum number of bags. Given a fixed k 4, we prove it is NP-Hard to compute a minimum size decomposition with width at most k in the class of graphs with treewidth at least 4. We design polynomial time algorithms to compute minimum size tree decompositions in some classes of graphs with treewidth at most 3 (including trees). Part of these results will be presented in ICGT 2014. In Chapter 3, we study the chordality (longest induced cycle) of graphs and introduce the notion of good tree decomposition (where each bag must satisfy some particular structure). Precisely, we study the Cops and Robber games in graphs with no long induced cycles. Our main result is the design of a polynomial-Time algorithm that either returns an induced cycle of length at least k+1 of a graph G or compute a k-Good tree decomposition of G. These results have been published in ICALP 2012 and Algorithmica. In the second part of the thesis, we focus on routing problems.

The Steiner Tree problem has appeared in the Karp's list of the first 21 NP-hard problems and is well known as one of the most fundamental problems in Network Design area. We study the Node-Weighted version of the Prize Collecting Steiner Tree problem. In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our goal is to find a subtree T of the graph minimizing the cost of the nodes in T plus penalty of the nodes not in T. By a reduction from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1-o(1))ln n unless NP has quasi-polynomial time algorithms, where n is the number of vertices of the graph. Moss and Rabani claimed an O(log n)-approximation algorithm for the problem using a Primal-Dual approach in their STOC'01 paper \cite{moss2001}. We show that their algorithm is incorrect by providing a counter example in which there is an O(n) gap between the dual solution constructed by their algorithm and the optimal solution. Further, evidence is given that their algorithm probably does not have a simple fix. We propose a new algorithm which is more involved and introduces novel ideas in primal dual approach for network design problems. Also, our algorithm is a Lagrangian Multiplier Preserving algorithm and we show how this property can be utilized to design an O(log n)-approximation algorithm for the Node-Weighted Quota Steiner Tree problem using the Lagrangian Relaxation method. We also show an application of the Node Weighted Quota Steiner Tree problem in designing algorithm with better approximation factor for Technology Diffusion problem, a problem proposed by Goldberg and Liu in \cite{goldberg2012} (SODA 2013). In Technology Diffusion, we are given a graph G and a threshold θ(v) associated with each vertex v and we are seeking a set of initial nodes called the seed set. Technology Diffusion is a dynamic process defined over time in which each vertex is either active or inactive. The vertices in the seed set are initially activated and each other vertex v gets activated whenever there are at least θ(v) active nodes connected to v through other active nodes. The Technology Diffusion problem asks to find the minimum seed set activating all nodes. Goldberg and Liu gave an O(rllog n)-approximation algorithm for the problem where r and l are the diameter of G and the number of distinct threshold values, respectively. We improve the approximation factor to O(min{r,l}log n) by establishing a close connection between the problem and the Node Weighted Quota Steiner Tree problem.

The Steiner Tree problem has appeared in the Karp's list of the first 21 NP-hard problems and is well known as one of the most fundamental problems in Network Design area. We study the Node-Weighted version of the Prize Collecting Steiner Tree problem. In this problem, we are given a simple graph with a cost and penalty value associated with each node. Our goal is to find a subtree T of the graph minimizing the cost of the nodes in T plus penalty of the nodes not in T. By a reduction from set cover problem it can be easily shown that the problem cannot be approximated in polynomial time within factor of (1-o(1))ln n unless NP has quasi-polynomial time algorithms, where n is the number of vertices of the graph. Moss and Rabani claimed an O(log n)-approximation algorithm for the problem using a Primal-Dual approach in their STOC'01 paper \cite{moss2001}. We show that their algorithm is incorrect by providing a counter example in which there is an O(n) gap between the dual solution constructed by their algorithm and the optimal solution. Further, evidence is given that their algorithm probably does not have a simple fix. We propose a new algorithm which is more involved and introduces novel ideas in primal dual approach for network design problems. Also, our algorithm is a Lagrangian Multiplier Preserving algorithm and we show how this property can be utilized to design an O(log n)-approximation algorithm for the Node-Weighted Quota Steiner Tree problem using the Lagrangian Relaxation method. We also show an application of the Node Weighted Quota Steiner Tree problem in designing algorithm with better approximation factor for Technology Diffusion problem, a problem proposed by Goldberg and Liu in \cite{goldberg2012} (SODA 2013). In Technology Diffusion, we are given a graph G and a threshold θ(v) associated with each vertex v and we are seeking a set of initial nodes called the seed set. Technology Diffusion is a dynamic process defined over time in which each vertex is either active or inactive. The vertices in the seed set are initially activated and each other vertex v gets activated whenever there are at least θ(v) active nodes connected to v through other active nodes. The Technology Diffusion problem asks to find the minimum seed set activating all nodes. Goldberg and Liu gave an O(rllog n)-approximation algorithm for the problem where r and l are the diameter of G and the number of distinct threshold values, respectively. We improve the approximation factor to O(min{r,l}log n) by establishing a close connection between the problem and the Node Weighted Quota Steiner Tree problem.

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[pt] Esta tese generaliza a variante multi-período do clássico problema da Árvore de Steiner com coleta de prêmios (PCST), que visa encontrar um subgrafo conexo que maximize os prêmios recuperados de nós conectados menos o custo de utilização das arestas conectadas. Este trabalho adicionalmente: (a) permite que vértices sejam conectados à árvore em diferentes períodos de tempo; (b) impõe um orçamento pré-definido em arestas selecionadas em um horizonte específico de períodos de tempo; e (c) limita o comprimento total de arestas que podem ser adicionadas em um período de tempo. Um algoritmo branch-and-cut é fornecido para este problema, avaliando satisfatoriamente instâncias benchmark da literatura, adaptadas para uma configuração multi-período, de até aproximadamente 2000 vértices e 200 terminais em tempo razoável. / [en] This thesis generalizes the multi-period variant of the classical Prizecollecting Steiner Tree Problem, which aims at finding a connected subgraph that maximizes the revenues collected from connected nodes minus the costs to utilize the connecting edges. This work additionally: (a) allows vertices to be added to the tree at different time periods; (b) imposes a predefined budget on edges selected over a specific horizon of time periods; and (c) limits the total length of edges that can be added over a time period. A branch-and-cut algorithm is provided for this problem, satisfactorily evaluating benchmark instances from the literature, adapted to a multi-period setting, up to approximately 2000 vertices and 200 terminals in reasonable time.