In this dissertation, we examine specific network connectivity problems, and achieve improved approximation algorithm and integrality gap results for them. We introduce an important new, highly useful and applicable, network connectivity problem - the Vital Core Connectivity Problem (VCC). Despite its many practical uses, this problem has not been previously studied. We present the first constant factor approximation algorithm for VCC, and provide an upper bound on the integrality gap of its linear programming relaxation. We also introduce a new, useful, extension of the minimum spanning tree problem, called the Extended Minimum Spanning Tree Problem (EMST), that is based on a special case of VCC; and provide both a polynomial-time algorithm and a complete linear description for it. Furthermore, we show how to generalize this new problem to handle numerous disjoint vital cores, providing the first complete linear description of, and polynomial-time algorithm for, the generalized problem. We examine the Survivable Network Design Problem (SNDP) with multiple copies of edges allowed in the solution (multi-SNDP), and present a new approximation algorithm for which the approximation guarantee is better than that of the current best known for certain cases of multi-SNDP. With our method, we also obtain improved bounds on the integrality gap of the linear programming relaxation of the problem. Furthermore, we show the application of these results to variations of SNDP. We investigate cases where the optimal values of multi-SNDP and SNDP are equal; and we present an improvement on the previously best known integrality gap bound and approximation guarantee for the special case of SNDP with metric costs and low vertex connectivity requirements, as well as for the similar special case of the Vertex Connected Survivable Network Design Problem (VC-SNDP). The quality of the results that one can obtain for a given network design problem often depends on its integer linear programming formulation, and, in particular, on its linear programming relaxation. In this connection, we investigate formulations for the Steiner Tree Problem (ST). We propose two new formulations for ST, and investigate their strength in terms of their associated integrality gaps.

In this dissertation, we examine specific network connectivity problems, and achieve improved approximation algorithm and integrality gap results for them. We introduce an important new, highly useful and applicable, network connectivity problem - the Vital Core Connectivity Problem (VCC). Despite its many practical uses, this problem has not been previously studied. We present the first constant factor approximation algorithm for VCC, and provide an upper bound on the integrality gap of its linear programming relaxation. We also introduce a new, useful, extension of the minimum spanning tree problem, called the Extended Minimum Spanning Tree Problem (EMST), that is based on a special case of VCC; and provide both a polynomial-time algorithm and a complete linear description for it. Furthermore, we show how to generalize this new problem to handle numerous disjoint vital cores, providing the first complete linear description of, and polynomial-time algorithm for, the generalized problem. We examine the Survivable Network Design Problem (SNDP) with multiple copies of edges allowed in the solution (multi-SNDP), and present a new approximation algorithm for which the approximation guarantee is better than that of the current best known for certain cases of multi-SNDP. With our method, we also obtain improved bounds on the integrality gap of the linear programming relaxation of the problem. Furthermore, we show the application of these results to variations of SNDP. We investigate cases where the optimal values of multi-SNDP and SNDP are equal; and we present an improvement on the previously best known integrality gap bound and approximation guarantee for the special case of SNDP with metric costs and low vertex connectivity requirements, as well as for the similar special case of the Vertex Connected Survivable Network Design Problem (VC-SNDP). The quality of the results that one can obtain for a given network design problem often depends on its integer linear programming formulation, and, in particular, on its linear programming relaxation. In this connection, we investigate formulations for the Steiner Tree Problem (ST). We propose two new formulations for ST, and investigate their strength in terms of their associated integrality gaps.

In this dissertation, we examine specific network connectivity problems, and achieve improved approximation algorithm and integrality gap results for them. We introduce an important new, highly useful and applicable, network connectivity problem - the Vital Core Connectivity Problem (VCC). Despite its many practical uses, this problem has not been previously studied. We present the first constant factor approximation algorithm for VCC, and provide an upper bound on the integrality gap of its linear programming relaxation. We also introduce a new, useful, extension of the minimum spanning tree problem, called the Extended Minimum Spanning Tree Problem (EMST), that is based on a special case of VCC; and provide both a polynomial-time algorithm and a complete linear description for it. Furthermore, we show how to generalize this new problem to handle numerous disjoint vital cores, providing the first complete linear description of, and polynomial-time algorithm for, the generalized problem. We examine the Survivable Network Design Problem (SNDP) with multiple copies of edges allowed in the solution (multi-SNDP), and present a new approximation algorithm for which the approximation guarantee is better than that of the current best known for certain cases of multi-SNDP. With our method, we also obtain improved bounds on the integrality gap of the linear programming relaxation of the problem. Furthermore, we show the application of these results to variations of SNDP. We investigate cases where the optimal values of multi-SNDP and SNDP are equal; and we present an improvement on the previously best known integrality gap bound and approximation guarantee for the special case of SNDP with metric costs and low vertex connectivity requirements, as well as for the similar special case of the Vertex Connected Survivable Network Design Problem (VC-SNDP). The quality of the results that one can obtain for a given network design problem often depends on its integer linear programming formulation, and, in particular, on its linear programming relaxation. In this connection, we investigate formulations for the Steiner Tree Problem (ST). We propose two new formulations for ST, and investigate their strength in terms of their associated integrality gaps.

In this thesis, we consider the k-edge connected L-hop-constrained network design problem. Given a weighted graph G=(N,E), a set D of pairs of terminal nodes, and two integers k,L > 1, it consists in finding in G the minimum cost subgraph containing at least k edge-disjoint paths of at most L edges between each pair in D. This problem is of great interest in today's telecommunication industry, where highly survivable networks need to be constructed. We first study the particular case where the set of demands D is reduced to a single pair {s,t}. We propose an integer programming formulation for the problem, which consists in the st-cut and trivial inequalities, along with the so-called L-st-path-cut inequalities. We show that these three classes of inequalities completely describe the associated polytope when k=2 and L=2 or 3, and give necessary and sufficient conditions for them to be facet-defining. We also consider the dominant of the associated polytope, and discuss how the previous inequalities can be separated in polynomial time. We then extend the complete and minimal description obtained above to any number k of required edge-disjoint L-st-paths, but when L=2 only. We devise a cutting plane algorithm to solve the problem, using the previous polynomial separations, and present some computational results. After that, we consider the case where there is more than one demand in D. We first show that the problem is strongly NP-hard, for all L fixed, even when all the demands in D have one root node in common. For k=2 and L=2,3, we give an integer programming formulation, based on the previous constraints written for all pairs {s,t} in D. We then proceed by giving several new classes of facet-defining inequalities, valid for the problem in general, but more adapted to the rooted case. We propose separation procedures for these inequalities, which are embedded within a Branch-and-Cut algorithm to solve the problem when L=2,3. Extensive computational results from it are given and analyzed for both random and real instances. Since those results appear less satisfactory in the case of arbitrary demands (non necessarily rooted), we present additional families of valid inequalites in that situation. Again, separation procedures are devised for them, and added to our previous Branch-and-Cut algorithm, in order to see the practical improvement granted by them. Finally, we study the problem for greater values of L. In particular, when L=4, we propose new families of constraints for the problem of finding a subgraph that contains at least two L-st-paths either node-disjoint, or edge-disjoint. Using these, we obtain an integer programming formulation in the space of the design variables for each case. ------------------------------------------------ Dans cette thèse, nous considérons le problème de conception de réseau k-arete connexe à chemins L-bornés. Etant donné un graphe pondéré G=(N,E), un ensemble D de paires de noeuds terminaux, et deux entiers k,L > 1, ce problème consiste à trouver, dans G, un sous-graphe de cout minimum tel que, entre chaque paire dans D, il existe au moins k chemins arete-disjoints de longueur au plus L. Ce problème est d'un grand intéret dans l'industrie des télécommunications, où des réseaux hautement fiables doivent etre construits. Nous étudions tout d'abord le cas particulier où l'ensemble des demandes D est réduit à une seule paire de noeuds. Nous proposons une formulation du problème sous forme de programme linéaire en nombres entiers, laquelle consiste en les inégalités triviales et de coupe, ainsi que les inégalités dites de L-chemin-coupe. Nous montrons que ces trois types d'inégalités décrivent complètement le polytope associé lorsque k=2 et L=2,3, et donnons des conditions nécessaires et suffisantes pour que celles-ci en définissent des facettes. Nous considérons également le dominant du polytope associé et discutons de la séparation polynomiale des trois classes précédentes. Nous étendons alors cette description complète et minimale à tout nombre k de chemins arete-disjoints de longueur au plus 2. De plus, nous proposons un algorithme de plans coupants utilisant les précédentes séparations polynomiales, et en présentons quelques résultats calculatoires, pour tout k>1 et L=2,3. Nous considérons ensuite le cas où plusieurs demandes se trouvent dans D. Nous montrons d'abord que le problème est fortement NP-dur, pour tout L fixé et ce, meme si les demandes sont toutes enracinées en un noeud. Pour k=2 et L=2,3, nous donnons une formulation du problème sous forme de programme linéaire en nombres entiers. Nous proposons également de nouvelles classes d'inégalités valides, pour lesquelles nous réalisons une étude faciale. Celles-ci sont alors séparées dans le cadre d'un algorithme de coupes et branchements pour résoudre des instances aléatoires et réelles du problème. Enfin, nous étudions le problème pour de plus grandes valeurs de L. En particulier, lorsque L=4, nous donnons de nouvelles familles de contraintes pour le problème consistant à déterminer un sous-graphe contenant entre deux noeuds fixés au moins deux chemins de longueur au plus 4, que ceux-ci doivent etre arete-disjoints ou noeud-disjoints. Grace à ces dernières, nous parvenons à donner une formulation naturelle du problème dans chacun de ces deux cas.

Dans un graphe non orienté, le problème du sous-graphe k-sommet connexe consiste à déterminer un sous-graphe de poids minimum tel que entre chaque paires de sommets, il existe k chemins sommet-disjoints. Ce modèle a été étudié dans la littérature en termes d'arête connexité. Cependant, le cas de la sommet connexité n'a pas été traité jusqu'à présent. Nous décrivons de nouvelles inégalités valides et nous présentons un algorithme de Coupes et Branchements ainsi qu'une large étude expérimentale qui montrent l'efficacité des contraintes utilisées. Nous proposons ensuite une formulation étendue pour le même problème pour une connexité k=2, suivi d'un algorithme de Génération de Colonnes et Branchements pour résoudre cette formulation.Nous étudions ensuite la version avec chemins bornés du problème. Le problème consiste à trouver un sous-graphe de poids minimum, tel que entre chaque paire d'origine-destination, il existe k chemins sommet-disjoints de longueur au plus L. Nous proposons une formulation linéaire en nombres entiers pour L=2,3. Nous présentons de nouvelles inégalités valides et nous proposons des algorithmes de séparation pour ces contraintes. Nous présentons ensuite un algorithme de Coupes et Branchements qu'on a testé sur des instances de la TSPLIB. / Given a weighted undirected graph and an integer k, the k-node-connected subgraph problem is to find a minimum weight subgraph which contains k-node-disjoint paths between every pair of nodes. We introduce new classes of valid inequalities and discuss their facial aspect. We also devise separation routines, investigate the structural properties of the linear relaxation and discuss some reduction operations that can be used in a preprocessing phase for the separation. Using these results, we devise a Branch-and-Cut algorithm and present some computational results. Then we present a new extended formulation for the the k-node-connected subgraph problem, along with a Branch-and-Cut-and-Price algorithm for solving the problem.Next, we investigate the hop-constrained version of the problem. The k node-disjoint hop-constrained network design problem is to find a minimum weight subgraph such that between every origin and destination there exist at least k node-disjoint paths of length at most L. We propose an integer linear programming formulation for L=2,3 and investigate the associated polytope. We introduce valid inequalities and devise separation algorithms. Then, we propose a B\&C algorithm for solving the problem along with some computational results.

In this thesis, we consider the k-edge connected L-hop-constrained network design problem. Given a weighted graph G=(N,E), a set D of pairs of terminal nodes, and two integers k,L > 1, it consists in finding in G the minimum cost subgraph containing at least k edge-disjoint paths of at most L edges between each pair in D. This problem is of great interest in today's telecommunication industry, where highly survivable networks need to be constructed.<p><p>We first study the particular case where the set of demands D is reduced to a single pair {s,t}. We propose an integer programming formulation for the problem, which consists in the st-cut and trivial inequalities, along with the so-called L-st-path-cut inequalities. We show that these three classes of inequalities completely describe the associated polytope when k=2 and L=2 or 3, and give necessary and sufficient conditions for them to be facet-defining. We also consider the dominant of the associated polytope, and discuss how the previous inequalities can be separated in polynomial time.<p><p>We then extend the complete and minimal description obtained above to any number k of required edge-disjoint L-st-paths, but when L=2 only. We devise a cutting plane algorithm to solve the problem, using the previous polynomial separations, and present some computational results.<p><p>After that, we consider the case where there is more than one demand in D. We first show that the problem is strongly NP-hard, for all L fixed, even when all the demands in D have one root node in common. For k=2 and L=2,3, we give an integer programming formulation, based on the previous constraints written for all pairs {s,t} in D. We then proceed by giving several new classes of facet-defining inequalities, valid for the problem in general, but more adapted to the rooted case. We propose separation procedures for these inequalities, which are embedded within a Branch-and-Cut algorithm to solve the problem when L=2,3. Extensive computational results from it are given and analyzed for both random and real instances.<p><p>Since those results appear less satisfactory in the case of arbitrary demands (non necessarily rooted), we present additional families of valid inequalites in that situation. Again, separation procedures are devised for them, and added to our previous Branch-and-Cut algorithm, in order to see the practical improvement granted by them.<p><p>Finally, we study the problem for greater values of L. In particular, when L=4, we propose new families of constraints for the problem of finding a subgraph that contains at least two L-st-paths either node-disjoint, or edge-disjoint. Using these, we obtain an integer programming formulation in the space of the design variables for each case.<p><p>------------------------------------------------<p><p>Dans cette thèse, nous considérons le problème de conception de réseau k-arete connexe à chemins L-bornés. Etant donné un graphe pondéré G=(N,E), un ensemble D de paires de noeuds terminaux, et deux entiers k,L > 1, ce problème consiste à trouver, dans G, un sous-graphe de cout minimum tel que, entre chaque paire dans D, il existe au moins k chemins arete-disjoints de longueur au plus L. Ce problème est d'un grand intéret dans l'industrie des télécommunications, où des réseaux hautement fiables doivent etre construits.<p><p>Nous étudions tout d'abord le cas particulier où l'ensemble des demandes D est réduit à une seule paire de noeuds. Nous proposons une formulation du problème sous forme de programme linéaire en nombres entiers, laquelle consiste en les inégalités triviales et de coupe, ainsi que les inégalités dites de L-chemin-coupe. Nous montrons que ces trois types d'inégalités décrivent complètement le polytope associé lorsque k=2 et L=2,3, et donnons des conditions nécessaires et suffisantes pour que celles-ci en définissent des facettes. Nous considérons également le dominant du polytope associé et discutons de la séparation polynomiale des trois classes précédentes.<p><p>Nous étendons alors cette description complète et minimale à tout nombre k de chemins arete-disjoints de longueur au plus 2. De plus, nous proposons un algorithme de plans coupants utilisant les précédentes séparations polynomiales, et en présentons quelques résultats calculatoires, pour tout k>1 et L=2,3.<p><p>Nous considérons ensuite le cas où plusieurs demandes se trouvent dans D. Nous montrons d'abord que le problème est fortement NP-dur, pour tout L fixé et ce, meme si les demandes sont toutes enracinées en un noeud. Pour k=2 et L=2,3, nous donnons une formulation du problème sous forme de programme linéaire en nombres entiers. Nous proposons également de nouvelles classes d'inégalités valides, pour lesquelles nous réalisons une étude faciale. Celles-ci sont alors séparées dans le cadre d'un algorithme de coupes et branchements pour résoudre des instances aléatoires et réelles du problème.<p><p>Enfin, nous étudions le problème pour de plus grandes valeurs de L. En particulier, lorsque L=4, nous donnons de nouvelles familles de contraintes pour le problème consistant à déterminer un sous-graphe contenant entre deux noeuds fixés au moins deux chemins de longueur au plus 4, que ceux-ci doivent etre arete-disjoints ou noeud-disjoints. Grace à ces dernières, nous parvenons à donner une formulation naturelle du problème dans chacun de ces deux cas. <p> / Doctorat en sciences, Spécialisation Informatique / info:eu-repo/semantics/nonPublished