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The Multi-Terminal Vertex Separator Problem: Complexity, Polyhedra and Algorithms

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The Multi-Terminal Vertex Separator Problem: Complexity, Polyhedra and Algorithms The multi-terminal vertex separator problem : Complexity, Polyhedra and Algorithms Youcef Magnouche To cite this version: Youcef Magnouche. The multi-terminal vertex separator problem : Complexity, Polyhedra and Al- gorithms. Operations Research [cs.RO]. Université Paris sciences et lettres, 2017. English. NNT : 2017PSLED020. tel-01611046 HAL Id: tel-01611046 https://tel.archives-ouvertes.fr/tel-01611046 Submitted on 5 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THÈSE DE DOCTORAT de l’Université de recherche Paris Sciences et Lettres PSL Research University Préparée à l’Université Paris-Dauphine The multi-terminal vertex separator problem : Complexity, Polyhedra and Algorithms École Doctorale de Dauphine — ED 543 COMPOSITION DU JURY : Spécialité Informatique M. A. Ridha MAHJOUB Université Paris-Dauphine Directeur de thèse M. Denis CORNAZ Université Paris-Dauphine Membre du jury M. Mohamed DIDI BIHA Université de Caen Basse-Normandie Rapporteur M. Nelson MACULAN Université fédérale de Rio de Janeiro Rapporteur Soutenue le 26.06.2017 Mme Ivana LJUBIC par Youcef MAGNOUCHE ESSEC Business School de Paris Membre du jury M. Sébastien MARTIN Dirigée par A. Ridha MAHJOUB Université de Lorraine Membre du jury M. Frédéric SEMET École Centrale de Lille Président du jury Remerciements Je voudrais en premier lieu remercier Messieurs A. Ridha Mahjoub et Denis Cornaz de m’avoir permis d’effectuer cette thèse. Je remercie tout particulièrement Monsieur A. Ridha Mahjoub, Professeur à l’Université Paris-Dauphine, de m’avoir apporté son soutien, scientifique et moral, tout au long de ce travail. Il a su me transmettre les connaissances, la rigueur, la motivation et la passion pour la recherche en général et l’optimisation combinatoire en particulier. Il m’a également permis de surmonter les moments difficiles grâce à son très grand intérêt pour mon travail et sa constante disponibilité. Je tiens aussi à remercier très sincèrement Monsieur Denis Cornaz, maître de con- férences à l’Université Paris-Dauphine, pour ses idées et sa détermination qui m’ont permis d’élargir mon domaine de connaissances et le travail réalisé durant cette thèse. Il m’a également appris, grâce à sa haute exigence scientifique et à sa très grande estime de mes capacités, à ne jamais renoncer et à donner le meilleur de moi-même. Je remercie également vivement Monsieur Sébastien Martin, maître de conférences à l’université de Lorraine, pour toute l’aide qu’il m’a apportée durant cette thèse. Cette aide se situe bien sûr au niveau scientifique, grâce à nos collaborations et aux nombreuses discussions que nous avons eues sur mon travail et à ses conseils et idées. J’ai été très honoré que Monsieur Mohamed Didi Biha, Professeur des universités à l’Université de Caen Basse-Normandie, ait accepté de rapporter ma thèse. Je remercie également Monsieur Nelson Maculan, Professeur Émérite à l’Université fédérale de Rio de Janeiro, pour l’intérêt qu’il a bien voulu porter à ce travail et pour m’avoir fait l’honneur d’accepter la charge de rapporteur. Mes remerciements vont également à Madame Ivana Ljubic, Professeur à l’école ii Remerciements supérieure des sciences économiques et commerciales, pour avoir accepté d’examiner mes travaux. Ma gratitude va ensuite à Monsieur Frédéric Semet, Professeur à l’école Centrale de Lille, d’avoir accepté d’examiner ma thèse. Je tiens également à remercier très chaleureusement mes différents collègues et amis du LAMSADE qui ont partagé avec moi ces dernières années. Les discussions ani- mées et les moments de détente partagés, leur bonne humeur, leurs encouragements et leur amitié, mais aussi les échanges scientifiques et les travaux effectués ensemble, ont grandement contribué à mon épanouissement scientifique et personnel lors de ces années de thèse. Je tiens à remercier tout particulièrement Khalil, Myriam, Pedro, Thomas, Ian- Christopher, Ioannis et Linda avec qui j’ai partagé mon bureau C605 de Paris-Dauphine. Enfin, mes remerciements ne seraient pas complets si je ne remerciais pas ma famille et mes amis, qui m’ont supporté et appuyé toutes ces années et se sont réellement intéressés à mon travail et au domaine hautement incompréhensible de l’optimisation combinatoire en général. Abstract This thesis deals with the multi-terminal vertex separator problem. Given a graph G = (V [T;E) with V [T the set of vertices, where T is a set of terminals, and a weight function w : V ! Z, associated with nonterminal nodes, the multi-terminal vertex separator problem consists in partitioning V [ T into k + 1 subsets fS; V1;:::;Vkg such that there is no edge between two different subsets Vi and Vj, each Vi contains exactly one terminal and the weight of S is minimum. In this thesis, we consider the problem from a polyhedral point of view. We give two integer programming formulations for the problem, for one of them, we investigate the related polyhedron and discuss its polyhedral structure. We describe some valid inequalities and characterize when these inequalities define facets. We also derive separation algorithms for these inequalities. Using these results, we develop a Branch-and-Cut algorithm for the problem, along with an extensive computational study is presented. We also study the multi-terminal vertex separator polytope in the graphs decompos- able by one node cutsets. If G is a graph that decomposes into G1 and G2, we show that the multi-terminal vertex separator polytope in G can be described from two lin- ear systems related to G1 and G2. This gives rise to a technique for characterizing the multi-terminal vertex separator polytope in the graphs that are recursively decompos- able. We also obtain a procedure to describe facets for this polytope and show that the multi-terminal vertex separator problem can also be decomposed. Applications of this technique are also discussed. Moreover, we propose three extended formulations for the problem and derive Branch- and-Price and Branch-and-Cut-and-Price algorithms. For each formulation we present a column generation scheme to solve the linear relaxation, the way to compute the dual bound and the branching scheme. We present computational results and discuss the performance of each algorithm. Further, we discuss four variants of the multi-terminal vertex separator problem. We iv Abstract show that all these variants are NP-hard, we give an integer programming formulation and present some valid inequalities. Key words : Combinatorial optimization, Polytope, Facet, Branch-and-Cut algo- rithm, Branch-and-Price algorithm, Composition of polyhedra. Résumé Dans ce manuscrit nous considérons le problème du séparateur de poids minimum. Étant donné un graphe G = (V [ T;E), tel que V [ T représente l’ensemble des sommets où T est un ensemble de terminaux, et une fonction de poids w : V ! Z, associée aux sommets non terminaux, le problème du séparateur de poids minimum consiste à partitionner V [ T en k + 1 sous-ensembles fS; V1;:::;Vkg tel qu’il n’y a aucune arête entre deux sous-ensembles Vi et Vj, chaque Vi contient exactement un terminal et le poids de S est minimum. Dans cette thèse, nous étudions le problème d’un point de vue polyédral. Nous donnons deux formulations en nombres entiers pour le problème, et pour une de ces formulations, nous étudions le polyèdre associé. Nous présentons plusieurs inégalités valides, et décrivons leurs conditions de facette. Nous présentons également des algorithmes de séparation pour ces inégalités. En utilisant ces résultats, nous développons un algorithme de génération de coupes et branchement pour le problème, et présentons une analyse approfondie des résultats numériques. Nous étudions également le polytope des séparateurs dans les graphes décomposables par sommets d’articulation. Si G est un graphe qui se décompose en G1 et G2, alors nous montrons que le polytope des séparateurs dans G peut être décrit à partir des deux systèmes linéaires liés à G1 et G2. Ceci donne lieu à une technique permettant de caractériser le polytope des séparateurs dans les classes de graphes qui sont récur- sivement décomposables. Nous obtenons également une procédure de composition de facettes dans ce type de graphes et nous montrons que le problème du séparateur peut être aussi décomposé. Des applications de cette technique sont discutées. Ensuite, nous étudions des formulations étendues pour le problème et proposons des algorithmes de génération de colonnes et branchements ainsi que des algorithmes de génération de colonnes, de coupes et branchement. Pour chaque formulation, nous présentons un algorithme de génération de colonnes pour résoudre la relaxation linéaire, une procédure pour le calcul de la borne duale ainsi qu’une règle de branchement. Nous vi Résumé comparons ensuite la performance des algorithmes sur un ensemble d’instances à travers une étude expérimentale extensive. De plus, nous présentons quatre variantes du problème du séparateur. Nous montrons que les quatre variantes sont NP-difficiles, nous donnons une formulation en nombres entiers et présentons certaines inégalités valides. Mots clés : Optimisation combinatoire, Polytope, Facette, Algorithme de coupes et branchements, Algorithme de génération de colonnes et branchements, Composition de polyèdres. Résumé long Introduction L’optimisation combinatoire, et en particulier l’approche polyédral, sont très effi- caces pour traiter les problèmes combinatoires difficiles. Cette technique, initiée par Edmonds pour le problème de couplage, consiste à réduire le problème à un programme linéaire (ou une séquence de programmes linéaires) en donnant une description com- plète ou partielle du polytope associé par un système d’inégalités linéaires.
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