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Maximum Bounded Rooted-Tree Problem: Algorithms and Polyhedra Maximum Bounded Rooted-Tree Problem : Algorithms and Polyhedra Jinhua Zhao To cite this version: Jinhua Zhao. Maximum Bounded Rooted-Tree Problem : Algorithms and Polyhedra. Combinatorics [math.CO]. Université Clermont Auvergne, 2017. English. NNT : 2017CLFAC044. tel-01730182 HAL Id: tel-01730182 https://tel.archives-ouvertes.fr/tel-01730182 Submitted on 13 Mar 2018 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. Numéro d’Ordre : D.U. 2816 EDSPIC : 799 Université Clermont Auvergne École Doctorale Sciences Pour l’Ingénieur de Clermont-Ferrand THÈSE Présentée par Jinhua ZHAO pour obtenir le grade de Docteur d’Université Spécialité : Informatique Le Problème de l’Arbre Enraciné Borné Maximum : Algorithmes et Polyèdres Soutenue publiquement le 19 juin 2017 devant le jury M. ou Mme Mohamed DIDI-BIHA Rapporteur et examinateur Sourour ELLOUMI Rapporteuse et examinatrice Ali Ridha MAHJOUB Rapporteur et examinateur Fatiha BENDALI Examinatrice Hervé KERIVIN Directeur de Thèse Philippe MAHEY Directeur de Thèse iii Acknowledgments First and foremost I would like to thank my PhD advisors, Professors Philippe Mahey and Hervé Kerivin. Without Professor Philippe Mahey, I would never have had the chance to come to ISIMA or LIMOS here in France in the first place, and I am really grateful and honored to be accepted as a PhD student of his. My PhD could not be even started and needless to say finished without his support. I would also like to express my greatest appreciation and thanks to Professor Hervé Kerivin, who have taught me countless things over the course of my master and PhD, including but not limit to knowledge of all aspects of combinatorial optimization, scientific writing, methodology of research, means of tackle a complex problem, etc. We have had so many inspirational meetings during the last 5 years, and as a result of which, many ideas were inspected, implemented and eventually turned into new discoveries as a part of this dissertation. His patience, passion and step- by-step guidance was instrumental in helping me walk into the field of combinatorial optimization and go further in depth in order to crank out this dissertation. I would like to thank my committee members Professors Fatiha Bendali, Mohamed Didi- Biha, Sourour Elloumi, Ali Ridha Mahjoub for their time, interest, insightful questions and comments. I would also like to express my appreciation for the French Ministry of Higher Education and Research as the funding source of my PhD, and LIMOS as the lab that hosted me. My time at France was made enjoyable largely due to the many friends and colleagues I have met. Among them Maxime Chassaing and Benjamin Vincent are the first friends I have made after my arrival. They soon became my peer PhD students, and we shared a lot of common experiences in our academic and personal life. I would like to thank them for having my back all the time. I also thank Libo Ren as a forerunner in the PhD and researcher life in LIMOS, providing me valuable advises about research and how to live here in Clermont-Ferrand with his firsthand experience. I am have also appreciated the lovely atmosphere in the office I have worked for the past 6 years, which was created by all the colleagues I have worked with, Samuel Deleplanque, Jocelyn De Goer, Vanel Siyou, Danh Cong Nguyen, Suan Tay, and most importantly Professor Philippe Vaslin. I would like also express my thanks to all other colleagues I iv have met and worked with, to name a few, Professor Christophe Duhamel, Raksmey Phan, Bin Tian, Xiaofeng Zhao, Jean Connier. My special thanks also go to Professors Hong Sun, Jianjin Li and Kun Mean Hou, who were in charge of the ISIMA-WHU exchange program I was in. I will forever be grateful to my former internship advisor and 6-year co-worker Professor Philippe Vaslin. During my time in France, he helped me in every possible way, even more than I could ever hope for. As soon as I arrived in France, I was signed to an internship supervised by him. Since then, we have worked together in the same office for more than 6 years. During this time, he has helped me in all aspects of both scientific work and daily life, ranging from research suggestions, writing suggestions, dealing with all sorts of things in everyday life to teaching me French history, European geography and swimming. I could not have done anything without the support, love and encouragement of my friends and family members. I thank all my friends for providing support and friendship and having faith in me all the time. I thank my parents and parents in law for their unconditional love and supporting me in my pursuits abroad even if it means they have to suffer from the separation between them and their children and grandson. Last but definitely not the least, I would like to thank my loving and supportive wife Jie Guan and my beloved son Yicen Zhao for always being there for me. Thank you. Jinhua Zhao September 2017 So, we’ll go no more a-roving So late into the night – George Gordon Byron vii Abstract Given a simple undirected graph G =(V,E) with a so-called root node r ∈ V ,arooted tree, or an r-tree, of G is either the empty graph (∅, ∅), or a tree containing r. If a node- V capacity vector c ∈ Z+ is given, then a subgraph of G is said to be bounded if the degree of each node in the subgraph does not exceed its capacity. Let w be an edge-weight RE p RV r r vector in and a node-price vector in . The Maximum Bounded-Tree (MB T) problem consists of finding a bounded r-tree T =(U, F ) of G such that we + pv is e∈F v∈U maximized. If the capacity constraint from the MBrT problem is relaxed, we then obtain the Maximum r-Tree (MrT) problem. This dissertation contributes to the study of the MBrT problem and the MrTproblem. First we introduce the problems with their definitions and complexities. We define the associated polytopes along with a formulation for each of them. We present several polynomial-time combinatorial algorithms for both the MBrT problem (and thus the MrT problem) on trees, cycles and cactus graphs. Particularly, a dynamic-programming- based algorithm is used to solve the MBrT problem on trees, whereas on cycles we reduce it to some polynomially solvable problems in three different cases. For cactus graphs, we first show that the MBrTproblemcanbesolvedinpolynomialtimeonaso-calledcactus basis, then break down the problem on any cactus graph into a series of subproblems on trees and on cactus basis. The second part of this work investigates the polyhedral structure of three polytopes as- sociated with the MBrT problem and the MrT problem, namely Bxy(G, r, c), Bx(G, r, c) and Rx(G, r). Bxy(G, r, c) and Bx(G, r, c) are polytopes associated with the MBrTprob- lem, where Bxy(G, r, c) considers both edge- and node-indexed variables and Bx(G, r, c) considers only edge-indexed variables. Rx(G, r) is the polytope associated with the MrT problem that only considers edge-indexed variables. For each of the three polytopes, we study their dimensions, facets as well as possible ways of decomposition. We introduce some newly discovered constraints for each polytope, and show that these new constraints allow us to characterize them on several graph classes. Specifically, we provide character- ization for Bxy(G, r, c) on cactus graphs with the help of a decomposition through 1-sum. On the other hand, a TDI-system that characterizes Bx(G, r, c) isgivenineachcaseof viii trees and cycles. The characterization of Rx(G, r) on trees and cycles then follows as an immediate result. Finally, we discuss the separation problems for all the inequalities we have found so far, and present algorithms or cut-generation heuristics accordingly. A couple of branch- and-cut frameworks are implemented to solve the MBrT problem together with a greedy- based matheuristic. We compare the performances of the enhanced formulations with the original formulations through intensive computational test, where the results demonstrate convincingly the strength of the enhanced formulations. Keywords: combinatorial optimization, bounded rooted-tree, algorithm, polyhedral study, branch-and-cut ix Résumé Étant donnés un graphe simple non orienté G =(V,E) et un sommet particulier r ∈ V appelé racine,unarbre enraciné,our-arbre,deG est soit le graphe nul (∅, ∅) soit un arbre contenant r. Si un vecteur de capacités sur les sommets est donné, un sous-graphe de G est dit borné si le degré de chaque sommet dans le sous-graphe est inférieur ou égal à sa capacité. Soit w un vecteur de poids sur les arêtes dans RV et p un vecteur de profits sur les sommets dans RE. Leproblèmedur-arbre borné maximum (MBrT, de r r T =(U, F ) G l’anglais Maximum Bounded -Tree) consiste à trouver un -arbre borné de tel que we + pv soit maximisé. Si la contrainte de capacité du problème MBrT est e∈F v∈U relâchée, nous obtenons le problème du r-arbre maximum (MrT, de l’anglais Maximum r-Tree).
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