O problema da árvore geradora de comunicação ótima recebe um grafo com comprimentos não negativos nas arestas e um requerimento não negativo entre cada par de vértices; sendo o objetivo encontrar uma árvore geradora do grafo que minimize o custo de comunicação, que é a soma sobre cada par de vértice da distância entre eles na árvore vezes o requerimento entre eles. Este problema é NP-difícil, assim como vários casos particulares dele. Neste trabalho estudamos algumas variantes deste problema, introduzimos novos casos particulares que são também NP-difíceis e propomos esquemas de aproximação polinomial para alguns deles. / The optimum communication spanning tree problem receives a graph with non-negative lengths over the edges and non-negative requirements for each pair of nodes; being the objective to find a spanning tree of the graph that minimizes the communication cost, which is given by the sum, over each pair of nodes, of the distance, in the tree, between the nodes multiplied by the requirement between them. This problem and several of its particular cases are NP-hard. In this work we study some of the variants, also we introduce new NP-hard particular cases of the problem and propose polynomial approximation schemes for some of them.

Considering the Chinese Postman Problem, in which a mailman must deliver mail to houses in a neighborhood. The mailman must cover each side of the street that has houses, at least once. The focus of this paper is our attempt to discover the optimal path, or the least number of times each street is walked. The integration of algorithms from graph theory and operations research form the method used to explain solutions to the Chinese Postman Problem.

The fundamental theorem of algebra (FTA) is an important theorem in algebra. This theorem asserts that the complex field is algebracially closed. This thesis will include historical research of proofs of the fundamental theorem of algebra and provide information about the first proof given by Gauss of the theorem and the time when it was proved.

The main purpose of this thesis is to describe methods of constructing computer-free proofs of existence of finite groups and give useful techniques to perform double coset enumeration of groups with symmetric presentations over their control groups.

Deux problèmes robustes d'optimisation NP-difficiles sont étudiés dans cette thèse: le problème min-max regret de couverture pondérée (min-max regret WSCP) et le problème min-max regret de couverture et localisation maximale (min-max regret MCLP). Les données incertaines dans ces problèmes sont modélisées par des intervalles et seules les valeurs minimales et maximales pour chaque intervalle sont connues. Le min-max regret WSCP a été investigué notamment dans le cadre théorique, alors que le min-max regret MCLP a des applications en logistique des catastrophes étudiées dans cette thèse. Deux autres critères d'optimisation robuste ont été dérivés pour le MCLP: le max-max MCLP et le min-max MCLP. En matière de méthodes, formulations mathématiques, algorithmes exacts et heuristiques ont été développés et appliqués aux deux problèmes. Des expérimentations computationnelles ont montré que les algorithmes exacts ont permis de résoudre efficacement 14 des 75 instances générées par le min-max regret WSCP et toutes les instances réalistes pour le min-max regret MCLP. Pour les cas simulés qui n'ont pas été résolus de manière optimale dans les deux problèmes, les heuristiques développées dans cette thèse ont trouvé des solutions aussi bien ou mieux que le meilleur algorithme exact dans presque tous les cas. En ce qui concerne l'application en logistique des catastrophes, les modèles robustes ont trouvé des solutions similaires pour les scénarios réalistes des tremblements de terre qui a eu lieu à Katmandu au Népal en 2015. Cela indique que nous avons une solution robuste / Two robust optimization NP-Hard problems are studied in this thesis: the min-max regret Weighted Set Covering Problem (min-max regret WSCP) and the min-max regret Maximal Coverage Location Problem (min-max regret MCLP). The min-max regret WSCP and min-max regret MCLP are, respectively, the robust optimization counterparts of the Set Covering Problem and of the Maximal Coverage Location Problem. The uncertain data in these problems is modeled by intervals and only the minimum and maximum values for each interval are known. However, while the min-max regret WSCP is mainly studied theoretically, the min-max regret MCLP has an application in disaster logistics which is also investigated in this thesis. Two other robust optimization criteria were derived for the MCLP: the max-max MCLP and the min-max MCLP. In terms of methods, mathematical formulations, exact algorithms and heuristics were developed and applied to both problems. Computational experiments showed that the exact algorithms efficiently solved 14 out of 75 instances generated to the min-max regret WSCP and all realistic instances created to the min-max regret MCLP. For the simulated instances that was not solved to optimally in both problems, the heuristics developed in this thesis found solutions, as good as, or better than the exact algorithms in almost all instances. Concerning the application in disaster logistics, the robust models found similar solutions for realistic scenarios of the earthquakes that hit Kathmandu, Nepal in 2015. This indicates that we have got a robust solution, according to all optimization models

Cloud-Radio Access Network (C-RAN) est une architecture prometteuse pour faire face à l’augmentation exponentielle des demandes de trafic de données et surmonter les défis des réseaux de prochaine génération (5G). Le principe de base de CRAN consiste à diviser la station de base traditionnelle en deux entités : les unités de bande de base (BaseBand Unit, BBU) et les têtes radio distantes (Remote Radio Head, RRH) et à mettre en commun les BBUs de plusieurs stations dans des centres de données centralisés (pools de BBU). Ceci permet la réduction des coûts d’exploitation, l’amélioration de la capacité du réseau ainsi que des gains en termes d’utilisation des ressources. Pour atteindre ces objectifs, les opérateurs réseaux ont besoin d’investiguer de nouveaux algorithmes pour les problèmes d’allocation de ressources permettant ainsi de faciliter le déploiement de l’architecture C-RAN. La plupart de ces problèmes sont très complexes et donc très difficiles à résoudre. Par conséquent, nous utilisons l’optimisation combinatoire qui propose des outils puissants pour adresser ce type des problèmes.Un des principaux enjeux pour permettre le déploiement du C-RAN est de déterminer une affectation optimale des RRHs (antennes) aux centres de données centralisés (BBUs) en optimisant conjointement la latence sur le réseau de transmission fronthaul et la consommation des ressources. Nous modélisons ce problème à l’aide d’une formulation mathématique basée sur une approche de programmation linéaire en nombres entiers permettant de déterminer les stratégies optimales pour le problème d’affectation des ressources entre RRH-BBU et nous proposons également des heuristiques afin de pallier la difficulté au sens de la complexité algorithmique quand des instances larges du problème sont traitées, permettant ainsi le passage à l’échelle. Une affectation optimale des antennes aux BBUs réduit la latence de communication attendue et offre des gains en termes d’utilisation des ressources. Néanmoins, ces gains dépendent fortement de l’augmentation des niveaux d’interférence inter-cellulaire causés par la densité élevée des antennes déployées dans les réseaux C-RANs. Ainsi, nous proposons une formulation mathématique exacte basée sur les méthodes Branch-and-Cut qui consiste à consolider et ré-optimiser les rayons de couverture des antennes afin de minimiser les interférences inter-cellulaires et de garantir une couverture maximale du réseau conjointement. En plus de l’augmentation des niveaux d’interférence, la densité élevée des cellules dans le réseau CRAN augmente le nombre des fonctions BBUs ainsi que le trafic de données entre les antennes et les centres de données centralisés avec de fortes exigences en termes de latence sur le réseau fronthaul. Par conséquent, nous discutons dans la troisième partie de cette thèse comment placer d’une manière optimale les fonctions BBUs en considérant la solution split du 3GPP afin de trouver le meilleur compromis entre les avantages de la centralisation dans C-RAN et les forts besoins en latence et bande passante sur le réseau fronthaul. Nous proposons des algorithmes (exacts et heuristiques) issus de l’optimisation combinatoire afin de trouver rapidement des solutions optimales ou proches de l’optimum, même pour des instances larges du problèmes. / Cloud Radio Access Network (C-RAN) has been proposed as a promising architecture to meet the exponential growth in data traffic demands and to overcome the challenges of next generation mobile networks (5G). The main concept of C-RAN is to decouple the BaseBand Units (BBU) and the Remote Radio Heads (RRH), and place the BBUs in common edge data centers (BBU pools) for centralized processing. This gives a number of benefits in terms of cost savings, network capacity improvement and resource utilization gains. However, network operators need to investigate scalable and cost-efficient algorithms for resource allocation problems to enable and facilitate the deployment of C-RAN architecture. Most of these problems are very complex and thus very hard to solve. Hence, we use combinatorial optimization which provides powerful tools to efficiently address these problems.One of the key issues in the deployment of C-RAN is finding the optimal assignment of RRHs (or antennas) to edge data centers (BBUs) when jointly optimizing the fronthaul latency and resource consumption. We model this problem by a mathematical formulation based on an Integer Linear Programming (ILP) approach to provide the optimal strategies for the RRH-BBU assignment problem and we propose also low-complexity heuristic algorithms to rapidly reach good solutions for large problem instances. The optimal RRH-BBU assignment reduces the expected latency and offers resource utilization gains. Such gains can only be achieved when reducing the inter-cell interference caused by the dense deployment of cell sites. We propose an exact mathematical formulation based on Branch-and-Cut methods that enables to consolidate and re-optimize the antennas radii in order to jointly minimize inter-cell interference and guarantee a full network coverage in C-RAN. In addition to the increase of inter-cell interference, the high density of cells in C-RAN increases the amount of baseband processing as well as the amount of data traffic demands between antennas and centralized data centers when strong latency requirements on fronthaul network should be met. Therefore, we discuss in the third part of this thesis how to determine the optimal placement of BBU functions when considering 3GPP split option to find optimal tradeoffs between benefits of centralization in C-RAN and transport requirements. We propose exact and heuristic algorithms based on combinatorial optimization techniques to rapidly provide optimal or near-optimal solutions even for large network sizes.

We present a novel theorem of Borel Combinatorics that sheds light on the types of continuous functions that can be defined on the Cantor space. We specifically consider the part X=F(2ᴳ) from the Cantor space, where the group G is the additive group of integer pairs ℤ². That is, X is the set of aperiodic {0,1} labelings of the two-dimensional infinite lattice graph. We give X the Bernoulli shift action, and this action induces a graph on X in which each connected component is again a two-dimensional lattice graph. It is folklore that no continuous (indeed, Borel) function provides a two-coloring of the graph on X, despite the fact that any finite subgraph of X is bipartite. Our main result offers a much more complete analysis of continuous functions on this space. We construct a countable collection of finite graphs, each consisting of twelve "tiles", such that for any property P (such as "two-coloring") that is locally recognizable in the proper sense, a continuous function with property P exists on X if and only if a function with a corresponding property P' exists on one of the graphs in the collection. We present the theorem, and give several applications.

京都大学 / 0048 / 新制・課程博士 / 博士(人間・環境学) / 甲第21863号 / 人博第892号 / 新制||人||213(附属図書館) / 2018||人博||892(吉田南総合図書館) / 京都大学大学院人間・環境学研究科共生人間学専攻 / (主査)教授 立木 秀樹, 教授 日置 尋久, 准教授 櫻川 貴司, 特定講師 DE BRECHT Matthew / 学位規則第4条第1項該当 / Doctor of Human and Environmental Studies / Kyoto University / DGAM

MiniZinc is a modelling language for constraint satisfaction and optimisation problems. It can be used to solve difficult problems by declaratively modelling them and giving them to a generic solver. A linter, a tool for static analysis, is implemented for MiniZinc to provide analysis for improving models. Suggesting rewrites that will speed up solving, removing unnecessary constructs, and pointing out potential problems are examples of the analysis this tool provides. A method for finding points of interest in abstract syntax trees (parsed models) is designed and implemented. The linter is tested and evaluated against models in the MiniZinc Benchmarks, a collection of models used to benchmark solvers. The result from running the linter on one of the models from the benchmarks is more closely inspected and evaluated. The suggestions were correct and made the model simpler, but, unfortunately, there was no noticeable impact on the solving speed.

Constraint-Based Local Search (CBLS) is a technology used to solve computationally hard optimisation problems. A model written in a solver-independent modelling language needs to be processed before it can be solved by a CBLS solver. In this processing step, it is necessary to identify invariants and create an invariant structure. How to best obtain such a structure, or even how to identify a good structure, is not clear. The purpose of this project is to develop a framework for evaluating invariant structures and structure identification schemes. To do this, we introduce a set of metrics, which are also evaluated. The evaluation shows that these metrics are useful for evaluating invariant structures and structure identification schemes. We introduce a notion of optimal invariant structures and show that these can in many cases be produced by simple structure identification schemes. Finally, we present a strategy that improves on these schemes and yields optimal invariant structures in even more cases.