In this thesis, we study two problems. The first one is the renorming problem in Banach lattices. We state the problem and give some known results related to it. Then we pass to construct a positive doubly power bounded operator with a nonpositive inverse on an infinite dimensional AL-space which generalizes the result of [10]. The second problem is related to the mean ergodicity of positive operators on KBspaces. We prove that any positive power bounded operator T in a KB-space E which satisfies lim n!1 dist1 n n&amp / #8722 / 1 Xk=0 Tkx, [&amp / #8722 / g, g] + BE= 0 (8x 2 E, kxk 1), () where BE is the unit ball of E, g 2 E+, and 0 &lt / 1, is mean ergodic and its fixed space Fix(T) is finite dimensional. This generalizes the main result of [12]. Moreover, under the assumption that E is a -Dedekind complete Banach lattice, we prove that if, for any positive power bounded operator T, the condition () implies that T is mean ergodic then E is a KB-space.

No description available.

p. 1-174 / Submitted by Santiago Fabio (fabio.ssantiago@hotmail.com) on 2013-03-19T17:46:05Z No. of bitstreams: 1 11114.pdf: 1715038 bytes, checksum: 8250a093019aa65753b7de3ff892e5d3 (MD5) / Approved for entry into archive by Tatiana Lima(tatianasl@ufba.br) on 2013-04-12T18:55:41Z (GMT) No. of bitstreams: 1 11114.pdf: 1715038 bytes, checksum: 8250a093019aa65753b7de3ff892e5d3 (MD5) / Made available in DSpace on 2013-04-12T18:55:41Z (GMT). No. of bitstreams: 1 11114.pdf: 1715038 bytes, checksum: 8250a093019aa65753b7de3ff892e5d3 (MD5) Previous issue date: 2005 / A análise do Processo de Contratação Estruturada de Serviços é um estudo de caso da BRASKEM - indústria pertencente ao setor petroquímico nacional, com estabelecimentos fabris em quatro estados brasileiros, inclusive no estado da Bahia -, em cuja porção inicial desenvolve um exame do arcabouço teórico da Nova Economia das Instituições focada em seu alicerce, a Economia dos Custos de Transação, visualizando a definição das estruturas de governança e desse modo estabelecendo a revisão de literatura necessária ao entendimento almejado. Considera um lapso temporal dividido em duas fases, a primeira delas com início em 1998 e indo até 2000, ainda como a COPENE, e a segunda de 2002 a 2004, entre as quais observou-se um processo de integração vertical a partir da fusão de empresas de primeira e segunda gerações da indústria petroquímica para a formação da BRASKEM, período no qual se procura mostrar a importância das estruturas de governança presentes em ambas as fases, fundamentais para promover a redução de custos de transação, além de obter economias estruturais relevantes no sentido de viabilizar a agilização dos resultados consistentes e sustentáveis demandados pelas organizações contemporâneas. Espera-se que a abordagem deste trabalho possa lançar foco à oportunidade que a concentração das atividades estruturadas de Contratação de Serviços assim como a adequada gestão destes pode representar, ao proporcionarem resultados positivos na busca por competitividade, mostrando-se, de fato, uma contribuição neste particular, seja no crivo das decisões empresariais ou no tocante à evolução dos estudos acadêmicos. / Salvador

FERREIRA, Francicleber Martins. Expressividade e complexidade em lógicas preferenciais, híbridas e de grau limitado. 2012. 130 f. Tese (Doutorado em ciência da computação)- Universidade Federal do Ceará, Fortaleza-CE, 2012. / Submitted by Elineudson Ribeiro (elineudsonr@gmail.com) on 2016-07-20T11:58:10Z No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) / Approved for entry into archive by Rocilda Sales (rocilda@ufc.br) on 2016-07-25T11:47:07Z (GMT) No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) / Made available in DSpace on 2016-07-25T11:47:07Z (GMT). No. of bitstreams: 1 2012_tese_fmferreira.pdf: 772301 bytes, checksum: b1e1a404221e38ffb8e0712513749a4c (MD5) Previous issue date: 2012 / We investigate the model theory of Preferential Logics, Hybrid Logic and fragments of Second-Order Logic with respect to finite models. The semantics of these logics differ from the semantics of classical logics either by using relations between models or by restricting the cardinality of the models considered. This work has three main parts. In the first part of this work we study the model theory of preferential logics. Preferential logics arise in the context of nonmonotonic reasoning in Artificial Intelligence. The main characteristic of those logics is the existence of a relation between models. It allows the definition of a nonmonotonic consequence relation by considering the minimal models of a set of sentences. Using the approach of Abstract Model Theory we generalize some expressiveness results to classes of preferential logics. We show that whenever a class of minimal models of a finite set of sentences is axiomatizable, without considering the preference relation, then it is finitely axiomatizable. We also show that when such class of minimal models implicitly defines a symbol, then the finite axiomatization can be put in a very specic form, namely, the initial set of sentences plus a explicit definition for the symbol. In the second part of this work, we investigate the finite model theory of Hybrid Logic. Hybrid Logics are extensions of modal logics with hybrid terms which refer to single states in a Kripke model. We study the complexity of the model- and frame-checking problems for Hybrid Logic. We show that for each graph problem in the Polynomial Hierarchy and each natural number n there is a formula which expresses this problem for graphs of cardinality n. We also show that the size of such formulas is bounded by a polynomial in n. We show that one can disregard the global modalities if one consider only connected graphs with loops. We define fragments which correspond to each degree of the Polynomial Hierarchy. This leads to an alternative proof of the NP-hardness of the model-checking problem for an specic fragment of Full Hybrid Logic. In the last part of this work, we explore the descriptive complexity of the logic obtained by restricting second-order quantication to relations of bounded degree. Based on previous work from Schwentick et al. and Grandjean and Olive, we introduce the Bounded-Degree Second-Order Logic and show that it captures the class ALIN of classes of unary structures accepted by a alternating random access machine in linear time and bounded number of alternations. We also extend this logic with the transitive closure operator on high-order relations on bounded-degree relations. We show that the Bounded-Degree Second-Order Logic with Transitive Closure Operator captures linear number of registers in a nondeterministic random access machine provided that registers store values bounded by a linear function in the cardinality of the input structure. / Nós investigamos a teoria dos modelos de Lógicas Preferenciais, Lógica Híbrida e fragmentos da Lógica de Segunda-Ordem com relação a modelos finitos. A semânticas dessas lógicas diferem da abordagem clássica pelo uso de relações entre modelos ou por restringir a cardinalidade dos modelos a cardinais finitos. Este trabalho tem três partes. Na primeira parte deste trabalho nós estudamos a teoria dos modelos de lógicas preferenciais. Lógicas preferenciais surgem no contexto do raciocínio não-monotônico em Inteligência Artificial. A principal característica dessas lógicas é a existência de uma relação entre modelos. Isso permite a definição de uma relação de consequência não monotônica considerando-se os modelos minimais de um conjunto de sentenças. Usando a abordagem da Teoria dos Modelos Abstrata, nós generalizamos alguns resultados de expressividade para classes de lógicas preferenciais. Nós mostramos que sempre que uma classe de modelos minimais de um conjunto finito de sentenças é axiomatizável, então tal classe é finitamente axiomatizável. Nós mostramos que se tal classe define implicitamente um símbolo do vocabulário, existe uma axiomatização finita de uma forma particular, a saber, o conjunto finito de sentenças inicial mais uma definição explícita para o símbolo definido. Na segunda parte desse trabalho, nós investigamos a teoria dos modelos finitos da Lógica Híbrida. Lógicas Híbridas são extensões da lógica modal através de termos híbridos que se referem a estados individuais em um modelo de Kripke. Nós estudamos a complexidade computacional dos problemas de model- e frame- checking para a Lógica Híbrida. Nós mostramos que para cada problema de grafos na Hierarquia Polinomial e cada número n, existe uma fórmula que exprime esse problema para grafos de cardinalidade n. Nós mostramos que o tamanho das fórmulas é limitado por um polinômio em n. Nós mostramos que podemos abrir mão das modalidades globais se nos limitarmos a grafos conexos com loops. Nós definimos fragmentos da Lógica Híbrida que correspondem a cada nível da Hierarquia Polinomial. Isso nos leva a uma prova alternativa da NP-dificuldade do problema de model-checking para um fragmento específico de da Lógica Híbrida. Na última parte desse trabalho, nós exploramos a complexidade descritiva da lógica obtida ao restringirmos a quantificação de segunda-ordem a relações de grau limitado. Baseados em trabalhos anteriores de Schwentick et al. e de Grandjean e Olive, nós introduzimos a Lógica de Segunda-Ordem de Grau Limitado e mostramos que ela captura a classe ALIN de classes de estruturas unárias aceitas por uma máquina de acesso randômico em tempo linear e um número fixo de alternâncias dependente apenas do problema. Nós estendemos essa lógica com o operador de fecho transitivo sobre relações de ordem superior sobre relações de grau limitado. Nós mostramos que a Lógica de Segunda- Ordem de Grau Limitado com Fecho Transitivo captura quantidade linear de registradores em uma máquina de acesso randômico não-determinística onde os valores armazenados em cada registrador durante a computação são limitados por uma função linear na cardinalidade da estrutura de entrada.

Problems in formal verification are often stated in terms of finite automata and extensions thereof. In this thesis we investigate several such algorithmic problems. In the first part of the thesis we develop a theory of completeness thresholds in Bounded Model Checking. A completeness threshold for a given model M and a specification φ is a bound k such that, if no counterexample to φ of length k or less can be found in M, then M in fact satisfies φ. We settle a problem of Kroening et al. [KOS⁺11] in the affirmative, by showing that the linearity problem for both regular and ω-regular specifications (provided as finite automata and Buchi automata respectively) is PSPACE-complete. Moreover, we establish the following dichotomies: for regular specifications, completeness thresholds are either linear or exponential, whereas for ω-regular specifications, completeness thresholds are either linear or at least quadratic in the recurrence diameter of the model under consideration. Given a formula in a temporal logic such as LTL or MTL, a fundamental problem underpinning automata-based model checking is the complexity of evaluating the formula on a given finite word. For LTL, the complexity of this task was recently shown to be in NC [KF09]. In the second part of the thesis we present an NC algorithm for MTL, a quantitative (or metric) extension of LTL, and give an AC¹ algorithm for UTL, the unary fragment of LTL. We then establish a connection between LTL path checking and planar circuits which, among others, implies that the complexity of LTL path checking depends on the Boolean connectives allowed: adding Boolean exclusive or yields a temporal logic with P-complete path-checking problem. In the third part of the thesis we study the decidability of the reachability problem for parametric timed automata. The problem was introduced over 20 years ago by Alur, Henzinger, and Vardi [AHV93]. It is known that for three or more parametric clocks the problem is undecidable. We translate the problem to reachability questions in certain extensions of parametric one-counter machines. By further reducing to satisfiability in Presburger arithmetic with divisibility, we obtain decidability results for several classes of parametric one-counter machines. As a corollary, we show that, in the case of a single parametric clock (with arbitrarily many nonparametric clocks) the reachability problem is NEXP-complete, improving the nonelementary decision procedure of Alur et al. The case of two parametric clocks is open. Here, we show that the reachability is decidable in this case of automata with a single parameter.

abstract: Recently, automation, shared use, and electrification are proposed and viewed as the “three revolutions” in the future transportation sector to significantly relieve traffic congestion, reduce pollutant emissions, and increase transportation system sustainability. Motivated by the three revolutions, this research targets on the passenger-focused scheduled transportation systems, where (1) the public transit systems provide high-quality ridesharing schedules/services and (2) the upcoming optimal activity planning systems offer the best vehicle routing and assignment for household daily scheduled activities. The high quality of system observability is the fundamental guarantee for accurately predicting and controlling the system. The rich information from the emerging heterogeneous data sources is making it possible. This research proposes a modeling framework to systemically account for the multi-source sensor information in urban transit systems to quantify the estimated state uncertainty. A system of linear equations and inequalities is proposed to generate the information space. Also, the observation errors are further considered by a least square model. Then, a number of projection functions are introduced to match the relation between the unique information space and different system states, and its corresponding state estimate uncertainties are further quantified by calculating its maximum state range. In addition to optimizing daily operations, the continuing advances in information technology provide precious individual travel behavior data and trip information for operational planning in transit systems. This research also proposes a new alternative modeling framework to systemically account for boundedly rational decision rules of travelers in a dynamic transit service network with tight capacity constraints. An agent-based single-level integer linear formulation is proposed and can be effectively by the Lagrangian decomposition. The recently emerging trend of self-driving vehicles and information sharing technologies starts creating a revolutionary paradigm shift for traveler mobility applications. By considering a deterministic traveler decision making framework, this research addresses the challenges of how to optimally schedule household members’ daily scheduled activities under the complex household-level activity constraints by proposing a set of integer linear programming models. Meanwhile, in the microscopic car-following level, the trajectory optimization of autonomous vehicles is also studied by proposing a binary integer programming model. / Dissertation/Thesis / Doctoral Dissertation Civil, Environmental and Sustainable Engineering 2018

This article shows the relationship between market and firms through the theory of transactional cost and the relationsthat determine the structure and boundaries of the firm. Using the assumptions from the transactional cost approach,this article proposes some variables that determine optimal organizational structures and their boundaries. / Este artículo muestra la vinculación entre mercado y empresas por medio del enfoque de los costos de transacción, y lasrelaciones que influyen en la determinación de la estructura y los límites de la empresa. Mediante el uso de los supuestosde la teoría de los costos transaccionales, el artículo propone algunas variables que determinan estructuras organizacionalesóptimas y sus respectivos límites.

Submitted by Kamila Costa (kamilavasconceloscosta@gmail.com) on 2015-08-03T12:38:15Z No. of bitstreams: 2 Dissertação - Francisco de A P Januário.pdf: 1215702 bytes, checksum: 7f02a7976f19b94633a48b22a4990adf (MD5) ficha_catalografica.pdf: 1919 bytes, checksum: cbf0df9103c43df7202f18e8010435b9 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-08-04T15:34:24Z (GMT) No. of bitstreams: 2 Dissertação - Francisco de A P Januário.pdf: 1215702 bytes, checksum: 7f02a7976f19b94633a48b22a4990adf (MD5) ficha_catalografica.pdf: 1919 bytes, checksum: cbf0df9103c43df7202f18e8010435b9 (MD5) / Approved for entry into archive by Divisão de Documentação/BC Biblioteca Central (ddbc@ufam.edu.br) on 2015-08-04T15:38:25Z (GMT) No. of bitstreams: 2 Dissertação - Francisco de A P Januário.pdf: 1215702 bytes, checksum: 7f02a7976f19b94633a48b22a4990adf (MD5) ficha_catalografica.pdf: 1919 bytes, checksum: cbf0df9103c43df7202f18e8010435b9 (MD5) / Made available in DSpace on 2015-08-04T15:38:25Z (GMT). No. of bitstreams: 2 Dissertação - Francisco de A P Januário.pdf: 1215702 bytes, checksum: 7f02a7976f19b94633a48b22a4990adf (MD5) ficha_catalografica.pdf: 1919 bytes, checksum: cbf0df9103c43df7202f18e8010435b9 (MD5) Previous issue date: 2015-04-01 / CNPq - Conselho Nacional de Desenvolvimento Científico e Tecnológico / The development of programs written in Lua programming language, which is largely used in applications for digital TV and games, can cause errors, deadlocks, arithmetic overflow, and division by zero. This work aims to propose a methodology for checking programs written in Lua programming language using the Efficient SMT-Based Context-BoundedModel Checker (ESBMC) tool, which represents the state-of-the-art context-bounded model checker. It is used for ANSI-C/C++ programs and has the ability to verify array out-of-bounds, division by zero, and user-defined assertions. The proposed approach consists in translating programs written in Lua to an intermediate language, which are further verified by ESBMC. The translator is developed with the ANTLR (ANother Tool for Language Recognition) tool, which is used for developing the lexer and parser, based on the Lua language grammar. This work is motivated by the need for extending the benefits of bounded model checking, based on satisfiability modulotheories, to programs written in Lua programming language. The experimental results show that the proposed methodology can be very effective, regarding model checking (safety) of Luaprogramming language properties. / O desenvolvimento de programas escritos na linguagem de programação Lua, que é muito utilizada em aplicações para TV digital e jogos, pode gerar erros, deadlocks, estouro aritmético e divisão por zero. Este trabalho tem como objetivo propor uma metodologia de verificação para programas escritos na linguagem de programação Lua usando a ferramenta Efficient SMT-Based Context-Bounded Model Checker (ESBMC), que representa o estado da arte em verificação de modelos de contexto limitado. O ESBMC é aplicado a programas embarcados ANSI-C/C++ e possui a capacidade de verificar estouro de limites de vetores, divisão por zero e assertivas definidas pelo usuário. A abordagem proposta consiste na tradução de programas escritos em Lua para uma linguagem intermediária, que é posteriormente verificada pelo ESBMC. O tradutor foi desenvolvido com a ferramenta ANTLR (do inglês “ANother Tool for Language Recognition”), que é utilizada na construção de analisadores léxicos e sintáticos, a partir da gramática da linguagem Lua. Este trabalho é motivado pela necessidade de se estender os benefícios da verificação de modelos, baseada nas teorias de satisfatibilidade, a programas escritos na linguagem de programação Lua. Os resultados experimentais mostram que a metodologia proposta pode ser muito eficaz, no que diz respeito à verificação de propriedades (segurança) da linguagem de programação Lua.

Cette thèse est consacrée à l’étude de problèmes inverses associés à des équations aux dérivées partielles hyperboliques et de type Schrödinger.La première partie de la thèse est consacrée à l’étude de problèmes inverses pour l’équation des ondes. Il s’agit d’examiner les propriétés de stabilité et d’unicité dans l’identification de certains coefficients apparaissant dans l’équation des ondes, à partir de différents types d’observation.La deuxième partie de cette thèse, traite du problème de l’identification du champ magnétique et du potentiel électrique apparaissant dans l’équation du Schrödinger. Nous prouvons que ces coefficients peuvent être déterminés de façon stable dans tout le domaine, à partir de données de type Neumann. La dérivation de ces résultats est basée sur la construction d’un ensemble de solutions de type optique géométrique, adaptées au système étudié. Il existe une méthode alternative pour l’analyse de ce type de problèmes inverses, celle de Bukhgeim-Klibanov, qui utilise une estimation de Carleman spécifique à l’opérateur con-sidéré. Elle nous a permis de montrer qu’il est possible de récupérer de façon stable et simultanée, la partie spatiale des potentiels électrique et magnétique de l’équation de Schrödinger magnétique, à partir d’un nombre fini de mesures partielles de la solution. / This thesis is devoted to the study of inverse problems associated to hyperbolic and Schrödinger equations. The first part of the thesis is devoted to the study of inverse problemsfor the wave equation. The aim is to examine the stability andthe uniqueness issues in the identification of certain coefficients appearing in the wave equation from different types of observation. The second part of this thesis deals with the problem of the identification of a magnetic field and an electric potential appearing in the Schrödinger equation. We prove that these coefficients can be stably determined throughout the domain, using Neumann data. The derivation of these results is based on the construction of a set of geometric optics solutions adapted to the system studied. There is an alternative method for the analysis of this type of inverse problem, which is due to Bukhgeim-Klibanov, and which uses a Carleman estimate. We show that it is possible to stably and simultaneously recover the spatial part of the electrical and magnetic potentialsappearing in the magnetic Schrödinger equation, from a finite number of measurements.

Étant donnés un graphe simple non orienté G = (V, E) et un sommet particulier r dans V appelé racine, un arbre enraciné, ou r-arbre, de G 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 et p un vecteur de profits sur les sommets. Le problème du r-arbre borné maximum (MBrT, de l’anglais Maximum Bounded r-Tree) consiste à trouver un r-arbre borné T = (U, F) de G tel que son poids soit maximisé. Si la contrainte de capacité du problème MBrT est relâchée, nous obtenons le problème du r-arbre maximum (MrT, de l’anglais Maximum r-Tree). Cette thèse contribue à l’étude des problèmes MBrT et MrT.Tout d’abord, ces deux problèmes sont formellement définis et leur complexité est étudiée. Nous présentons ensuite des polytopes associés ainsi qu’une formulation pour chacun d’entre eux. Par la suite, nous proposons plusieurs algorithmes combinatoires pour résoudre le problème MBrT (et donc le problème MrT) en temps polynomial sur les arbres, les cycles et les cactus. En particulier, un algorithme de programmation dynamique est utilisé pour résoudre le problème MBrT sur les arbres. Pour les cycles, nous sommes amenés a considérer trois cas différents pour lesquels le problem MBrT se réduit à certains problèmes polynomiaux. Pour les cactus, nous montrons tout d’abord que le problème MBrT peut être résolu en temps polynomial sur un type de graphes appelé cactus basis. En utilisant une série de décompositions en sous-problèmes sur les arbres et les cactus basis, nous obtenons un algorithme pour les graphes de type cactus.La deuxième partie de ce travail étudie la structure polyédrale de trois polytopes associés aux problèmes MBrT et MrT. Les deux premiers polytopes, Bxy(G,r,c) et Bx(G,r,c) sont associés au problème MBrT. Tous deux considèrent des variables sur les arêtes de G, mais seuls Bxy(G,r,c) possède également des variables sur les sommets de G. Le troisième polytope, Rx(G,r), est associé au problème MrT et repose uniquement sur les variables sur les arêtes. Pour chacun de ces trois polytopes, nous étudions sa dimension, caractérisons certaines inégalités définissant des facettes, et présentons les moyens possibles de décomposition. Nous introduisons également de nouvelles familles de contraintes. L’ajout de ces contraintes nous permettent de caractériser ces trois polytopes dans plusieurs classes de graphes.Pour finir, nous étudions les problèmes de séparation pour toutes les inégalités que nous avons trouvées jusqu’ici. Des algorithmes polynomiaux de séparation sont présentés, et lorsqu’un problème de séparation est NP-difficile, nous donnons des heuristiques de séparation. Tous les résultats théoriques développés dans ce travail sont implémentés dans plusieurs algorithmes de coupes et branchements auxquels une matheuristique est également jointe pour générer rapidement des solutions réalisables. Des expérimentations intensives ont été menées via le logiciel CPLEX afin de comparer les formulations renforcées et originales. Les résultats obtenus montrent de manière convaincante la force des formulations renforcées. / Given a simple undirected graph G = (V, E) with a so-called root node r in V, a rooted tree, or an r-tree, of G is either the empty graph, or a tree containing r. If a node-capacity vector c 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 vector and p a node-price vector. The Maximum Bounded r-Tree (MBrT) problem consists of finding a bounded r-tree T = (U, F) of G such that its weight is 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 MrT problem.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 MBrT problem can be solved in polynomial time on a so-called cactus 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 associated 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 MBrT problem, 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 characterization 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) is given in each case of 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.