Orientador: Orlando Lee / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-25T17:05:58Z (GMT). No. of bitstreams: 1 Freitas_LucasIsmailyBezerra_M.pdf: 2067916 bytes, checksum: 77f11deab9d862fe9a10de2df94b447c (MD5) Previous issue date: 2014 / Resumo: Neste trabalho estudamos a conjectura de Tuza, que relaciona cobertura mínima de triângulos por arestas com empacotamento máximo de triângulos aresta-disjuntos em grafos. Em 1981, Tuza conjecturou que para todo grafo, o número máximo de triângulos aresta-disjuntos é no máximo duas vezes o tamanho de uma cobertura mínima de triângulos por arestas. O caso geral da conjectura continua aberta. Contudo, diversas tentativas de prová-la surgiram na literatura, obtendo resultados para várias classes de grafos. Nesta dissertação, nós apresentamos os principais resultados obtidos da conjectura de Tuza. Atualmente, existem várias versões da conjectura. Contudo, ressaltamos que nosso foco está na conjectura aplicada a grafos simples. Apresentamos também uma conjectura que se verificada, implica na veracidade da conjectura de Tuza. Demonstramos ainda que se G é um contra-exemplo mínimo para a conjectura de Tuza, então G é 4-conexo. Deduzimos desse resultado que a conjectura de Tuza é válida para grafos sem minor do K_5 / Abstract: In this thesis we study the conjecture of Tuza, which relates covering of triangles (by edges) with packing of edge-disjoint triangles in graphs. In 1981, Tuza conjectured that for any graph, the maximum number of edge-disjoint triangles is at most twice the size of a minimum cover of triangles by edges. The general case of the conjecture remains open. However, several attempts to prove it appeared in the literature, which contain results for several classes of graphs. In this thesis, we present the main known results for the conjecture of Tuza. Currently, there are several versions of Tuza's conjecture. Nevertheless, we emphasize that our focus is on conjecture applied to simple graphs. We also present a conjecture that, if verified, implies the validity of the conjecture of Tuza. We also show that if G is a mininum counterexample to the conjecture of Tuza, then G is 4-connected. We can deduce from this result that the conjecture of Tuza is valid for graphs with no K_5 minor / Mestrado / Ciência da Computação / Mestre em Ciência da Computação

We focus on space-efficient, namely succinct, representations of static ordinal unlabeled trees. These structures have space complexity which is optimal up to a lower-order term, yet they support a reasonable set of operations in constant time. This topic has been studied in the last 27 years by numerous authors who came with several distinct solutions to this problem. It is not only of an academic interest, the succinct tree data structures has been used in several data-intensive applications, such as XML processing and representation of suffix trees. In this thesis, we describe the current state of knowledge in this area, compare the many different approaches, and propose several either new or alternative algorithms for operations in the representations alongside. Powered by TCPDF (www.tcpdf.org)

Municipalities in the Czech Republic deal with their budget, which among others consists of granted subsidies from the region, State, European Union or other organizations. Nowadays the budget transactions are not being under the control. In the future, it is appropriate to introduce some external view to control their spending. Establishment of an audit service for each municipality would be financially inevitable. Therefore it is suggested to provide the State audit services only in some municipalities and to share them with more municipalities within one region. Deployment of the audit centers and assigning municipalities lead to solving a linear problem which falls under the covering problem class. The establishment of audit centers is only illustrative, the employment of more shared state services could follow a similar principle.

A good strategy to test a software component involves the generation of the whole set of cases that participate in its operation. While testing only individual values may not be enough, exhaustive testing of all possible combinations is not always feasible. An alternative technique to accomplish this goal is called combinato- rial testing. Combinatorial testing is a method that can reduce cost and increase the effectiveness of software testing for many applications. It is based on con- structing functional test-suites of economical size, which provide coverage of the most prevalent configurations. Covering arrays are combinatorial objects, that have been applied to do functional tests of software components. The use of cov- ering arrays allows to test all the interactions, of a given size, among the input parameters using the minimum number of test cases. For software testing, the fundamental problem is finding a covering array with the minimum possible number of rows, thus reducing the number of tests, the cost, and the time expended on the software testing process. Because of the importance of the construction of (near) optimal covering arrays, much research has been carried out in developing effective methods for constructing them. There are several reported methods for constructing these combinatorial models, among them are: (1) algebraic methods, recursive methods, (3) greedy methods, and (4) metaheuristics methods. Metaheuristic methods, particularly through the application of simulated anneal- ing has provided the most accurate results in several instances to date. Simulated annealing algorithm is a general-purpose stochastic optimization method that has proved to be an effective tool for approximating globally optimal solutions to many optimization problems. However, one of the major drawbacks of the simulated an- nealing is the time it requires to obtain good solutions. In this thesis, we propose the development of an improved simulated annealing algorithm / Avila George, H. (2012). Constructing Covering Arrays using Parallel Computing and Grid Computing [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/17027 / Palancia

Une classe de graphes est close par mineurs si, pour tout graphe dans la classe et tout mineur de ce graphe, le mineur est ́egalement dans la classe. Par un fameux th ́eor`eme de Robertson et Seymour, nous savons que car- act ́eriser une telle classe peut ˆetre fait `a l’aide d’un nombre fini de mineurs exclus minimaux. Ceux-ci sont des graphes qui n’appartiennent pas `a la classe et qui sont minimaux dans le sens des mineurs pour cette propri ́et ́e.Dans cette thèse, nous étudions trois problèmes à propos de classes de graphes closes par mineurs. Les deux premiers sont reliés à la caractérisation de certaines classes de graphes, alors que le troisième étudie une relation de “packing-covering” dans des graphes excluant un mineur.Pour le premier problème, nous étudions des plongements isométriques de graphes dont les arêtes sont pondérées dans des espaces métriques. Principalement, nous nous intêressons aux espaces ell_2 et ell_∞. E ́tant donné un graphe pondéré, un plongement isométrique associe à chaque sommet du graphe un vecteur dans l’autre espace de sorte que pour chaque arête du graphe le poids de celle-ci est égal à la distance entre les vecteurs correspondant à ses sommets. Nous disons qu’une fonction de poids sur les arêtes est une fonction de distances réalisable s’il existe un tel plongement. Le paramètre f_p(G) détermine la dimension k minimale d’un espace ell_p telle que toute fonction de distances réalisable de G peut être plongée dans ell_p^k. Ce paramètre est monotone dans le sens des mineurs. Nous caractérisons les graphes tels que f_p(G) a une grande valeur en termes de mineurs inévitables pour p = 2 et p = ∞. Une famille de graphes donne des mineurs inévitables pour un invariant monotone pour les mineurs, si ces graphes “expliquent” pourquoi l’invariant est grand.Le deuxième problème étudie les mineurs exclus minimaux pour la classe de graphes avec φ(G) borné par une constante k, où φ(G) est un paramètre lié au dominant des coupes d’un graphe G. Ce polyèdre contient tous les points qui, composante par composante, sont plus grands ou égaux à une combination convexe des vecteurs d’incidence de coupes dans G. Le paramètre φ(G) est égal au membre de droite maximum d’une description linéaire du dominant des coupes de G en forme entière minimale. Nous étudions les mineurs exclus minimaux pour la propriété φ(G) <= 4 et montrons une nouvelle borne sur φ(G) en termes du “vertex cover number”.Le dernier problème est d’un autre type. Nous étudions une relation de “packing-covering” dans les classes de graphes excluant un mineur. Étant donné un graphe G, une boule de centre v et de rayon r est l’ensemble de tous les sommets de G qui sont à distance au plus r de v. Pour un graphe G et une collection de boules donnés nous pouvons définir un hypergraphe H dont les sommets sont ceux de G et les arêtes correspondent aux boules de la collection. Il est bien connu que dans l’hypergraphe H, le “transversal number” τ(H) vaut au moins le “packing number” ν(H). Nous montrons une borne supérieure sur ν(H) qui est linéaire en τ(H), résolvant ainsi un problème ouvert de Chepoi, Estellon et Vaxès. / A class of graphs is closed under taking minors if for each graph in the class and each minor of this graph, the minor is also in the class. By a famous result of Robertson and Seymour, we know that characterizing such a class can be done by identifying a finite set of minimal excluded minors, that is, graphs which do not belong to the class and are minor-minimal for this property.In this thesis, we study three problems in minor-closed classes of graphs. The first two are related to the characterization of some graph classes, while the third one studies a packing-covering relation for graphs excluding a minor.In the first problem, we study isometric embeddings of edge-weighted graphs into metric spaces. In particular, we consider ell_2- and ell_∞-spaces. Given a weighted graph, an isometric embedding maps the vertices of this graph to vectors such that for each edge of the graph the weight of the edge equals the distance between the vectors representing its ends. We say that a weight function on the edges of the graph is a realizable distance function if such an embedding exists. The minor-monotone parameter f_p(G) determines the minimum dimension k of an ell_p-space such that any realizable distance function of G is realizable in ell_p^k. We characterize graphs with large f_p(G) value in terms of unavoidable minors for p = 2 and p = ∞. Roughly speaking, a family of graphs gives unavoidable minors for a minor-monotone parameter if these graphs “explain” why the parameter is high.The second problem studies the minimal excluded minors of the class of graphs such that φ(G) is bounded by some constant k, where φ(G) is a parameter related to the cut dominant of a graph G. This unbounded polyhedron contains all points that are componentwise larger than or equal to a convex combination of incidence vectors of cuts in G. The parameter φ(G) is equal to the maximum right-hand side of a facet-defining inequality of the cut dominant of G in minimum integer form. We study minimal excluded graphs for the property φ(G) <= 4 and provide also a new bound of φ(G) in terms of the vertex cover number.The last problem has a different flavor as it studies a packing-covering relation in classes of graphs excluding a minor. Given a graph G, a ball of center v and radius r is the set of all vertices in G that are at distance at most r from v. Given a graph and a collection of balls, we can define a hypergraph H such that its vertices are the vertices of G and its edges correspond to the balls in the collection. It is well-known that, in the hypergraph H, the transversal number τ(H) is at least the packing number ν(H). We show that we can bound τ(H) from above by a linear function of ν(H) for every graphs G and ball collections H if the graph G excludes a minor, solving an open problem by Chepoi, Estellon et Vaxès. / Doctorat en Sciences / info:eu-repo/semantics/nonPublished

abstract: The construction of many families of combinatorial objects remains a challenging problem. A t-restriction is an array where a predicate is satisfied for every t columns; an example is a perfect hash family (PHF). The composition of a PHF and any t-restriction satisfying predicate P yields another t-restriction also satisfying P with more columns than the original t-restriction had. This thesis concerns three approaches in determining the smallest size of PHFs. Firstly, hash families in which the associated property is satisfied at least some number lambda times are considered, called higher-index, which guarantees redundancy when constructing t-restrictions. Some direct and optimal constructions of hash families of higher index are given. A new recursive construction is established that generalizes previous results and generates higher-index PHFs with more columns. Probabilistic methods are employed to obtain an upper bound on the optimal size of higher-index PHFs when the number of columns is large. A new deterministic algorithm is developed that generates such PHFs meeting this bound, and computational results are reported. Secondly, a restriction on the structure of PHFs is introduced, called fractal, a method from Blackburn. His method is extended in several ways; from homogeneous hash families (every row has the same number of symbols) to heterogeneous ones; and to distributing hash families, a relaxation of the predicate for PHFs. Recursive constructions with fractal hash families as ingredients are given, and improve upon on the best-known sizes of many PHFs. Thirdly, a method of Colbourn and Lanus is extended in which they horizontally copied a given hash family and greedily applied transformations to each copy. Transformations of existential t-restrictions are introduced, which allow for the method to be applicable to any t-restriction having structure like those of hash families. A genetic algorithm is employed for finding the "best" such transformations. Computational results of the GA are reported using PHFs, as the number of transformations permitted is large compared to the number of symbols. Finally, an analysis is given of what trade-offs exist between computation time and the number of t-sets left not satisfying the predicate. / Dissertation/Thesis / Doctoral Dissertation Computer Science 2019

A k × n array with entries from an "alphabet" A = { 0, 1, ..., q - 1 } of size q is said to form a t-covering array (resp. orthogonal array) if each t × n submatrix of the array contains, among its columns, at least one (resp. exactly one) occurrence of each t-letter word from A (we must thus have n = qt for an orthogonal array to exist and n ≥ qt for a t -covering array). In this paper, we continue the agenda laid down in Godbole et al. (2009) in which the notion of consecutive covering arrays was defined and motivated; a detailed study of these arrays for the special case q = 2, has also carried out by the same authors. In the present article we use first a Markov chain embedding method to exhibit, for general values of q, the probability distribution function of the random variable W = Wk, n, t defined as the number of sets of t consecutive rows for which the submatrix in question is missing at least one word. We then use the Chen-Stein method (Arratia et al., 1989, 1990) to provide upper bounds on the total variation error incurred while approximating L (W) by a Poisson distribution Po (λ) with the same mean as W. Last but not least, the Poisson approximation is used as the basis of a new statistical test to detect run-based discrepancies in an array of q-ary data.

The classical Erdös-Ko-Rado theorem states that if k≤ ⌊n/2⌋ then the largest family of pairwise intersecting k-subsets of [n]={1,. ,n} is of size (n-1 k-1). A family of k subsets satisfying this pairwise intersecting property is called an EKR family. We generalize the EKR property and provide asymptotic lower bounds on the size of the largest family A of k-subsets of [n] that satis es the following property: For each A,B,CεA, each of the four sets A∩B∩C; A∩B∩C̃; A∩B̃∩C;Ã ∩B∩C are non-empty. This generalized EKR (GEKR) property is motivated, generalizations are suggested, and a comparison is made with fixed weight 3-covering arrays. Our techniques are probabilistic, and reminiscent of those used in (A. Godbole, D. Skipper, and R. Sunley, Comb Prob Computing 5 (1996), 105-118) and in the work of Roux, as cited in (N. J. A. Sloane, J Comb Designs 1 (1993), 51-63).

Covering arrays for words of length (Formula presented.) over a (Formula presented.) -letter alphabet are (Formula presented.) arrays with entries from the alphabet so that for each choice of (Formula presented.) columns, each of the (Formula presented.) (Formula presented.) -letter words appears at least once among the rows of the selected columns. We study two schemes in which all words are not considered to be different. In the first case known as partitioning hash families, words are equivalent if they induce the same partition of a (Formula presented.) element set. In the second case, words of the same weight are equivalent. In both cases, we produce logarithmic upper bounds on the minimum size (Formula presented.) of a covering array. Definitive results for (Formula presented.), as well as general results, are provided.

Since the early 1960s location problems have been used throughout various industries and in various countries. During recent years the field of location problems has become increasingly popular due to the fact that it is applicable in real life situations – especially in emergency services such as hospital, police station and ambulance locations to name a few. Despite the fact that location problems are so widely used with great success, it is still not being used to full potential in industries where it can have a major impact. One of these industries is the private security industry in South Africa. This dissertation addresses various mathematical models that can assist the management of privately owned security companies to determine strategic locations for their reaction vehicles, these locations will increase both resource utilization and improve the level of service they provide to customers. These models are used in different scenarios to see how the models adapt to input changes. / Dissertation (MSc)--University of Pretoria, 2014. / Industrial and Systems Engineering / MSc / Restricted