Indiana University-Purdue University Indianapolis (IUPUI) / Text Mining is process of extracting high-quality knowledge from analysis of textual data. Rapidly growing interest and focus on research in many fields is resulting in an overwhelming amount of research literature. This literature is a vast source of knowledge. But due to huge volume of literature, it is practically impossible for researchers to manually extract the knowledge. Hence, there is a need for automated approach to extract knowledge from unstructured data. Text mining is right approach for automated extraction of knowledge from textual data. The objective of this thesis is to mine documents pertaining to research literature, to find novel associations among entities appearing in that literature using Incremental Mining. Traditional text mining approaches provide binary associations. But it is important to understand context in which these associations occur. For example entity A has association with entity B in context of entity C. These contexts can be visualized as multi-way associations among the entities which are represented by a Hypergraph. This thesis work talks about extracting such multi-way associations among the entities using Frequent Itemset Mining and application of a new concept called Output space sampling to extract such multi-way associations in space and time efficient manner. We incorporated concept of personalization in Output space sampling so that user can specify his/her interests as the frequent hyper-associations are extracted from the text.

A multimodal network (MMN) is a novel mathematical construct that captures the structure of biological networks, computational network models, and relationships from biological databases. An MMN subsumes the structure of graphs and hypergraphs, either undirected or directed. Formally, an MMN is a triple (V,E,M) where V is a set of vertices, E is a set of modal hyperedges, and M is a set of modes. A modal hyperedge e=(T,H,A,m) in E is an ordered 4-tuple, in which T,H,A are subsets of V and m is an element of M. The sets T, H, and A are the tail, head, and associate of e, while m is its mode. In the context of biology, each vertex is a biological entity, each hyperedge is a relationship, and each mode is a type of relationship (e.g., 'forms complex' and 'is a'). Within the space of multimodal networks, structural operations such as union, intersection, hyperedge contraction, subnetwork selection, and graph or hypergraph projections can be performed. A denotational semantics approach is used to specify the semantics of each hyperedge in MMN in terms of interaction among its vertices. This is done by mapping each hyperedge e to a hyperedge code algo:V(e), an algorithm that details how the vertices in V(e) get used and updated. A semantic MMN-based model is a function of a given schedule of evaluation of hyperedge codes and the current state of the model, a set of vertex-value pairs. An MMN-based computational system is implemented as a proof of concept to determine empirically the benefits of having it. This system consists of an MMN database populated by data from various biological databases, MMN operators implemented as database functions, graph operations implemented in C++ using LEDA, and mmnsh, a shell scripting language that provides a consistent interface to both data and operators. It is demonstrated that computational network models may enrich the MMN database and MMN data may be used as input to other computational tools and environments. A simulator is developed to compute from an initial state and a schedule of hyperedge codes the resulting state of a semantic MMN model. / Ph. D.

This thesis is divided into two parts. The first part is concerned with the commutative algebra of certain combinatorial structures arising from uniform hypergraphs. The main focus lies on two particular classes of hypergraphs called chordal hypergraphs and complete hypergraphs, respectively. Both these classes arise naturally as generalizations of the corresponding well known classes of simple graphs. The classes of chordal and complete hypergraphs are introduced and studied in Chapter 2 and Chapter 3 respectively. Chapter 4, that is the content of \cite{E5}, answers a question posed at the P.R.A.G.MAT.I.C. summer school held in Catania, Italy, in 2008. In Chapter 5 we study hypergraph analogues of line graphs and cycle graphs. Chapter 6 is concerned with a connectedness notion for hypergraphs and in Chapter 7 we study a weak version of shellability.The second part is concerned with affine monoids and their monoid rings. Chapter 8 provide a combinatorial study of a class of positive affine monoids that behaves in some sense like numerical monoids. Chapter 9 is devoted to the class of numerical monoids of maximal embedding dimension. A combinatorial description of the graded Betti numbers of the corresponding monoid rings in terms of the minimal generators of the monoids is provided. Chapter 10 is concerned with monomial subrings generated by edge sets of complete hypergraphs.

Master of Science / Department of Industrial & Manufacturing Systems Engineering / Todd W. Easton / Integer programming (IP) is a powerful technique used by many companies and organizations to determine optimal strategies for making decisions and managing resources to achieve their goals. One class of IP problems is the multiple knapsack (MK) problem. However, MK and other IP problems, are extremely complicated since they are ${\cal NP}$-hard problems. Furthermore, there exist numerous instances that can not be solved. One technique commonly used to reduce the solution time for IP problems is lifting. This method, introduced by Gomory, takes an existing valid inequality and strengthens it. Lifting has the potential to form facet defining inequalities, which are the strongest inequalities to solve an IP problem. As a result, lifting is frequently used in integer programming applications. This research takes a broad approach to simultaneous lifting and provides its theoretical background for. The underlying hypergraphic structure for simultaneous lifting in an MK problem is identified and called a constellation. A constellation contains two hypercliques and multiple hyperstars from various conflict hypergraphs. Theoretical results demonstrate that a constellation induces valid inequalities that could be obtained by simultaneous lifting. Moreover, these constellation inequalities can be facet defining. The primary advancements, constellations and the associated valid inequalities, of this thesis are theoretical in nature. By providing the theory behind simultaneous lifting, researchers should be able to apply this knowledge to develop new algorithms that enable simultaneous lifting to be performed faster and over more complex integer programs.

The capacity of a directed hypergraph is a particular numerical quantity associated with a hypergraph. It is of interest because of certain important connections to longstanding conjectures in theoretical computer science related to fast matrix multiplication and perfect hashing as well as various longstanding conjectures in extremal combinatorics. We give an overview of the concept of the capacity of a hypergraph and survey a few basic results regarding this quantity. Furthermore, we discuss the Lovász number of an undirected graph, which is known to upper bound the capacity of the graph (and in practice appears to be the best such general purpose bound). We then elaborate on some attempted generalizations/modifications of the Lovász number to undirected hypergraphs that we have tried. It is not currently known whether these attempted generalizations/modifications upper bound the capacity of arbitrary hypergraphs. An important method for proving lower bounds on hypergraph capacity is to exhibit a large independent set in a strong power of the hypergraph. We examine methods for this and show a barrier to attempts to usefully generalize certain of these methods to hypergraphs. We then look at cap sets: independent sets in powers of a certain hypergraph. We examine certain structural properties of them with the hope of finding ones that allow us to prove upper bounds on their size. Finally, we consider two interesting generalizations of capacity and use one of them to formulate several conjectures about connections between cap sets and sunflower-free sets.

O estudo do comportamento assintótico de densidades de algumas subestruturas é uma das principais áreas de estudos em combinatória. Na Teoria das Permutações, fixadas permutações ?1 e ?2 e um inteiro n > 0, estamos interessados em estudar o comportamento das densidades de ?1 e ?2 na família de permutações de tamanho n. Assim, existem duas direções naturais que podemos seguir. Na primeira direção, estamos interessados em achar a permutação de tamanho n que maximiza a densidade das permutações ?1 e ?2 simultaneamente. Para n suficientemente grande, explicitamos a densidade máxima que uma família de permutações podem assumir dentre todas as permutações de tamanho n. Na segunda direção, estamos interessados em achar a permutação de tamanho n que minimiza a densidade de ?1 e ?2 simultaneamente. Quando ?1 é a permutação identidade com k elementos e ?2 é a permutação reversa com l elementos, Myers conjecturou que o mínimo é atingido quando tomamos o mínimo dentre as permutações que não possuem a ocorrência de ?1 ou ?2. Mostramos que se restringirmos o espaço de busca somente ao conjunto de permutações em camadas, então a Conjectura de Myers é verdadeira. Por outro lado, na Teoria dos Grafos, o problema de encontrar um circuito Hamiltoniano é um problema NP-completo clássico e está entre os 21 problemas Karp. Dessa forma, uma abordagem comum na literatura para atacar esse problema é encontrar condições que um grafo deve satisfazer e que garantem a existência de um circuito Hamiltoniano em tal grafo. O célebre resultado de Dirac afirma que se um grafo G de ordem n possui grau mínimo pelo menos n/2, então G possui um circuito Hamiltoniano. Seguindo a linha de Dirac, mostramos que, dados inteiros 1 6 l 6 k/2 e ? > 0 existe um inteiro n0 > 0 tal que, se um hipergrafo k-uniforme H de ordem n satisfaz ?k-2(H) > ((4(k - l) - 1)/(4(k - l)2) + ?) (n 2), então H possui um l-circuito Hamiltoniano. / The study of asymptotic behavior of densities of some substructures is one of the main areas in combinatorics. In Permutation Theory, fixed permutations ?1 and ?2 and an integer n > 0, we are interested in the behavior of densities of ?1 and ?2 among the permutations of size n. Thus, there are two natural directions we can follow. In the first direction, we are interested in finding the permutation of size n that maximizes the density of the permutations ?1 and ?2 simultaneously. We explicit the maximum density of a family of permutations between all the permutations of size n. In the second direction, we are interested in finding the permutation of size n that minimizes the density of ?1 and ?2 simultaneously. When ?1 is the identity permutation with l elements and ?2 is the reverse permutation with k elements, Myers conjectured that the minimum is achieved when we take the minimum among the permutations which do not have the occurrence of ?1 or ?2. We show that if we restrict the search space only to set of layered permutations and k > l, then the Myers\' Conjecture is true. On the other hand, in Graph Theory, the problem of finding a Hamiltonian cycle is a NP-complete problem and it is among the 21 Karp problems. Thus, one approach to attack this problem is to find conditions that a graph must meet to ensure the existence of a Hamiltonian cycle on it. The celebrated result of Dirac shows that a graph G of order n that has minimum degree at least n/2 has a Hamiltonian cycle. Following the line of Dirac, we show that give integers 1 6 l 6 k/2 and gamma > 0 there is an integer n0 > 0 such that if a hypergraph k-Uniform H of order n satisfies ?k-2(H) > ((4(k-l)-1)/(4(k-l)2)+?) (n 2), then H has a Hamiltonian l-cycle.

We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes. We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every $s$, every STS of high enough order is not $s$-list-colourable. The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability $p$ where the random graph $G_{n,p}$ asymptotically almost surely contains every tree of bounded degree. This $p$ is nearly optimal in terms of the maximum degree of the trees that we embed. Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph.

We study several extremal problems in graphs and hypergraphs. The first one is on list-colouring hypergraphs, which is a generalization of the ordinary colouring of hypergraphs. We discuss two methods for determining the list-chromatic number of hypergraphs. One method uses hypergraph polynomials, which invokes Alon's combinatorial nullstellensatz. This method usually requires computer power to complete the calculations needed for even a modest-sized hypergraph. The other method is elementary, and uses the idea of minimum improper colourings. We apply these methods to various classes of hypergraphs, including the projective planes. We focus on solving the list-colouring problem for Steiner triple systems (STS). It is not hard using either method to determine that Steiner triple systems of orders 7, 9 and 13 are 3-list-chromatic. For systems of order 15, we show that they are 4-list-colourable, but they are also ``almost'' 3-list-colourable. For all Steiner triple systems, we prove a couple of simple upper bounds on their list-chromatic numbers. Also, unlike ordinary colouring where a 3-chromatic STS exists for each admissible order, we prove using probabilistic methods that for every $s$, every STS of high enough order is not $s$-list-colourable. The second problem is on embedding nearly-spanning bounded-degree trees in sparse graphs. We determine sufficient conditions based on expansion properties for a sparse graph to embed every nearly-spanning tree of bounded degree. We then apply this to random graphs, addressing a question of Alon, Krivelevich and Sudakov, and determine a probability $p$ where the random graph $G_{n,p}$ asymptotically almost surely contains every tree of bounded degree. This $p$ is nearly optimal in terms of the maximum degree of the trees that we embed. Finally, we solve a problem that arises from quantum computing, which can be formulated as an extremal question about maximizing the size of a type of acyclic directed graph.

Graph polynomials are polynomials associated to graphs that encode the number of subgraphs with given properties. We list different frameworks used to define graph polynomials in the literature. We present the edge elimination polynomial and introduce several graph polynomials equivalent to it. Thereby, we connect a recursive definition to the counting of colorings and to the counting of (spanning) subgraphs. Furthermore, we define a graph polynomial that not only generalizes the mentioned, but also many of the well-known graph polynomials, including the Potts model, the matching polynomial, the trivariate chromatic polynomial and the subgraph component polynomial. We proof a recurrence relation for this graph polynomial using edge and vertex operation. The definitions and statements are given in such a way that most of them are also valid in the case of hypergraphs.

Let F0 be a fixed k-uniform hypergraph, and let H be a given k-uniform hypergraph on n vertices. An F0-packing of H is a family F of edge-disjoint copies of F0 which are subhypergraphs in H. Let nF0(H) denote the maximum size |F| of an F0-packing F of H. It is well-known that computing nF0(H) is NP-hard for nearly any choice of F0. In this thesis, we consider the special case when F0 is a linear hypergraph, that is, when no two edges of F0 overlap in more than one vertex. We establish for z > 0 and n &ge n0(z) sufficiently large, an algorithm which, in time polynomial in n, constructs an F0-packing F of H of size |F| ≥ nF0(H) - znk. A central direction in our proof uses so-called fractional F0-packings of H which are known to approximate nF0(H). The driving force of our argument, however, is the use and development of several tools within the theory of hypergraph regularity.