Dissertations / Theses on the topic 'Structural Combinatorics'
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Ferra, Gomes de Almeida Girão António José. "Extremal and structural problems of graphs." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/285427.
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In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy.
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Borenstein, Evan. "Additive stucture, rich lines, and exponential set-expansion." Diss., Atlanta, Ga. : Georgia Institute of Technology, 2009. http://hdl.handle.net/1853/29664.
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Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.Committee Chair: Croot, Ernie; Committee Member: Costello, Kevin; Committee Member: Lyall, Neil; Committee Member: Tetali, Prasad; Committee Member: Yu, XingXing. Part of the SMARTech Electronic Thesis and Dissertation Collection.
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Tanigawa, Shinichi. "Combinatorial Rigidity and Generation of Discrete Structures." 京都大学 (Kyoto University), 2010. http://hdl.handle.net/2433/120806.
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Brignall, Robert. "Simplicity in relational structures and its application to permutation classes." Thesis, St Andrews, 2007. http://hdl.handle.net/10023/431.
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Rockney, Alissa Ann. "A Predictive Model Which Uses Descriptors of RNA Secondary Structures Derived from Graph Theory." Digital Commons @ East Tennessee State University, 2011. https://dc.etsu.edu/etd/1300.
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The secondary structures of ribonucleic acid (RNA) have been successfully modeled with graph-theoretic structures. Often, simple graphs are used to represent secondary RNA structures; however, in this research, a multigraph representation of RNA is used, in which vertices represent stems and edges represent the internal motifs. Any type of RNA secondary structure may be represented by a graph in this manner. We define novel graphical invariants to quantify the multigraphs and obtain characteristic descriptors of the secondary structures. These descriptors are used to train an artificial neural network (ANN) to recognize the characteristics of secondary RNA structure. Using the ANN, we classify the multigraphs as either RNA-like or not RNA-like. This classification method produced results similar to other classification methods. Given the expanding library of secondary RNA motifs, this method may provide a tool to help identify new structures and to guide the rational design of RNA molecules.
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Amiouny, Samir V. "Combinatorial mechanics." Diss., Georgia Institute of Technology, 1993. http://hdl.handle.net/1853/25576.
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Barnard, Kristen M. "Some Take-Away Games on Discrete Structures." UKnowledge, 2017. http://uknowledge.uky.edu/math_etds/44.
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The game of Subset Take-Away is an impartial combinatorial game posed by David Gale in 1974. The game can be played on various discrete structures, including but not limited to graphs, hypergraphs, polygonal complexes, and partially ordered sets. While a universal winning strategy has yet to be found, results have been found in certain cases. In 2003 R. Riehemann focused on Subset Take-Away on bipartite graphs and produced a complete game analysis by studying nim-values. In this work, we extend the notion of Take-Away on a bipartite graph to Take-Away on particular hypergraphs, namely oddly-uniform hypergraphs and evenly-uniform hypergraphs whose vertices satisfy a particular coloring condition. On both structures we provide a complete game analysis via nim-values. From here, we consider different discrete structures and slight variations of the rules for Take-Away to produce some interesting results. Under certain conditions, polygonal complexes exhibit a sequence of nim-values which are unbounded but have a well-behaved pattern. Under other conditions, the nim-value of a polygonal complex is bounded and predictable based on information about the complex itself. We introduce a Take-Away variant which we call “Take-As-Much-As-You-Want”, and we show that, again, nim-values can grow without bound, but fortunately they are very easily described for a given graph based on the total number of vertices and edges of the graph. Finally we consider Take-Away on a specific type of partially ordered set which we call a rank-complete poset. We have results, again via an analysis of nim-values, for Take-Away on a rank-complete poset for both ordinary play as well as misère play.
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Allen, Peter. "Finding combinatorial structures." Thesis, London School of Economics and Political Science (University of London), 2008. http://etheses.lse.ac.uk/60/.
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In this thesis we answer questions in two related areas of combinatorics: Ramsey theory and asymptotic enumeration. In Ramsey theory we introduce a new method for finding desired structures. We find a new upper bound on the Ramsey number of a path against a kth power of a path. Using our new method and this result we obtain a new upper bound on the Ramsey number of the kth power of a long cycle. As a corollary we show that, while graphs on n vertices with maximum degree k may in general have Ramsey numbers as large as ckn, if the stronger restriction that the bandwidth should be at most k is given, then the Ramsey numbers are bounded by the much smaller value. We go on to attack an old conjecture of Lehel: by using our new method we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new method replaces their use of the Regularity Lemma, and allows us to prove that for any n > 218000, whenever the edges of the complete graph on n vertices are two-coloured there exist disjoint monochromatic cycles covering all n vertices. In asymptotic enumeration we examine first the class of bipartite graphs with some forbidden induced subgraph H. We obtain some results for every H, with special focus on the cases where the growth speed of the class is factorial, and make some comments on a connection to clique-width. We then move on to a detailed discussion of 2-SAT functions. We find the correct asymptotic formula for the number of 2-SAT functions on n variables (an improvement on a result of Bollob´as, Brightwell and Leader [13], who found the dominant term in the exponent), the first error term for this formula, and some bounds on smaller error terms. Finally we obtain various expected values in the uniform model of random 2-SAT functions.
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Spiegel, Christoph. "Additive structures and randomness in combinatorics." Doctoral thesis, Universitat Politècnica de Catalunya, 2020. http://hdl.handle.net/10803/669327.
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Arithmetic Combinatorics, Combinatorial Number Theory, Structural Additive Theory and Additive Number Theory are just some of the terms used to describe the vast field that sits at the intersection of Number Theory and Combinatorics and which will be the focus of this thesis. Its contents are divided into two main parts, each containing several thematically related results.
The first part deals with the question under what circumstances solutions to arbitrary linear systems of equations usually occur in combinatorial structures..The properties we will be interested in studying in this part relate to the solutions to linear systems of equations.
A first question one might ask concerns the point at which sets of a given size will typically contain a solution. We will establish a threshold and also study the distribution of the number of solutions at that threshold, showing that it converges to a Poisson distribution in certain cases.
Next, Van der Waerden’s Theorem, stating that every finite coloring of the integers contains monochromatic arithmetic progression of arbitrary length, is by some considered to be the first result in Ramsey Theory. Rado generalized van der Waerden’s result by characterizing those linear systems whose solutions satisfy a similar property and Szemerédi strengthened it to a statement concerning density rather than colorings. We will turn our attention towards versions of Rado’s and Szemerédi’s Theorem in random sets, extending previous work of Friedgut, Rödl, Rucin´ski and Schacht in the case of the former and of Conlon, Gowers and Schacht for the latter to include a larger variety of systems and solutions.
Lastly, Chvátal and Erdo¿s suggested studying Maker-Breaker games. These games have deep connections to the theory of random structures and we will build on work of Bednarska and Luczak to establish the threshold for how much a large variety of games need to be biased in favor of the second player. These include games in which the first player wants to occupy a solution to some given linear system, generalizing the van der Waerden games introduced by Beck.
The second part deals with the extremal behavior of sets with interesting additive properties. In particular, we will be interested in bounds or structural descriptions for sets exhibiting some restrictions with regards to either their representation function or their sumset.
First, we will consider Sidon sets, that is sets of integers with pairwise unique differences. We will study a generalization of Sidon sets proposed very recently by Kohayakawa, Lee, Moreira and Rödl, where the pairwise differences are not just distinct, but in fact far apart by a certain measure. We will obtain strong lower bounds for such infinite sets using an approach of Cilleruelo. As a consequence of these bounds, we will also obtain the best current lower bound for Sidon sets in randomly generated infinite sets of integers of high density.
Next, one of the central results at the intersection of Combinatorics and Number Theory is the Freiman–Ruzsa Theorem stating that any finite set of integers of given doubling can be efficiently covered by a generalized arithmetic progression. In the case of particularly small doubling, more precise structural descriptions exist. We will first study results going beyond Freiman’s well-known 3k–4 Theorem in the integers. We will then see an application of these results to sets of small doubling in finite cyclic groups.
Lastly, we will turn our attention towards sets with near-constant representation functions. Erdo¿s and Fuchs established that representation functions of arbitrary sets of integers cannot be too close to being constant. We will first extend the result of Erdo¿s and Fuchs to ordered representation functions. We will then address a related question of Sárközy and Sós regarding weighted representation function.La combinatòria aritmètica, la teoria combinatòria dels nombres, la teoria additiva estructural i la teoria additiva de nombres són alguns dels termes que es fan servir per descriure una branca extensa i activa que es troba en la intersecció de la teoria de nombres i de la combinatòria, i que serà el motiu d'aquesta tesi doctoral. La primera part tracta la qüestió de sota quines circumstàncies es solen produir solucions a sistemes lineals d’equacions arbitràries en estructures additives. Una primera pregunta que s'estudia es refereix al punt en que conjunts d’una mida determinada contindran normalment una solució. Establirem un llindar i estudiarem també la distribució del nombre de solucions en aquest llindar, tot demostrant que en certs casos aquesta distribució convergeix a una distribució de Poisson. El següent tema de la tesis es relaciona amb el teorema de Van der Waerden, que afirma que cada coloració finita dels nombres enters conté una progressió aritmètica monocromàtica de longitud arbitrària. Aquest es considera el primer resultat en la teoria de Ramsey. Rado va generalitzar el resultat de van der Waerden tot caracteritzant en aquells sistemes lineals les solucions de les quals satisfan una propietat similar i Szemerédi la va reforçar amb una versió de densitat del resultat. Centrarem la nostra atenció cap a versions del teorema de Rado i Szemerédi en conjunts aleatoris, ampliant els treballs anteriors de Friedgut, Rödl, Rucinski i Schacht i de Conlon, Gowers i Schacht. Per últim, Chvátal i Erdos van suggerir estudiar estudiar jocs posicionals del tipus Maker-Breaker. Aquests jocs tenen una connexió profunda amb la teoria de les estructures aleatòries i ens basarem en el treball de Bednarska i Luczak per establir el llindar de la quantitat que necessitem per analitzar una gran varietat de jocs en favor del segon jugador. S'inclouen jocs en què el primer jugador vol ocupar una solució d'un sistema lineal d'equacions donat, generalitzant els jocs de van der Waerden introduïts per Beck. La segona part de la tesis tracta sobre el comportament extrem dels conjunts amb propietats additives interessants. Primer, considerarem els conjunts de Sidon, és a dir, conjunts d’enters amb diferències úniques quan es consideren parelles d'elements. Estudiarem una generalització dels conjunts de Sidons proposats recentment per Kohayakawa, Lee, Moreira i Rödl, en que les diferències entre parelles no són només diferents, sinó que, en realitat, estan allunyades una certa proporció en relació a l'element més gran. Obtindrem límits més baixos per a conjunts infinits que els obtinguts pels anteriors autors tot usant una construcció de conjunts de Sidon infinits deguda a Cilleruelo. Com a conseqüència d'aquests límits, obtindrem també el millor límit inferior actual per als conjunts de Sidon en conjunts infinits generats aleatòriament de nombres enters d'alta densitat. A continuació, un dels resultats centrals a la intersecció de la combinatòria i la teoria dels nombres és el teorema de Freiman-Ruzsa, que afirma que el conjunt suma d'un conjunt finit d’enters donats pot ser cobert de manera eficient per una progressió aritmètica generalitzada. En el cas de que el conjunt suma sigui de mida petita, existeixen descripcions estructurals més precises. Primer estudiarem els resultats que van més enllà del conegut teorema de Freiman 3k-4 en els enters. Llavors veurem una aplicació d’aquests resultats a conjunts de dobles petits en grups cíclics finits. Finalment, dirigirem l’atenció cap a conjunts amb funcions de representació gairebé constants. Erdos i Fuchs van establir que les funcions de representació de conjunts arbitraris d’enters no poden estar massa a prop de ser constants. Primer estendrem el resultat d’Erdos i Fuchs a funcions de representació ordenades. A continuació, abordarem una pregunta relacionada de Sárközy i Sós sobre funció de representació ponderada.
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Burris, Christina Suzann. "Analytic Combinatorics Applied to RNA Structures." Thesis, Virginia Tech, 2018. http://hdl.handle.net/10919/83888.
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In recent years it has been shown that the folding pattern of an RNA molecule plays an important role in its function, likened to a lock and key system. γ-structures are a subset of RNA pseudoknot structures filtered by topological genus that lend themselves nicely to combinatorial analysis. Namely, the coefficients of their generating function can be approximated for large n. This paper is an investigation into the length-spectrum of the longest block in random γ-structures. We prove that the expected length of the longest block is on the order n - O(n^1/2). We further compare these results with a similar analysis of the length-spectrum of rainbows in RNA secondary structures, found in Li and Reidys (2018). It turns out that the expected length of the longest block for γ-structures is on the order the same as the expected length of rainbows in secondary structures.Master of Science
Ribonucleic acid (RNA), similar in composition to well-known DNA, plays a myriad of roles within the cell. The major distinction between DNA and RNA is the nature of the nucleotide pairings. RNA is single stranded, to mean that its nucleotides are paired with one another (as opposed to a unique complementary strand). Consequently, RNA exhibits a knotted 3D structure. These diverse structures (folding patterns) have been shown to play important roles in RNA function, likened to a lock and key system. Given the cost of gathering data on folding patterns, little is known about exactly how structure and function are related. The work presented centers around building the mathematical framework of RNA structures in an effort to guide technology and further scientific discovery. We provide insight into the prevalence of certain important folding patterns.
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Roberts, Barnaby. "Structure and randomness in extremal combinatorics." Thesis, London School of Economics and Political Science (University of London), 2017. http://etheses.lse.ac.uk/3592/.
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In this thesis we prove several results in extremal combinatorics from areas including Ramsey theory, random graphs and graph saturation. We give a random graph analogue of the classical Andr´asfai, Erd˝os and S´os theorem showing that in some ways subgraphs of sparse random graphs typically behave in a somewhat similar way to dense graphs. In graph saturation we explore a ‘partite’ version of the standard graph saturation question, determining the minimum number of edges in H-saturated graphs that in some way resemble H themselves. We determine these values for K4, paths, and stars and determine the order of magnitude for all graphs. In Ramsey theory we give a construction from a modified random graph to solve a question of Conlon, determining the order of magnitude of the size-Ramsey numbers of powers of paths. We show that these numbers are linear. Using models from statistical physics we study the expected size of random matchings and independent sets in d-regular graphs. From this we give a new proof of a result of Kahn determining which d-regular graphs have the most independent sets. We also give the equivalent result for matchings which was previously unknown and use this to prove the Asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow and Markstrom. Using these methods we give an alternative proof of Shearer’s upper bound on off-diagonal Ramsey numbers.
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Patel, Viresh. "Partitions of combinatorial structures." Thesis, London School of Economics and Political Science (University of London), 2009. http://etheses.lse.ac.uk/3006/.
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In this thesis we explore extremal, structural, and algorithmic problems involving the partitioning of combinatorial structures. We begin by considering problems from the theory of graph cuts. It is well known that every graph has a cut containing at least half its edges. We conjecture that (except for one example), given any two graphs on the same vertex set, we can partition the vertices so that at least half the edges of each graph go across the partition. We give a simple algorithm that comes close to proving this conjecture. We also prove, using probabilistic methods, that the conjecture holds for certain classes of graphs. We consider an analogue of the graph cut problem for posets and determine which graph cut results carry over to posets. We consider both extremal and algorithmic questions, and in particular, we show that the analogous maxcut problem for posets is polynomial-time solvable in contrast to the maxcut problem for graphs, which is NP-complete. Another partitioning problem we consider is that of obtaining a regular partition (in the sense of the Szemeredi Regularity Lemma) for posets, where the partition respects the order of the poset. We prove the existence of such order-preserving, regular partitions for both the comparability graph and the covering graph of a poset, and go on to derive further properties of such partitions. We give a new proof of an old result of Frankl and Furedi, which characterises all 3-uniform hypergraphs for which every set of 4 vertices spans exactly 0 or 2 edges. We use our new proof to derive a corresponding stability result. We also look at questions concerning an analogue of the graph linear extension problem for posets.
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Dovgal, Sergey. "An interdisciplinary image of Analytic Combinatorics." Thesis, Paris 13, 2019. http://www.theses.fr/2019PA131065.
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Cette thèse est consacrée au développement des outils et à l’utilisation des méthodes de la combinatoire analytique, notamment l’énumération exacte et asymptotique, les propriétés statistiques des objets aléatoires et la génération aléatoire. L’ingrédient clé est la multidisciplinarité du domaine, qui est soulignée par des exemples tirés de la programmation logique, de la mécanique statistique, de la biologie, de la statistique mathématique, des réseaux et de la théorie des files d’attenteThis thesis is devoted to the development of tools and the use of methods from Analytic Combinatorics, including exact and asymptotic enumeration, statistical properties of random objects, and random generation.The key ingredient is the multidisciplinarity of the domain, which is emphasised by using examples from computational logic, statistical mechanics, biology, mathematical statistics, networks and queueing theory
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Da, Silva Pereira Luis Miguel. "Combinatorics of Singular Cardinals and PCF structures." Paris 7, 2007. http://www.theses.fr/2007PA077100.
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Ce travail est centré sur la théorie PCF de Shelah. Nous étudions la connexion entre la topologie des espaces PCF et les notions standard de la théorie PCF. Nous démontrons une généralisation du résultat qui dit que les espaces PCF séparables sont séquentiels et nous obtenons comme corollaire qu'il existe beaucoup de suites qui ont une vrai cofmalité modulo l'idéal des ensembles finis. Nous démontrons que ce corollaire est optimal. Nous donnons aussi une démonstration topologique d'estimatives cardinales obtenues précédemment par l'utilisation de la norme de Galvin-Hajnal. Nous étudions une conséquence de la négation de la conjecture PCF de Shelah appelé la Propriété des Ensembles Livres Approximables (PELA). Nous notons que la PELA est incompatible avec l'existence d'échelles continus en forme d'arbre et nous prouvons la consistance de ces échelles avec l'existence des plus grands des grands cardinaux établissant ainsi que la PELA n'est pas impliqué par les grands cardinaux. Nous étudions l'existence d'échelles continus en forme d'arbre et la négation de la PELA en plusieurs extensions de Prikry de l'universThis work is centered on Shelah's PCF theory. We study the connection between the topology of PCF spaces and standard PCF theory notions. We prove a generalization of the result that says that separable PCF spaces are sequential and obtain as a corollary that there exist many sequences that have true cofinality modulo the ideal of finite sets. We prove that this corollary is optimal. We also give a topological proof of cardinal estimates previously obtained through the use of the Galvin-Hajnal norm. We study a consequence of the negation of Shelah's PCF conjecture called the Approachable Free Subset Property (AFSP). We note that AFSP is incompatible with the existence of tree-like continuous scales and prove the consistency of these scales with the largest large cardinal axioms thus establishing that AFSP is not implied by large cardinals. We study the existence of tree-like continuous scales and the negation of AFSP in several Prikry extensions of the universe
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Hamel, Mariah. "Arithmetic structures in random sets." Thesis, University of British Columbia, 2008. http://hdl.handle.net/2429/2838.
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We prove various results in additive combinatorics for subsets of random sets. In particular we extend Sarkozy's theorem and a theorem of Green on long arithmetic progressions in sumsets to dense subsets of random sets with asymptotic density 0. Our proofs require a transference argument due to Green and Green-Tao which enables us to apply known results for sets of positive upper density to subsets of random sets which have positive relative density. We also prove a density result which states that if a subset of a random set has positive relative density, then the sumset of the subset must have positive upper density in the integers.
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McShine, Lisa Maria. "Random sampling of combinatorial structures." Diss., Georgia Institute of Technology, 2000. http://hdl.handle.net/1853/28771.
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Falgas-Ravry, Victor. "Thresholds in probabilistic and extremal combinatorics." Thesis, Queen Mary, University of London, 2012. http://qmro.qmul.ac.uk/xmlui/handle/123456789/8827.
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This thesis lies in the field of probabilistic and extremal combinatorics: we study discrete structures, with a focus on thresholds, when the behaviour of a structure changes from one mode into another. From a probabilistic perspective, we consider models for a random structure depending on some parameter. The questions we study are then: When (i.e. for what values of the parameter) does the probability of a given property go from being almost 0 to being almost 1? How do the models behave as this transition occurs? From an extremal perspective, we study classes of structures depending on some parameter. We are then interested in the following questions: When (for what value of the parameter) does a particular property become unavoidable? What do the extremal structures look like? The topics covered in this thesis are random geometric graphs, dependent percolation, extremal hypergraph theory and combinatorics in the hypercube.
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Okoth, Isaac Owino. "Combinatorics of oriented trees and tree-like structures." Thesis, Stellenbosch : Stellenbosch University, 2015. http://hdl.handle.net/10019.1/96860.
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Thesis (PhD)--Stellenbosch University, 2015.ENGLISH ABSTRACT : In this thesis, a number of combinatorial objects are enumerated. Du and Yin as well as Shin and Zeng (by a different approach) proved an elegant formula for the number of labelled trees with respect to a given in degree sequence, where each edge is oriented from a vertex of lower label towards a vertex of higher label. We refine their result to also take the number of sources (vertices of in degree 0) or sinks (vertices of out degree 0) into account. We find formulas for the mean and variance of the number of sinks or sources in these trees. We also obtain a differential equation and a functional equation satisfied by the generating function for these trees. Analogous results for labelled trees with two marked vertices, related to functional digraphs, are also established. We extend the work to count reachable vertices, sinks and leaf sinks in these trees. Among other results, we obtain a counting formula for the number of labelled trees on n vertices in which exactly k vertices are reachable from a given vertex v and also the average number of vertices that are reachable from a specified vertex in labelled trees of order n. In this dissertation, we also enumerate certain families of set partitions and related tree-like structures. We provide a proof for a formula that counts connected cycle-free families of k set partitions of {1, . . . , n} satisfying a certain coherence condition and then establish a bijection between these families and the set of labelled free k-ary cacti with a given vertex-degree distribution. We then show that the formula also counts coloured Husimi graphs in which there are no blocks of the same colour that are incident to one another. We extend the work to count coloured oriented cacti and coloured cacti. Noncrossing trees and related tree-like structures are also considered in this thesis. Specifically, we establish formulas for locally oriented noncrossing trees with a given number of sources and sinks, and also with given indegree and outdegree sequences. The work is extended to obtain the average number of reachable vertices in these trees. We then generalise the concept of noncrossing trees to find formulas for the number of noncrossing Husimi graphs, cacti and oriented cacti. The study is further extended to find formulas for the number of bicoloured noncrossing Husimi graphs and the number of noncrossing connected cycle-free pairs of set partitions.
AFRIKAANSE OPSOMMING : In hierdie tesis word ’n aantal kombinatoriese objekte geenumereer. Du en Yin asook Shin en Zeng (deur middel van ’n ander benadering) het ’n elegante formule vir die aantal geëtiketteerde bome met betrekking tot ’n gegewe ingangsgraadry, waar elke lyn van die nodus met die kleiner etiket na die nodus met die groter etiket toe georiënteer word. Ons verfyn hul resultaat deur ook die aantal bronne (nodusse met ingangsgraad 0) en putte (nodusse met uitgangsgraad 0) in ag te neem. Ons vind formules vir die gemiddelde en variansie van die aantal putte of bronne in hierdie bome. Ons bepaal verder ’n differensiaalvergelyking en ’n funksionaalvergelyking wat deur die voortbringende funksie van hierdie bome bevredig word. Analoë resultate vir geëtiketteerde bome met twee gemerkte nodusse (wat verwant is aan funksionele digrafieke), is ook gevind. Ons gaan verder voort deur ook bereikbare nodusse, bronne en putte in hierdie bome at te tel. Onder andere verkry ons ’n formule vir die aantal geëtiketteerde bome met n nodusse waarin presies k nodusse vanaf ’n gegewe nodus v bereikbaar is asook die gemiddelde aantal nodusse wat bereikbaar is vanaf ’n gegewe nodus. Ons enumereer in hierdie tesis verder sekere families van versamelingsverdelings en soortgelyke boom-vormige strukture. Ons gee ’n bewys vir ’n formule wat die aantal van samehangende siklus-vrye families van k versamelingsverdelings op {1, . . . , n} wat ’n sekere koherensie-vereiste bevredig, en ons beskryf ’n bijeksie tussen hierdie familie en die versameling van geëtiketteerde vrye k-êre kaktusse met ’n gegewe nodus-graad-verdeling. Ons toon ook dat hierdie formule ook gekleurde Husimi-grafieke tel waar blokke van dieselfde kleur nie insident met mekaar mag wees nie. Ons tel verder ook gekleurde georiënteerde kaktusse en gekleurde kaktusse. Nie-kruisende bome en soortgelyke boom-vormige strukture word in hierdie tesis ook beskou. On bepaal spesifiek formules vir lokaal georiënteerde nie-kruisende bome wat ’n gegewe aantal bronne en putte het asook nie-kruisende bome met gegewe ingangs- en uitgangsgraadrye. Ons gaan voort deur die gemiddelde aantal bereikbare nodusse in hierdie bome te bepaal. Ons veralgemeen dan die konsep van nie-kruisende bome en vind formules vir die aantal nie-kruisende Husimi-grafieke, kaktusse en georiënteerde kaktusse. Laastens vind ons ’n formule vir die aantaal tweegekleurde nie-kruisende Husimi-grafieke en die aantal nie-kruisende samehangende siklus-vrye pare van versamelingsverdelings.
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Delagrave, Simon. "Combinatorial structure and function studies." Thesis, Massachusetts Institute of Technology, 1995. http://hdl.handle.net/1721.1/38736.
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Fountoulakis, N. "Thresholds and the structure of sparse random graphs." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275502.
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In this thesis, we obtain approximations to the non-3-colourability threshold of sparse random graphs and we investigate the structure of random graphs near the region where the transition from 3-colourability to non-3-colourability seems to occur. It has been observed that, as for many other properties, the property of non-3-colourability of graphs exhibits a sharp threshold behaviour. It is conjectured that there exists a critical average degree such that when the average degree of a random graph is around this value the probability of the random graph being non-3-colourable changes rapidly from near 0 to near 1. The difficulty in calculating the critical value arises because the number of proper 3-colourings of a random graph is not concentrated: there is a `jackpot' effect. In order to reduce this effect, we focus on a sub-family of proper 3-colourings, which are called rigid 3-colourings. We give precise estimates for their expected number and we deduce that when the average degree of a random graph is bigger than 5, then the graph is asymptotically almost surely not 3-colourable. After that, we investigate the non-$k$-colourability of random regular graphs for any $k \geq 3$. Using a first moment argument, for each $k \geq 3$ we provide a bound so that whenever the degree of the random regular graph is bigger than this, then the random regular graph is asymptotically almost surely not $k$-colourable. Moreover, in a (failed!) attempt to show that almost all 5-regular graphs are not 3-colourable, we analyse the expected number of rigid 3-colourings of a random 5-regular graph. Motivated by the fact that the transition from 3-colourability to non-3-colourability occurs inside the subgraph of the random graph that is called the 3-core, we investigate the structure of this subgraph after its appearance. Indeed, we do this for the $k$-core, for any $k \geq 2$; and by extending existing techniques we obtain the asymptotic behaviour of the proportion of vertices of each fixed degree. Finally, we apply these results in order to obtain a more clear view of the structure of the 2-core (or simply the core) of a random graph after the emergence of its giant component. We determine the asymptotic distributions of the numbers of isolated cycles in the core as well as of those cycles that are not isolated there having any fixed length. Then we focus on its giant component, and in particular we give the asymptotic distributions of the numbers of 2-vertex and 2-edge-connected components.
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Limnios, Stratis. "Graph Degeneracy Studies for Advanced Learning Methods on Graphs and Theoretical Results Edge degeneracy: Algorithmic and structural results Degeneracy Hierarchy Generator and Efficient Connectivity Degeneracy Algorithm A Degeneracy Framework for Graph Similarity Hcore-Init: Neural Network Initialization based on Graph Degeneracy." Thesis, Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAX038.
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L'extraction de sous-structures significatives a toujours été un élément clé de l’étude des graphes. Dans le cadre de l'apprentissage automatique, supervisé ou non, ainsi que dans l'analyse théorique des graphes, trouver des décompositions spécifiques et des sous-graphes denses est primordial dans de nombreuses applications comme entre autres la biologie ou les réseaux sociaux.Dans cette thèse, nous cherchons à étudier la dégénérescence de graphe, en partant d'un point de vue théorique, et en nous appuyant sur nos résultats pour trouver les décompositions les plus adaptées aux tâches à accomplir. C'est pourquoi, dans la première partie de la thèse, nous travaillons sur des résultats structurels des graphes à arête-admissibilité bornée, prouvant que de tels graphes peuvent être reconstruits en agrégeant des graphes à degré d’arête quasi-borné. Nous fournissons également des garanties de complexité de calcul pour les différentes décompositions de la dégénérescence, c'est-à-dire si elles sont NP-complètes ou polynomiales, selon la longueur des chemins sur lesquels la dégénérescence donnée est définie.Dans la deuxième partie, nous unifions les cadres de dégénérescence et d'admissibilité en fonction du degré et de la connectivité. Dans ces cadres, nous choisissons les plus expressifs, d'une part, et les plus efficaces en termes de calcul d'autre part, à savoir la dégénérescence 1-arête-connectivité pour expérimenter des tâches de dégénérescence standard, telle que la recherche d’influenceurs.Suite aux résultats précédents qui se sont avérés peu performants, nous revenons à l'utilisation du k-core mais en l’intégrant dans un cadre supervisé, i.e. les noyaux de graphes. Ainsi, en fournissant un cadre général appelé core-kernel, nous utilisons la décomposition k-core comme étape de prétraitement pour le noyau et appliquons ce dernier sur chaque sous-graphe obtenu par la décomposition pour comparaison. Nous sommes en mesure d'obtenir des performances à l’état de l’art sur la classification des graphes au prix d’une légère augmentation du coût de calcul.Enfin, nous concevons un nouveau cadre de dégénérescence de degré s’appliquant simultanément pour les hypergraphes et les graphes biparties, dans la mesure où ces derniers sont les graphes d’incidence des hypergraphes. Cette décomposition est ensuite appliquée directement à des architectures de réseaux de neurones pré-entrainés étant donné qu'elles induisent des graphes biparties et utilisent le core d'appartenance des neurones pour réinitialiser les poids du réseaux. Cette méthode est non seulement plus performant que les techniques d'initialisation de l’état de l’art, mais il est également applicable à toute paire de couches de convolution et linéaires, et donc adaptable à tout type d'architectureExtracting Meaningful substructures from graphs has always been a key part in graph studies. In machine learning frameworks, supervised or unsupervised, as well as in theoretical graph analysis, finding dense subgraphs and specific decompositions is primordial in many social and biological applications among many others.In this thesis we aim at studying graph degeneracy, starting from a theoretical point of view, and building upon our results to find the most suited decompositions for the tasks at hand.Hence the first part of the thesis we work on structural results in graphs with bounded edge admissibility, proving that such graphs can be reconstructed by aggregating graphs with almost-bounded-edge-degree. We also provide computational complexity guarantees for the different degeneracy decompositions, i.e. if they are NP-complete or polynomial, depending on the length of the paths on which the given degeneracy is defined.In the second part we unify the degeneracy and admissibility frameworks based on degree and connectivity. Within those frameworks we pick the most expressive, on the one hand, and computationally efficient on the other hand, namely the 1-edge-connectivity degeneracy, to experiment on standard degeneracy tasks, such as finding influential spreaders.Following the previous results that proved to perform poorly we go back to using the k-core but plugging it in a supervised framework, i.e. graph kernels. Thus providing a general framework named core-kernel, we use the k-core decomposition as a preprocessing step for the kernel and apply the latter on every subgraph obtained by the decomposition for comparison. We are able to achieve state-of-the-art performance on graph classification for a small computational cost trade-off.Finally we design a novel degree degeneracy framework for hypergraphs and simultaneously on bipartite graphs as they are hypergraphs incidence graph. This decomposition is then applied directly to pretrained neural network architectures as they induce bipartite graphs and use the coreness of the neurons to re-initialize the neural network weights. This framework not only outperforms state-of-the-art initialization techniques but is also applicable to any pair of layers convolutional and linear thus being applicable however needed to any type of architecture
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Montejano, Cantoral Amanda. "Colored combinatorial structures: homomorphisms and counting." Doctoral thesis, Universitat Politècnica de Catalunya, 2009. http://hdl.handle.net/10803/32031.
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A partir del problema de los cuatro colores se inicia el estudio de coloraciones en grafos; teoría que ha existido ya por más de 150 años, y se ocupa del problema fundamental de partir un conjunto en clases de acuerdo con ciertas reglas. Con esta modesta base, dicha teoría se sitúa en un punto central de las matemáticas discretas con una gran cantidad de generalizaciones y aplicaciones contemporáneas. En esta tesis, centramos nuestro interés en dos áreas muy activas de investigación, que provienen de problemas en coloraciones: la teoría de Homomorfismos, y la teoría de Ramsey. La teoría de homomorfismos se ocupa del estudio de los morfismos naturales entre objetos pertenecientes a clases de estructuras combinatorias. El número cromático de un grafo simple G, se puede definir, en este contexto, como el orden mínimo de un grafo completo que admita un homomorfismo desde G. Por esta razón, la teoría de homomorfismos se ha estudiado extensamente como generalización de la teoría de coloraciones. Una excelente referencia en este tema es el libro de Hell y Nesetril {Graphs and homomorphisms, Oxf. Univ. Press, 2004}. La teoría de Ramsey estudia la existencia de ciertos patrones de color en estructuras coloreadas. A partir de los teoremas de Ramsey, Hilbert, Schur y van der Waerden, dicha teoría se ha desarrollado como una bella y amplia área de la combinatoria, donde se usan una gran variedad de técnicas provenientes de diversas ramas de la matemática, y cuyos resultados forman una parte muy importante de la teoría de grafos y la combinatoria en general. Una buena referencia en esta área es el libro de Langman y Robertson {Ramsey Theory on the Integers, Stud. Math. Lib. 24, AMS, 2003}. El trabajo en esta tesis está organizado en dos partes. La primera, se ocupa del estudio de homomorfismos de grafos mixtos coloreados, que son grafos con vértices unidos tanto por arcos coloreados como por aristas coloreadas. El número cromático de un grafo mixto coloreado G, se define como el mínimo orden do otro grafo mixto coloreado H, con la propiedad de que existe un homomorfismo (que preserva colores) de G en H. Estas nociones fueron introducidas por Nesetril y Raspaud en {Colored homomorphisms of colored mixed graphs, J. C. T. Ser. B 80 (2000)}. Generalizando algunos resultados de grafos orientados, estudiamos el número cromático mixto coloreado de las siguientes clases de grafos: caminos, árboles, grafos con número cromático acíclico acotado, k-árboles parciales, grafos planos, grafos outerplanos y grafos escasos en aristas. Motivados por la conjetura de la dicotomía en sistemas relacionales, nos interesamos en el estudio de la clases de grafos 2-coloreados en aristas y su relación con los grafos orientados. La segunda parte de la tesis se ubica dentro de la teoría de Ramsey aritmética. Estudiamos la existencia y enumeración de estructuras coloreadas, sobre todo monocromáticas y heterocromáticas, en coloraciones de grupos abelianos. Las estructuras que consideramos son soluciones de sistemas de ecuaciones en grupos, siendo los ejemplos más importantes las progresiones aritméticas y las ternas de Schur. En esta parte damos una descripción estructural de aquellas coloraciones en grupos abelianos que no contengan progresiones aritmeticas heterocromáticas de tamaño 3. Dicha descripción prueba una conjetura de Jungic et al. {Rainbow Ramsey Theory. Integers: E. J. C. N. T. 5(2) A9. (2005)} concerniente al tamaño de la clase cromática mas pequeña en dichas coloraciones de grupos cíclicos.Starting with the four color problem, the theory of graph coloring has existed for more than 150 years. It deals with the fundamental problem of partitioning a set of objects into classes according to certain rules. From this modest beginning, the theory has become central in discrete mathematics, with many contemporary generalizations and applications. In this thesis, our particular interest is in two very active areas of research which have emerged from coloring problems: Graph Homomorphism Theory and Arithmetic Ramsey Theory. Graph Homomorphism Theory can be described as the study of classes of combinatorial structures under natural morphisms. The chromatic number of a simple graph G can be stated, in this context, as the smallest complete graph to which G admits a homomorphism. Thus Graph Homomorphism Theory has been extensively studied as a generalization of colorings. An excellent reference in the subject is the book by Hell and Nesetril {Graphs and homomorphisms, Oxf. Univ. Press, 2004}. Ramsey Theory studies the existence of particular color patterns in colored structures. Starting with the Theorems of Ramsey, Hilbert, Schur and van der Waerden, the theory has developed as a wide and beautiful area of combinatorics, in which a great variety of techniques are used from many branches of mathematics. Many of the classical results in the area are arithmetic versions of the theory and we are interested in this particular branch of Ramsey Theory. A good reference in the area is the book of Langman and Robertson {Ramsey Theory on the Integers, Stud. Math. Lib. 24, AMS, 2003}. This thesis is organized in two parts. The first part deals with the study of homomorphisms in the class of colored mixed graphs, which are graphs with vertices linked by both colored arcs and colored edges. The chromatic number of such a graph G is defined as the smallest order of a colored mixed graph H such that there exists a (color preserving) homomorphism from G to H. These notions were introduced by Nesetril and Raspaud in {Colored homomorphisms of colored mixed graphs, J. C. T. Ser. B 80 (2000)}. Generalizing known results for the class of oriented graphs we study the colored mixed chromatic number of paths, trees, graphs with bounded acyclic chromatic number, graphs of bounded treewidth, planar graphs, outerplanar graphs and sparse graphs. In particular we give the exact chormatic number of planar graphs and of partial 2-trees with appropriately large girth. Motivated by the dichotomy conjecture for relational structures we focuss on the class of 2-edge colored graphs and study its relationship with the class of oriented graphs. In particular we consider the characterization of cores and of duality pairs in this class. The second part of the thesis is related to Arithmetic Ramsey Theory. We consider the existence and the enumeration of colored structures, mainly monochromatic or rainbow structures, in colorings of finite groups. The structures under consideration can be described as solutions of systems of equations in the group, the main examples being arithmetic progressions and Schur triples. We give a structural description of those colorings in abelian groups which do not contain 3-term arithmetic progressions with its members having pairwise distinct colors. This structural description proves a conjecture of Jungic et al. {Rainbow Ramsey Theory. Integers: E. J. C. N. T. 5(2) A9. (2005)} on the size of the smallest chromatic class of such colorings in cyclic groups.
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Stokes, Klara. "Combinatorial structures for anonymous database search." Doctoral thesis, Universitat Rovira i Virgili, 2011. http://hdl.handle.net/10803/52799.
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This thesis treats a protocol for anonymous database search (or if one prefer, a protocol for user-private information retrieval), that is based on the use of combinatorial configurations. The protocol is called P2P UPIR. It is proved that the (v,k,1)-balanced incomplete block designs (BIBD) and in particular the finite projective planes are optimal configurations for this protocol. The notion of n-anonymity is applied to the configurations for P2P UPIR protocol and the transversal designs are proved to be n-anonymous configurations for P2P UPIR, with respect to the neighborhood points of the points of the configuration. It is proved that to the configurable tuples one can associate a numerical semigroup. This theorem implies results on existence of combinatorial configurations. The proofs are constructive and can be used as algorithms for finding combinatorial configurations. It is also proved that to the triangle-free configurable tuples one can associate a numerical semigroup. This implies results on existence of triangle-free combinatorial configurations.
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Rattan, Amarpreet. "Parking Functions and Related Combinatorial Structures." Thesis, University of Waterloo, 2001. http://hdl.handle.net/10012/1028.
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The central topic of this thesis is parking functions. We give a survey of some of the current literature concerning parking functions and focus on their interaction with other combinatorial objects; namely noncrossing partitions, hyperplane arrangements and tree inversions. In the final chapter, we discuss generalizations of both parking functions and the above structures.
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Tyomkyn, Mykhaylo. "Packings and embeddings of combinatorial structures." Thesis, University of Cambridge, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.609410.
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Goldberg, Leslie Ann. "Efficient algorithms for listing combinatorial structures." Thesis, University of Edinburgh, 1991. http://hdl.handle.net/1842/10917.
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This thesis studies the problem of designing efficient algorithms for listing combinatorial structures. The main notion of efficiency that we use is due to Johnson, Yannakakis, and Papadimitriou. It is called polynomial delay. A listing algorithm is said to have delay d if and only if it satisfies the following conditions whenever it is run with any input p: 1. It executes at most d(p) machine instructions before either producing the first output or halting. 2. After any output it executes at most d(p) machine instructions before either producing the next output or halting. An algorithm is said to have polynomial delay if its delay is bounded from above by a polynomial in the length of the input. In the thesis we also define a weaker notion of efficiency which we call cumulative polynomial delay. There are some families of combinatorial structures for which it is easy to design a polynomial delay listing algorithm. For example, it is easy to design a polynomial delay algorithm that takes as input a unary integer n and lists all n-vertex graphs. In this thesis we focus on more difficult problems such as the following. Problem 1 - Listing unlabeled graphs - Design a polynomial delay algorithm that takes as input a unary integer n and lists exactly one representative from each isomorphism class in the set of n-vertex graphs. Problem 2 - Listing Hamiltonian graphs - Design a polynomial delay algorithm that takes as input a unary integer n and lists all Hamiltonian n-vertex graphs. We start the thesis by developing general methods for solving listing problems such as 1 and 2. Then we apply the methods to specific combinatorial families obtaining various listing algorithms including the following. 1. A polynomial space polynomial delay listing algorithm for unlabeled graphs 2. A polynomial space polynomial delay listing algorithm for any first order one property* 3. A polynomial delay listing algorithm for Hamiltonian graphs 4. A polynomial space polynomial delay listing algorithm for graphs with cliques of specified sizes 5. A polynomial space cumulative polynomial delay listing algorithm for k-colorable graphs. * A first order graph property is called a one property if and only if it is the case that almost every graph has the property.
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Atminas, Aistis. "Well-quasi-ordering of combinatorial structures." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67023/.
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In this work we study the notion of well-quasi-ordering for various partial orders and its relation to some other notions, such as clique-width. In particular, we prove decidability of well-quasi-ordering for factorial languages, subquadratic properties of graphs and classes of graphs with finite distinguishing number. In addition, we reveal some new classes of graphs and permutations which are or are not well-quasi-ordered. We also prove that subquadratic properties or classes of graphs with finite distinguishing number that are well-quasi-ordered have bounded clique-width and we identify two new minimal classes of graphs of unbounded clique-width.
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Little, David P. "Q-enumeration of classical combinatorial structures /." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2000. http://wwwlib.umi.com/cr/ucsd/fullcit?p9989758.
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Emtander, Eric. "Chordal and Complete Structures in Combinatorics and Commutative Algebra." Doctoral thesis, Stockholms universitet, Matematiska institutionen, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-48241.
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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.
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Mustafa, Nabil. "Approximations of Points: Combinatorics and Algorithms." Habilitation à diriger des recherches, Université Paris-Est, 2013. http://tel.archives-ouvertes.fr/tel-01062825.
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At the core of successful manipulation and computation over large geometric data is the notion of approximation, both structural and computational. The focus of this thesis will be on the combinatorial and algorithmic aspects of approximations of point-set data P in d-dimensional Euclidean space. It starts with a study of geometric data depth where the goal is to compute a point which is the 'combinatorial center' of P. Over the past 50 years several such measures of combinatorial centers have been proposed, and we will re-examine several of them: Tukey depth, Simplicial depth, Oja depth and Ray-Shooting depth. This can be generalized to approximations with a subset, leading to the notion of epsilon-nets. There we will study the problem of approximations with respect to convexity. Along the way, this requires re-visiting and generalizing some basic theorems of convex geometry, such as the Caratheodory's theorem. Finally we will turn to the algorithmic aspects of these problems. We present a polynomial-time approximation scheme for computing hitting-sets for disks in the plane. Of separate interest is the technique, an analysis of local-search via locality graphs. A further application of this technique is then presented in computing independent sets in intersection graphs of rectangles in the plane.
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Leung, Yiu-cho. "Counting combinatorial structures in recursively constructible graphs /." View abstract or full-text, 2007. http://library.ust.hk/cgi/db/thesis.pl?CSED%202007%20LEUNG.
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Quinn, Kathleen Anne Sara. "Combinatorial structures with applications to information theory." Thesis, Royal Holloway, University of London, 1991. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.261791.
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Gupta, Swati Ph D. Massachusetts Institute of Technology. "Combinatorial structures in online and convex optimization." Thesis, Massachusetts Institute of Technology, 2017. http://hdl.handle.net/1721.1/112014.
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Thesis: Ph. D., Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2017.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Cataloged from student-submitted PDF version of thesis.
Includes bibliographical references (pages 157-163).
Motivated by bottlenecks in algorithms across online and convex optimization, we consider three fundamental questions over combinatorial polytopes. First, we study the minimization of separable strictly convex functions over polyhedra. This problem is motivated by first-order optimization methods whose bottleneck relies on the minimization of a (often) separable, convex metric, known as the Bregman divergence. We provide a conceptually simple algorithm, Inc-Fix, in the case of submodular base polyhedra. For cardinality-based submodular polytopes, we show that Inc-Fix can be speeded up to be the state-of-the-art method for minimizing uniform divergences. We show that the running time of Inc-Fix is independent of the convexity parameters of the objective function. The second question is concerned with the complexity of the parametric line search problem in the extended submodular polytope P: starting from a point inside P, how far can one move along a given direction while maintaining feasibility. This problem arises as a bottleneck in many algorithmic applications like the above-mentioned Inc-Fix algorithm and variants of the Frank-Wolfe method. One of the most natural approaches is to use the discrete Newton's method, however, no upper bound on the number of iterations for this method was known. We show a quadratic bound resulting in a factor of n6 reduction in the worst-case running time from the previous state-of-the-art. The analysis leads to interesting extremal questions on set systems and submodular functions. Next, we develop a general framework to simulate the well-known multiplicative weights update algorithm for online linear optimization over combinatorial strategies U in time polynomial in log /U/, using efficient approximate general counting oracles. We further show that efficient counting over the vertex set of any 0/1 polytope P implies efficient convex minimization over P. As a byproduct of this result, we can approximately decompose any point in a 0/1 polytope into a product distribution over its vertices. Finally, we compare the applicability and limitations of the above results in the context of finding Nash-equilibria in combinatorial two-player zero-sum games with bilinear loss functions. We prove structural results that can be used to find certain Nash-equilibria with a single separable convex minimization.
by Swati Gupta.
Ph. D.
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Collins, Jarred T. "Moore - Greig designs - a new combinatorial structure /." View online ; access limited to URI, 2005. http://0-wwwlib.umi.com.helin.uri.edu/dissertations/dlnow/3186900.
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Li, Quan Ph D. Massachusetts Institute of Technology. "Algorithms and algorithmic obstacles for probabilistic combinatorial structures." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/115765.
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Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 209-214).
We study efficient average-case (approximation) algorithms for combinatorial optimization problems, as well as explore the algorithmic obstacles for a variety of discrete optimization problems arising in the theory of random graphs, statistics and machine learning. In particular, we consider the average-case optimization for three NP-hard combinatorial optimization problems: Large Submatrix Selection, Maximum Cut (Max-Cut) of a graph and Matrix Completion. The Large Submatrix Selection problem is to find a k x k submatrix of an n x n matrix with i.i.d. standard Gaussian entries, which has the largest average entry. It was shown in [13] using non-constructive methods that the largest average value of a k x k submatrix is 2(1 + o(1) [square root] log n/k with high probability (w.h.p.) when k = O(log n/ log log n). We show that a natural greedy algorithm called Largest Average Submatrix LAS produces a submatrix with average value (1+ o(1)) [square root] 2 log n/k w.h.p. when k is constant and n grows, namely approximately [square root] 2 smaller. Then by drawing an analogy with the problem of finding cliques in random graphs, we propose a simple greedy algorithm which produces a k x k matrix with asymptotically the same average value (1+o(1) [square root] 2log n/k w.h.p., for k = o(log n). Since the maximum clique problem is a special case of the largest submatrix problem and the greedy algorithm is the best known algorithm for finding cliques in random graphs, it is tempting to believe that beating the factor [square root] 2 performance gap suffered by both algorithms might be very challenging. Surprisingly, we show the existence of a very simple algorithm which produces a k x k matrix with average value (1 + o[subscript]k(1) + o(1))(4/3) [square root] 2log n/k for k = o((log n)¹.⁵), that is, with asymptotic factor 4/3 when k grows. To get an insight into the algorithmic hardness of this problem, and motivated by methods originating in the theory of spin glasses, we conduct the so-called expected overlap analysis of matrices with average value asymptotically (1 + o(1))[alpha][square root] 2 log n/k for a fixed value [alpha] [epsilon] [1, fixed value a E [1, [square root]2]. The overlap corresponds to the number of common rows and common columns for pairs of matrices achieving this value. We discover numerically an intriguing phase transition at [alpha]* [delta]= 5[square root]2/(3[square root]3) ~~ 1.3608.. [epsilon] [4/3, [square root]2]: when [alpha] < [alpha]* the space of overlaps is a continuous subset of [0, 1]², whereas [alpha] = [alpha]* marks the onset of discontinuity, and as a result the model exhibits the Overlap Gap Property (OGP) when [alpha] > [alpha]*, appropriately defined. We conjecture that OGP observed for [alpha] > [alpha]* also marks the onset of the algorithmic hardness - no polynomial time algorithm exists for finding matrices with average value at least (1+o(1)[alpha][square root]2log n/k, when [alpha] > [alpha]* and k is a growing function of n. Finding a maximum cut of a graph is a well-known canonical NP-hard problem. We consider the problem of estimating the size of a maximum cut in a random Erdős-Rényi graph on n nodes and [cn] edges. We establish that the size of the maximum cut normalized by the number of nodes belongs to the interval [c/2 + 0.47523[square root]c,c/2 + 0.55909[square root]c] w.h.p. as n increases, for all sufficiently large c. We observe that every maximum size cut satisfies a certain local optimality property, and we compute the expected number of cuts with a given value satisfying this local optimality property. Estimating this expectation amounts to solving a rather involved multi-dimensional large deviations problem. We solve this underlying large deviation problem asymptotically as c increases and use it to obtain an improved upper bound on the Max-Cut value. The lower bound is obtained by application of the second moment method, coupled with the same local optimality constraint, and is shown to work up to the stated lower bound value c/2 + 0.47523[square root]c. We also obtain an improved lower bound of 1.36000n on the Max-Cut for the random cubic graph or any cubic graph with large girth, improving the previous best bound of 1.33773n. Matrix Completion is the problem of reconstructing a rank-k n x n matrix M from a sampling of its entries. We propose a new matrix completion algorithm using a novel sampling scheme based on a union of independent sparse random regular bipartite graphs. We show that under a certain incoherence assumption on M and for the case when both the rank and the condition number of M are bounded, w.h.p. our algorithm recovers an [epsilon]-approximation of M in terms of the Frobenius norm using O(nlog² (1/[epsilon])) samples and in linear time O(nlog² (1/[epsilon])). This provides the best known bounds both on the sample complexity and computational cost for reconstructing (approximately) an unknown low-rank matrix. The novelty of our algorithm is two new steps of thresholding singular values and rescaling singular vectors in the application of the "vanilla" alternating minimization algorithm. The structure of sparse random regular graphs is used heavily for controlling the impact of these regularization steps.
by Quan Li.
Ph. D.
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Sinclair, Alistair John. "Randomised algorithms for counting and generating combinatorial structures." Thesis, University of Edinburgh, 1988. http://hdl.handle.net/1842/11392.
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Sobkowiak, Jessica. "Some combinatorial structures constructed from modular Leonard triples." [Tampa, Fla] : University of South Florida, 2009. http://purl.fcla.edu/usf/dc/et/SFE0002978.
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ÓMáille, Paul E. "Combinatorial protein engineering by structure-based gene shuffling /." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1486572165277805.
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Gay, Joël. "Representation of Monoids and Lattice Structures in the Combinatorics of Weyl Groups." Thesis, Université Paris-Saclay (ComUE), 2018. http://www.theses.fr/2018SACLS209/document.
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La combinatoire algébrique est le champ de recherche qui utilise des méthodes combinatoires et des algorithmes pour étudier les problèmes algébriques, et applique ensuite des outils algébriques à ces problèmes combinatoires. L’un des thèmes centraux de la combinatoire algébrique est l’étude des permutations car elles peuvent être interprétées de bien des manières (en tant que bijections, matrices de permutations, mais aussi mots sur des entiers, ordre totaux sur des entiers, sommets du permutaèdre…). Cette riche diversité de perspectives conduit alors aux généralisations suivantes du groupe symétrique. Sur le plan géométrique, le groupe symétrique engendré par les transpositions élémentaires est l’exemple canonique des groupes de réflexions finis, également appelés groupes de Coxeter. Sur le plan monoïdal, ces même transpositions élémentaires deviennent les opérateurs du tri par bulles et engendrent le monoïde de 0-Hecke, dont l’algèbre est la spécialisation à q=0 de la q-déformation du groupe symétrique introduite par Iwahori. Cette thèse se consacre à deux autres généralisations des permutations. Dans la première partie de cette thèse, nous nous concentrons sur les matrices de permutations partielles, en d’autres termes les placements de tours ne s’attaquant pas deux à deux sur un échiquier carré. Ces placements de tours engendrent le monoïde de placements de tours, une généralisation du groupe symétrique. Dans cette thèse nous introduisons et étudions le 0-monoïde de placements de tours comme une généralisation du monoïde de 0-Hecke. Son algèbre est la dégénérescence à q=0 de la q-déformation du monoïde de placements de tours introduite par Solomon. On étudie par la suite les propriétés monoïdales fondamentales du 0-monoïde de placements de tours (ordres de Green, propriété de treillis du R-ordre, J-trivialité) ce qui nous permet de décrire sa théorie des représentations (modules simples et projectifs, projectivité sur le monoïde de 0-Hecke, restriction et induction le long d’une fonction d’inclusion).Les monoïdes de placements de tours sont en fait l’instance en type A de la famille des monoïdes de Renner, définis comme les complétés des groupes de Weyl (c’est-à-dire les groupes de Coxeter cristallographiques) pour la topologie de Zariski. Dès lors, dans la seconde partie de la thèse nous étendons nos résultats du type A afin de définir les monoïdes de 0-Renner en type B et D et d’en donner une présentation. Ceci nous conduit également à une présentation des monoïdes de Renner en type B et D, corrigeant ainsi une présentation erronée se trouvant dans la littérature depuis une dizaine d’années. Par la suite, nous étudions comme en type A les propriétés monoïdales de ces nouveaux monoïdes de 0-Renner de type B et D : ils restent J-triviaux, mais leur R-ordre n’est plus un treillis. Cela ne nous empêche pas d’étudier leur théorie des représentations, ainsi que la restriction des modules projectifs sur le monoïde de 0-Hecke qui leur est associé. Enfin, la dernière partie de la thèse traite de différentes généralisations des permutations. Dans une récente séries d’articles, Châtel, Pilaud et Pons revisitent la combinatoire algébrique des permutations (ordre faible, algèbre de Hopf de Malvenuto-Reutenauer) en terme de combinatoire sur les ordres partiels sur les entiers. Cette perspective englobe également la combinatoire des quotients de l’ordre faible tels les arbres binaires, les séquences binaires, et de façon plus générale les récents permutarbres de Pilaud et Pons. Nous généralisons alors l’ordre faibles aux éléments des groupes de Weyl. Ceci nous conduit à décrire un ordre sur les sommets des permutaèdres, associaèdres généralisés et cubes dans le même cadre unifié. Ces résultats se basent sur de subtiles propriétés des sommes de racines dans les groupes de Weyl qui s’avèrent ne pas fonctionner pour les groupes de Coxeter qui ne sont pas cristallographiquesAlgebraic combinatorics is the research field that uses combinatorial methods and algorithms to study algebraic computation, and applies algebraic tools to combinatorial problems. One of the central topics of algebraic combinatorics is the study of permutations, interpreted in many different ways (as bijections, permutation matrices, words over integers, total orders on integers, vertices of the permutahedron…). This rich diversity of perspectives leads to the following generalizations of the symmetric group. On the geometric side, the symmetric group generated by simple transpositions is the canonical example of finite reflection groups, also called Coxeter groups. On the monoidal side, the simple transpositions become bubble sort operators that generate the 0-Hecke monoid, whose algebra is the specialization at q=0 of Iwahori’s q-deformation of the symmetric group. This thesis deals with two further generalizations of permutations. In the first part of this thesis, we first focus on partial permutations matrices, that is placements of pairwise non attacking rooks on a n by n chessboard, simply called rooks. Rooks generate the rook monoid, a generalization of the symmetric group. In this thesis we introduce and study the 0-Rook monoid, a generalization of the 0-Hecke monoid. Its algebra is a proper degeneracy at q = 0 of the q-deformed rook monoid of Solomon. We study fundamental monoidal properties of the 0-rook monoid (Green orders, lattice property of the R-order, J-triviality) which allow us to describe its representation theory (simple and projective modules, projectivity on the 0-Hecke monoid, restriction and induction along an inclusion map).Rook monoids are actually type A instances of the family of Renner monoids, which are completions of the Weyl groups (crystallographic Coxeter groups) for Zariski’s topology. In the second part of this thesis we extend our type A results to define and give a presentation of 0-Renner monoids in type B and D. This also leads to a presentation of the Renner monoids of type B and D, correcting a misleading presentation that appeared earlier in the litterature. As in type A we study the monoidal properties of the 0-Renner monoids of type B and D : they are still J-trivial but their R-order are not lattices anymore. We study nonetheless their representation theory and the restriction of projective modules over the corresponding 0-Hecke monoids. The third part of this thesis deals with different generalizations of permutations. In a recent series of papers, Châtel, Pilaud and Pons revisit the algebraic combinatorics of permutations (weak order, Malvenuto-Reutenauer Hopf algebra) in terms of the combinatorics of integer posets. This perspective encompasses as well the combinatorics of quotients of the weak order such as binary trees, binary sequences, and more generally the recent permutrees of Pilaud and Pons. We generalize the weak order on the elements of the Weyl groups. This enables us to describe the order on vertices of the permutahedra, generalized associahedra and cubes in the same unified context. These results are based on subtle properties of sums of roots in Weyl groups, and actually fail for non-crystallographic Coxeter groups
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Perarnau, Llobet Guillem. "Random combinatorial structures with low dependencies : existence and enumeration." Doctoral thesis, Universitat Politècnica de Catalunya, 2013. http://hdl.handle.net/10803/362940.
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En aquesta tesi s'estudien diferents problemes en el camp de la combinatòria i la teoria de grafs, utilitzant el mètode probabilístic. Aquesta tècnica, introduïda per Erdős , ha esdevingut una eina molt potent per tal de donar proves existencials per certs problemes en diferents camps de les matemàtiques on altres mètodes no ho han aconseguit.
Un dels seus principals objectius és l'estudi del comportament de les variables aleatòries. El cas en que aquestes variables compten el nombre d'esdeveniments dolents que tenen lloc en una estructura combinatòria és de particular interès. La idea del Paradigma de Poisson és estimar la probabilitat que tots aquests esdeveniments dolents no succeeixin a la vegada, quan les dependències entre ells són febles o escasses. En tal cas, aquesta probabilitat s'hauria de comportar de forma similar al cas on tots els esdeveniments són independents. El Lema Local de Lovász o la Desigualtat de Suen són exemples d'aquesta idea.
L'objectiu de la tesi és estudiar aquestes tècniques ja sigui proveint-ne noves versions, refinant-ne les existents per casos particulars o donant-ne noves aplicacions. A continuació s'enumeren les principals contribucions de la tesi.
La primera part d'aquesta tesi estén un resultat d' Erdős i Spencer sobre transversals llatins. Els autors proven que qualsevol matriu d'enters on cap nombre apareix massa vegades, admet un transversal on tots els nombres són diferents. Això equival a estudiar els aparellaments multicolors en aresta-coloracions de grafs complets bipartits. Sota les mateixes hipòtesis que, es donen resultats sobre el nombre d'aquests aparellaments. Les tècniques que s'utilitzen estan basades en l'estratègia desenvolupada per Lu i Székely.
En la segona part d'aquesta tesi s'estudien els codis identificadors. Un codi identificador és un conjunt de vèrtexs tal que tots els vèrtexs del graf tenen un veïnatge diferent en el codi. Aquí s'estableixen cotes en la mida d'un codi identificador mínim en funció dels graus i es resol parcialment una conjectura de Foucaud et al.. En un altre capítol, es mostra que qualsevol graf suficientment dens conté un subgraf que admet un codi identificador òptim.
En alguns casos, provar l'existència d'un cert objecte és trivial. Tot i així, es poden utilitzar les mateixes tècniques per obtenir resultats d'enumeració. L'estudi de patrons en permutacions n'és un bon exemple. A la tercera part de la tesi es desenvolupa una nova tècnica per tal d'estimar el nombre de permutacions d'una certa llargada que eviten còpies consecutives d'un patró donat.
En particular, es donen cotes inferiors i superiors per a aquest nombre. Una de les conseqüències és la prova de la conjectura CMP enunciada per Elizalde i Noy així com nous resultats en el comportament de la majoria dels patrons.
En l'última part de la tesi s'estudia la Conjectura Lonely Runner, enunciada independentment per Wills i Cusick i que té múltiples aplicacions en diferents camps de les matemàtiques. Aquesta coneguda conjectura diu que per qualsevol conjunt de corredors que corren al llarg d'un cercle unitari, hi ha un moment on tots els corredors estan suficientment lluny de l'origen. Aquí, es millora un resultat de Chen ampliant la distància de tots els corredors a l'origen. També s'estén el teorema del corredor invisible de Czerwiński i Grytczuk .In this thesis we study different problems in combinatorics and in graph theory by means of the probabilistic method. This method, introduced by Erdös, has become an extremely powerful tool to provide existential proofs for certain problems in different mathematical branches where other methods had failed utterly. One of its main concerns is to study the behavior of random variables. In particular, one common situation arises when these random variables count the number of bad events that occur in a combinatorial structure. The idea of the Poisson Paradigm is to estimate the probability of these bad events not happening at the same time when the dependencies among them are weak or rare. If this is the case, this probability should behave similarly as in the case where all the events are mutually independent. This idea gets reflected in several well-known tools, such as the Lovász Local Lemma or Suen inequality. The goal of this thesis is to study these techniques by setting new versions or refining the existing ones for particular cases, as well as providing new applications of them for different problems in combinatorics and graph theory. Next, we enumerate the main contributions of this thesis. The first part of this thesis extends a result of Erdös and Spencer on latin transversals [1]. They showed that an integer matrix such that no number appears many times, admits a latin transversal. This is equivalent to study rainbow matchings of edge-colored complete bipartite graphs. Under the same hypothesis of, we provide enumerating results on such rainbow matchings. The second part of the thesis deals with identifying codes, a set of vertices such that all vertices in the graph have distinct neighborhood within the code. We provide bounds on the size of a minimal identifying code in terms of the degree parameters and partially answer a question of Foucaud et al. On a different chapter of the thesis, we show that any dense enough graph has a very large spanning subgraph that admits a small identifying code. In some cases, proving the existence of a certain object is trivial. However, the same techniques allow us to obtain enumerative results. The study of permutation patterns is a good example of that. In the third part of the thesis we devise a new approach in order to estimate how many permutations of given length avoid a consecutive copy of a given pattern. In particular, we provide upper and lower bounds for them. One of the consequences derived from our approach is a proof of the CMP conjecture, stated by Elizalde and Noy as well as some new results on the behavior of most of the patterns. In the last part of this thesis, we focus on the Lonely Runner Conjecture, posed independently by Wills and Cusick and that has multiple applications in different mathematical fields. This well-known conjecture states that for any set of runners running along the unit circle with constant different speeds and starting at the same point, there is a moment where all of them are far enough from the origin. We improve the result of Chen on the gap of loneliness by studying the time when two runners are close to the origin. We also show an invisible runner type result, extending a result of Czerwinski and Grytczuk.
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Asim, Muhammad Ahsan. "Network Testing in a Testbed Simulator using Combinatorial Structures." Thesis, Blekinge Tekniska Högskola, Avdelningen för för interaktion och systemdesign, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:bth-6058.
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This report covers one of the most demanding issues of network users i.e. network testing. Network testing in this study is about performance evaluation of networks, by putting traffic load gradually to determine the queuing delay for different traffics. Testing of such operations is becoming complex and necessary due to use of real time applications such as voice and video traffic, parallel to elastic data of ordinary applications over WAN links. Huge size elastic data occupies almost 80% resources and causes delay for time sensitive traffic. Performance parameters like service outage, delay, packet loss and jitter are tested to assure the reliability factor of provided Quality of Service (QoS) in the Service Level Agreements (SLAs). Normally these network services are tested after deployment of physical networks. In this case most of the time customers have to experience unavailability (outage) of network services due to increased levels of load and stress. According to user-centric point of view these outages are violation and must be avoided by the net-centric end. In order to meet these challenges network SLAs are tested on simulators in lab environment. This study provides a solution for this problem in a form of testbed simulator named Combinatorial TestBed Simulator (CTBS). Prototype of this simulator is developed for conducting experiment. It provides a systematic approach of combinatorial structures for finding such traffic patterns that exceeds the limit of queuing delay, committed in SLAs. Combinatorics is a branch of mathematics that deals with discrete and normally finite elements. In the design of CTBS, technique of combinatorics is used to generate a variety of test data that cannot be generated manually for testing the given network scenario. To validate the design of CTBS, results obtained by pilot runs are compared with the results calculated using timeline. After validation of CTBS design, actual experiment is conducted to determine the set of traffic patterns that exceeds the threshold value of queuing delay for Voice over Internet Protocol (VOIP) traffic.14:36 Folkparksvagan Ronneby 372 40 Sweden
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Henderson, Roger William. "Cryptanalysis of braid group cryptosystem and related combinatorial structures." Thesis, Royal Holloway, University of London, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.440519.
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Jones, Sian K. "On the enumeration of sudoku and similar combinatorial structures." Thesis, University of South Wales, 2011. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.541662.
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Conner, Andrew Brondos 1981. "A(infinity)-structures, generalized Koszul properties, and combinatorial topology." Thesis, University of Oregon, 2011. http://hdl.handle.net/1794/11559.
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x, 68 p. : ill. (some col.)Motivated by the Adams spectral sequence for computing stable homotopy groups, Priddy defined a class of algebras called Koszul algebras with nice homological properties. Many important algebras arising naturally in mathematics are Koszul, and the Koszul property is often tied to important structure in the settings which produced the algebras. However, the strong defining conditions for a Koszul algebra imply that such algebras must be quadratic. A very natural generalization of Koszul algebras called K 2 algebras was recently introduced by Cassidy and Shelton. Unlike other generalizations of the Koszul property, the class of K 2 algebras is closed under many standard operations in ring theory. The class of K 2 algebras includes Artin-Schelter regular algebras of global dimension 4 on three linear generators as well as graded complete intersections. Our work comprises two distinct projects. Each project was motivated by an aspect of the theory of Koszul algebras which we regard as sufficiently powerful or fundamental to warrant an interpretation for K 2 algebras. A very useful theorem due to Backelin and Fröberg states that if A is a Koszul algebra and I is a quadratic ideal of A which is Koszul as a left A -module, then the factor algebra A/I is a Koszul algebra. We prove that if A is Koszul algebra and A I is a K 2 module, then A/I is a K 2 algebra provided A/I acts trivially on Ext A ( A/I,k ). As an application of our theorem, we show that the class of sequentially Cohen-Macaulay Stanley-Reisner rings are K 2 algebras and we give examples that suggest the class of K 2 Stanley-Reisner rings is actually much larger. Another important recent development in ring theory is the use of A ∞ -algebras. One can characterize Koszul algebras as those graded algebras whose Yoneda algebra admits only trivial A ∞ -structure. We show that, in contrast to the situation for Koszul algebras, vanishing of higher A ∞ -structure on the Yoneda algebra of a K 2 algebra need not be determined in any obvious way by the degrees of defining relations. We also demonstrate that obvious patterns of vanishing among higher multiplications cannot detect the K 2 property. This dissertation includes previously unpublished co-authored material.
Committee in charge: Dr. Brad Shelton, Chair; Dr. Victor Ostrik, Member; Dr. Nicholas Proudfoot, Member; Dr. Arkady Vaintrob, Member; Dr. David Boush, Outside Member
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Fournier, Bradford M. "Towards a Theory of Recursive Function Complexity: Sigma Matrices and Inverse Complexity Measures." ScholarWorks@UNO, 2015. http://scholarworks.uno.edu/td/2072.
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This paper develops a data structure based on preimage sets of functions on a finite set. This structure, called the sigma matrix, is shown to be particularly well-suited for exploring the structural characteristics of recursive functions relevant to investigations of complexity. The matrix is easy to compute by hand, defined for any finite function, reflects intrinsic properties of its generating function, and the map taking functions to sigma matrices admits a simple polynomial-time algorithm . Finally, we develop a flexible measure of preimage complexity using the aforementioned matrix. This measure naturally partitions all functions on a finite set by characteristics inherent in each function's preimage structure.
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Risley, Rebecca N. "A Generalization of Sturmian Sequences: Combinatorial Structure and Transcendence." Thesis, University of North Texas, 1998. https://digital.library.unt.edu/ark:/67531/metadc278440/.
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We investigate a class of minimal sequences on a finite alphabet Ak = {1,2,...,k} having (k - 1)n + 1 distinct subwords of length n. These sequences, originally defined by P. Arnoux and G. Rauzy, are a natural generalization of binary Sturmian sequences. We describe two simple combinatorial algorithms for constructing characteristic Arnoux-Rauzy sequences (one of which is new even in the Sturmian case). Arnoux-Rauzy sequences arising from fixed points of primitive morphisms are characterized by an underlying periodic structure. We show that every Arnoux-Rauzy sequence contains arbitrarily large subwords of the form V^2+ε and, in the Sturmian case, arbitrarily large subwords of the form V^3+ε. Finally, we prove that an irrational number whose base b-digit expansion is an Arnoux-Rauzy sequence is transcendental.
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Phillips, Linzy. "Erasure-correcting codes derived from Sudoku & related combinatorial structures." Thesis, University of South Wales, 2013. https://pure.southwales.ac.uk/en/studentthesis/erasurecorrecting-codes-derived-from-sudoku--related-combinatorial-structures(b359130e-bfc2-4df0-a6f5-55879212010d).html.
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This thesis presents the results of an investigation into the use of puzzle-based combinatorial structures for erasure correction purposes. The research encompasses two main combinatorial structures: the well-known number placement puzzle Sudoku and a novel three component construction designed specifically with puzzle-based erasure correction in mind. The thesis describes the construction of outline erasure correction schemes incorporating each of the two structures. The research identifies that both of the structures contain a number of smaller sub-structures, the removal of which results in a grid with more than one potential solution - a detrimental property for erasure correction purposes. Extensive investigation into the properties of these sub-structures is carried out for each of the two outline erasure correction schemes, and results are determined that indicate that, although the schemes are theoretically feasible, the prevalence of sub-structures results in practically infeasible schemes. The thesis presents detailed classifications for the different cases of sub-structures observed in each of the outline erasure correction schemes. The anticipated similarities in the sub-structures of Sudoku and sub-structures of Latin Squares, an established area of combinatorial research, are observed and investigated, the proportion of Sudoku puzzles free of small sub-structures is calculated and a simulation comparing the recovery rates of small sub-structure free Sudoku and standard Sudoku is carried out. The analysis of sub-structures for the second erasure correction scheme involves detailed classification of a variety of small sub-structures; the thesis also derives probabilistic lower bounds for the expected numbers of case-specific sub-structures within the puzzle structure, indicating that specific types of sub-structure hinder recovery to such an extent that the scheme is infeasible for practical erasure correction. The consequences of complex cell inter-relationships and wider issues with puzzle-based erasure correction, beyond the structures investigated in the thesis are also discussed, concluding that while there are suggestions in the literature that Sudoku and other puzzle-based combinatorial structures may be useful for erasure correction, the work of this thesis suggests that this is not the case.
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Ghannadian, Farzad. "The structure of the solution space and its relation to execution time of evolutionary algorithms with applications." Diss., Georgia Institute of Technology, 1997. http://hdl.handle.net/1853/15687.
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Matsuura, Akihiro. "Combinatorial Structures in Finite Automata, CNF Satisfiability and Arithmetic Computation." 京都大学 (Kyoto University), 2002. http://hdl.handle.net/2433/149387.
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Sharma, Dushyant 1975. "Cyclic exchange and related neighborhood structures for combinatorial optimization problems." Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/8526.
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Thesis (Ph.D.)--Massachusetts Institute of Technology, Sloan School of Management, 2002.Includes bibliographical references (p. 122-126).
In this thesis, we concentrate on neighborhood search algorithms based on very large-scale neighborhood structures. The thesis consists of three parts. In the first part, we develop a cyclic exchange neighborhood search based approach for partitioning problems. A partitioning problem is to divide a set of n elements into K subsets S1,... ,SK so as to minimize f(S1)+...+f(SK) for some specified function f. A partition S'1,.. ,S'K is called a cyclic exchange neighbor of the partition S1,...,SK if [...]. The problem of searching the cyclic exchange neighborhood is NP-hard. We develop new exact and heuristic algorithms to search this neighborhood structure. We propose cyclic exchange based neighborhood search algorithms for specific partitioning problems. We provide computational results on these problems indicating that the cyclic exchange is very effective and can be implemented efficiently in practice. The second part deals with the Combined Through and Fleet Assignment Model (ctFAM). This model integrates two airline planning models: (i) Fleet Assignment Model and (ii) Through Assignment Model, which are currently solved in a sequential manner because the combined problem is too large. This leads to sub-optimal solutions for the combined problem we develop very large-scale neighborhood search algorithms for the ctFAM. We also extend our neighborhood search algorithms to solve the multi-criteria objective function version of the ctFAM. Our computational results using real-life data show that neighborhood search can be a useful supplement to the current integer-programming optimization methods in airline scheduling.
(cont.) In the third part, we investigate the structure of neighborhoods in general. We call two neighborhood structures LO-equivalent if they have the same set of local optima for all instances of a combinatorial optimization problem. We define the extended neighborhood of a neighborhood structure N as the largest neighborhood structure that is LO-equivalent to N. In this thesis, we develop some theoretical properties of the extended neighborhood and relate these properties to the performance of a neighborhood structure. In particular, we show that the well-known 2-opt neighborhood structure for the Traveling Salesman Problem has a very large extended neighborhood, providing justification for its favorable empirical performance.
by Dushyant Sharma.
Ph.D.