Wireless communication is a rapidly growing segment of the communication industry, with the potential to provide low-cost, high-quality, and high-speed information exchange between portable devices. To harvest the available bandwidth efficientlyin a wireless network, they employ multiple orthogonal channels over multiple ra-dios at the nodes. In addition, nodes in these networks employ directional antennasas radios to improve spatial throughput. This dissertation develops algorithms forrouting and broadcasting with channel assignment in such networks. First, we com-pute the minimum cost path between a given source-destination pair with channelassignment on each link in the path such that no two transmissions interfere witheach other. Such a path must satisfy the constraint that no two consecutive links onthe path are assigned the same channel, referred to as "channel discontinuity con-straint." To compute such a path, we develop two graph expansion techniques basedon minimum cost perfect matching and dijkstra's algorithm. Through extensive sim-ulations, we study the effectiveness of the routing algorithms developed based onthe two expansion techniques and the benefits of employing the minimum cost per-fect matching based solution. Secondly, we study the benefits of sharing channelbandwidth across multiple flows. We model the routing and channel assignmentproblem in two different ways to account for the presence and absence of inter-flowbandwidth sharing. Benefits of multiple paths between a source-destination pairmotivates the problem of computing multiple paths between a source-destinationpair with channel assignment such that all the paths can be active simultaneouslyto achieve maximal flow between the pair in the considered network. Since finding even two such paths is NP-hard, we formulate the problem as an integer linearprogram and develop efficient heuristic to find these paths iteratively. Thirdly, wecompute a broadcast tree from a given root with channel assignment such that all the links in the broadcast tree can be active simultaneously without interferingwith each other. Since finding such a tree is an NP-hard problem, we formulatethe problem as an integer linear program (ILP) and develop heuristics to find thebroadcast tree with channel assignment. We evaluate and compare the performanceof the developed heuristics with respect to their success rate, average depth of theobtained tree, and average path length from root to a node in the network. Thisdissertation also analyzes the blocking performance of a channel assignment schemein a multi-channel wireless line network. We assume that the existing calls in thenetwork may be rearranged on different channels to accommodate an incoming call.The analysis is limited to single-hop calls with different transmission ranges.Finally, this dissertation evaluates the performance of disjoint multipath routingapproaches for all-to-all routing in packet-switched networks with respect to packetoverhead, path lengths, and routing table size. We develop a novel approach basedon cycle-embedding to obtain two node-disjoint paths between all source-destinationpairs with reduced number of routing table entries maintained at a node (hence thereduced look up time), small average path lengths, and less packet overhead. Westudy the trade-off between the number of routing table entries maintained at anode and the average length of the two disjoint paths by: (a) formulating the cycle-embedding problem as an integer linear program; and (b) developing a heuristic.We show that the number of routing table entries at a node may be reduced toat most two per destination using cycle-embedding approach, if the length of thedisjoint paths are allowed to exceed the minimum by 25%.

Classical vertex coloring problems ask for the minimum number of colors needed to color the vertices of a graph, such that adjacent vertices use different colors. Vertex coloring does have quite a few practical applications in communication theory, industry engineering and computer science. Such examples can be found in the book of Hansen and Marcotte. Deciding whether a graph is 3-colorable or not is a well-known NP-complete problem, even for triangle-free graphs. Intuitively, large girth may help reduce the chromatic number. However, in 1959, Erdos used the probabilitic method to prove that for any two positive integers g and k, there exist graphs of girth at least g and chromatic number at least k. Thus, restricting girth alone does not help bound the chromatic number. However, if we forbid certain tree structure in addition to girth restriction, then it is possible to bound the chromatic number. Randerath determined several such tree structures, and conjectured that if a graph is fork-free and triangle-free, then it is 3-colorable, where a fork is a star K1,4 with two branches subdivided once. The main result of this thesis is that Randerath’s conjecture is true for graphs with odd girth at least 7. We also give a proof that Randerath’s conjecture holds for graphs with maximum degree 4.

Singly-periodic (SP) and doubly-periodic (DP) graphs arc infinite graphs which have translational symmetries in one and two dimensions, respectively. The problem of counting the number of connected components in such graphs is investigated. A method for determining whether or not an SP graph is k-colourable for a given positive integer k is given, and the question of deciding k-colourability of DP graphs is discussed. Colourings of SP and DP graphs can themselves be either periodic or aperiodic, and properties which determine the symmetries of their colourings arc also explored.

Thesis (Ph. D.)--Rutgers University, 2007. / "Graduate Program in Operations Research." Includes bibliographical references (p. 57-61).

Thesis (Ph.D.) -- University of Rhode Island, 2007 / Typescript. Includes bibliographical references (leaves 80-83).

Thesis (Master)--İzmir Institute of Technology, İzmir, 2004. / Includes bibliographical references (leaves. 38-39).

Thesis (Master)--İzmir Institute of Technology, İzmir, 2004. / Includes bibliographical references (leaves. 35-36).

abstract: Graph coloring is about allocating resources that can be shared except where there are certain pairwise conflicts between recipients. The simplest coloring algorithm that attempts to conserve resources is called first fit. Interval graphs are used in models for scheduling (in computer science and operations research) and in biochemistry for one-dimensional molecules such as genetic material. It is not known precisely how much waste in the worst case is due to the first-fit algorithm for coloring interval graphs. However, after decades of research the range is narrow. Kierstead proved that the performance ratio R is at most 40. Pemmaraju, Raman, and Varadarajan proved that R is at most 10. This can be improved to 8. Witsenhausen, and independently Chrobak and Slusarek, proved that R is at least 4. Slusarek improved this to 4.45. Kierstead and Trotter extended the method of Chrobak and Slusarek to one good for a lower bound of 4.99999 or so. The method relies on number sequences with a certain property of order. It is shown here that each sequence considered in the construction satisfies a linear recurrence; that R is at least 5; that the Fibonacci sequence is in some sense minimally useless for the construction; and that the Fibonacci sequence is a point of accumulation in some space for the useful sequences of the construction. Limitations of all earlier constructions are revealed. / Dissertation/Thesis / Ph.D. Mathematics 2010

In recent times an evident trend in hardware is to opt for multi-core CPUs. This has lead to a situation where an increasing number of sequential algorithms are parallelized to fit these new multi-core environments. The greedy Multi-Coloring algorithm is a strictly sequential algorithm that is used in a wide range of applications. The application in focus is on decomposition by graph coloring for preconditioning techniques suitable for iterative solvers like the and methods. In order to perform all phases of these iterative solvers in parallel the graph analysis phase needs to be parallelized. Albeit many attempts have been made to parallelize graph coloring non of these attempts have successfully put the greedy Multi-Coloring algorithm into obsolescence. In this work techniques for parallel graph coloring are proposed and studied. Quantitative results, which represent the number of colors, confirm that the best new algorithm, the Normann algorithm, is performing on the same level as the greedy Multi-Coloring algorithm. Furthermore, in multi-core environments the parallel Normann algorithm fully outperforms the classical greedy Multi-Coloring algorithm for all large test matrices. With the features of the Normann algorithm quantified and presented in this work it is now possible to perform all phases of iterative solvers like and methods in parallel.

Orientador: Célia Picinin de Mello / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Computação / Made available in DSpace on 2018-08-27T04:52:30Z (GMT). No. of bitstreams: 1 Vilas-Boas_AloisiodeMenezes_M.pdf: 3075345 bytes, checksum: 59f85d259c9a55bce9d3409c06dd71fa (MD5) Previous issue date: 2015 / Resumo: Seja G um grafo simples. Uma coloração de arestas semiforte de G é uma coloração de arestas de G onde para cada par de vértices adjacentes u,v de G, o conjunto das cores atribuídas às arestas de u é diferente do conjunto das cores atribuídas às arestas de v. O índice cromático semiforte de G, denotado por chi'a(G), é o menor número de cores necessário para construir uma coloração de arestas semiforte para G. Esta coloração foi proposta por Zhang et al. em 2002. Nesse mesmo artigo, os autores conjecturaram que todo grafo simples conexo G, G diferente de C_5, com pelo menos três vértices possui chi'a(G) menor ou igual a Delta(G)+2. Esta conjectura conhecida como conjectura da coloração de arestas semiforte está aberta para grafos arbitrários, mas é válida para algumas classes de grafos. Nesta dissertação, apresentamos alguns resultados sobre a coloração de arestas semiforte. Em seguida, focamos em grafos split. Provamos a conjectura da coloração de arestas semiforte para algumas famílias destes grafos, dentre elas, os split-completos e os split-indiferença. Além disso, determinamos o índice cromático semiforte dos grafos split-indiferença com vértice universal. Para grafos split-indiferença sem vértice universal, exibimos condições para que seu índice cromático semiforte seja igual a Delta(G)+1 e conjecturamos chi'a(G) = Delta(G)+2 caso contrário / Abstract: Let G be a simple graph. An adjacent strong edge-coloring of G is an edge-coloring of G such that for each pair of adjacent vertices u,v of G, the set of colors assigned to the edges incident with u differs from the set of colors assigned to the edges incident with v. The adjacent strong chromatic index, denoted chi'a(G), of G is the minimum number of colors required to produce an adjacent strong edge-coloring for G. This coloring was proposed by Z. Zhang et al. In the same article, the authors conjectured that every simple connected graph G with at least three vertices and G not equal to C_5 (a 5-cycle) has chi'a(G) less or equal then Delta(G)+2. This conjecture is open for arbitrary graphs, but it holds for some classes of graphs. In this dissertation, we present some results on adjacent strong edge-coloring. Then, we focus on split graphs. We prove the conjecture for some families of split graphs including split-complete graphs and split-indifference graphs. Moreover, we determine a necessary condition for split-complete graphs G to have chi'a(G) = Delta(G)+1 and we determine the adjacent strong chromatic index for split-indifference graphs with a universal vertex. For a split-indifference graph G without universal vertices, we give conditions for its adjacent strong chromatic index to be Delta(G)+1 and we conjecture that chi'a(G) = Delta(G)+2, otherwise / Mestrado / Ciência da Computação / Mestre em Ciência da Computação