In this thesis, we study two problems in combinatorial optimization, the dominating set problem and the robustness problem. In the first half of the thesis, we focus on the dominating set problem in grid graphs and present a distributed algorithm for finding near optimal dominating sets on grids. The dominating set problem is a well-studied mathematical problem in which the goal is to find a minimum size subset of vertices of a graph such that all vertices that are not in that set have a neighbor inside that set. We first provide a simpler proof for an existing centralized algorithm that constructs dominating sets on grids so that the size of the provided dominating set is upper-bounded by the ceiling of (m+2)(n+2)/5 for m by n grids and its difference from the optimal domination number of the grid is upper-bounded by five. We then design a distributed grid domination algorithm to locate mobile agents on a grid such that they constitute a dominating set for it. The basis for this algorithm is the centralized grid domination algorithm. We also generalize the centralized and distributed algorithms for the k-distance dominating set problem, where all grid vertices are within distance k of the vertices in the dominating set. In the second half of the thesis, we study the computational complexity of checking a graph property known as robustness. This property plays a key role in diffusion of information in networks. A graph G=(V,E) is r-robust if for all pairs of nonempty and disjoint subsets of its vertices A,B, at least one of the subsets has a vertex that has at least r neighbors outside its containing set. In the robustness problem, the goal is to find the largest value of r such that a graph G is r-robust. We show that this problem is coNP-complete. En route to showing this, we define some new problems, including the decision version of the robustness problem and its relaxed version in which B=V \ A. We show these two problems are coNP-hard by showing that their complement problems are NP-hard.

Online social networks (OSNs) have become a powerful medium of communicating, sharing and disseminating information. Because of popularity and availability of OSNs throughout the world, the connected users can spread information faster and thus propagate influence over each other constantly. Due to such impact, a lot of applications on OSNs focused on picking an initial set of users (seeds) to infuse their message in the OSN. Due to huge size of the network, the main challenge in picking the initial set is to maximize the resultant influence over the users in the network. The optimization problem of finding out the most influential set of members in an OSN for maximization of influence is an NP-hard problem. In this paper, we propose using the Positive Influential Dominating Set (PIDS) algorithm for the initial seed. PIDS is a well-known algorithm which determines the influential backbone nodes in the networks. We implemented PIDS-based influence maximization by using different propagation models. We compared PIDS performance to that of the existing approaches based on greedy and random heuristics. The experimental results from extensive simulation on real-world network data sets show that PIDS gives better influence spread than greedy and random for both Independent Cascade Model and Linear Threshold Model of influence propagation. PIDS is also scalable to large networks and in all size ranges, it performs well in influence maximization.

Wireless Sensor Networks (WSN) offer a flexible low-cost solution to the problem of event monitoring, especially in places with limited accessibility or that represent danger to humans. WSNs are made of resource-constrained wireless devices, which require energy efficient mechanisms, algorithms and protocols. One of these mechanisms is Topology Control (TC) composed of two mechanisms, Topology Construction and Topology Maintenance. This dissertation expands the knowledge of TC in many ways. First, it introduces a comprehensive taxonomy for topology construction and maintenance algorithms for the first time. Second, it includes four new topology construction protocols: A3, A3Lite, A3Cov and A3LiteCov. These protocols reduce the number of active nodes by building a Connected Dominating Set (CDS) and then turning off unnecessary nodes. The A3 and A3-Lite protocols guarantee a connected reduced structure in a very energy efficient manner. The A3Cov and A3LiteCov protocols are extensions of their predecessors that increase the sensing coverage of the network. All these protocols are distributed -they do not require localization information, and present low message and computational complexity. Third, this dissertation also includes and evaluates the performance of four topology maintenance protocols: Recreation (DGTRec), Rotation (SGTRot), Rotation and Recreation (HGTRotRec), and Dynamic Local-DSR (DLDSR). Finally, an event-driven simulation tool named Atarraya was developed for teaching, researching and evaluating topology control protocols, which fills a need in the area of topology control that other simulators cannot. Atarraya was used to implement all the topology construction and maintenance cited, and to evaluate their performance. The results show that A3Lite produces a similar number of active nodes when compared to A3, while spending less energy due to its lower message complexity. A3Cov and A3CovLite show better or similar coverage than the other distributed protocols discussed here, while preserving the connectivity and energy efficiency from A3 and A3Lite. In terms of network lifetime, depending on the scenarios, it is shown that there can be a substantial increase in the network lifetime of 450% when a topology construction method is applied, and of 3200% when both topology construction and maintenance are applied, compared to the case where no topology control is used.

For a graph G = (V, E), a set S ⊆ V is a total dominating set if every vertex in V is adjacent to some vertex in S. The smallest cardinality of any total dominating set is the total domination number γt(G). For an arbitrary edge e εE(Ḡ), γt(G) - 2 ≤ γt(G + e) ≤ γt(G); if the latter inequality is strict for each e ε E(Ḡ) ≠ φ, then G is said to be γt-critical. The criticality index of an edge e ε E(Ḡ) is γt(e) = γt(G) - γt(G + e). Let E(Ḡ) = [e1...,em} and S = ∑j=1m̄ci(ej). The criticality index of G is ci(G) = S/m̄. For any rational number k, 0 ≤ k ≤ 2, we construct a graph G with ci(G) = k. For 1 ≤ k ≤ 2, we construct graphs with this property that are γt-critical as well as graphs that are not γt-critical.

A set S of vertices in a graph G = (V, E) is a locating-dominating set if S is a dominating set of G, and every pair of distinct vertices {u, v} in V - S is located with respect to S, that is, if the set of neighbors of u that are in S is not equal to the set of neighbors of v that are in S. We give a construction of trees that have unique minimum locating-dominating sets.

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A mobile ad hoc network (MANET) is a wireless network that uses multi-hop peer-to-peer routing instead of static network infrastructure to provide network connectivity. MANETs have applications in rapidly deployed and dynamic military and civilian systems. The network topology in a MANET usually changes with time. Therefore, there are new challenges for routing protocols in MANETs since traditional routing protocols may not be suitable for MANETs. For example, some assumptions used by these protocols are not valid in MANETs or some protocols cannot efficiently handle topology changes. Researchers are designing new MANET routing protocols and comparing and improving existing MANET routing protocols before any routing protocols are standardized using simulations. However, the simulation results from different research groups are not consistent with each other. This is because of a lack of consistency in MANET routing protocol models and application environments, including networking and user traffic profiles. Therefore, the simulation scenarios are not equitable for all protocols and conclusions cannot be generalized. Furthermore, it is difficult for one to choose a proper routing protocol for a given MANET application. According to the aforementioned issues, my Ph.D. research focuses on MANET routing protocols. Specifically, my contributions include the characterization of differ- ent routing protocols using a novel systematic relay node set (RNS) framework, design of a new routing protocol for MANETs, a study of node mobility, including a quantitative study of link lifetime in a MANET and an adaptive interval scheme based on a novel neighbor stability criterion, improvements of a widely-used network simulator and corresponding protocol implementations, design and development of a novel emulation test bed, evaluation of MANET routing protocols through simulations, verification of our routing protocol using emulation, and development of guidelines for one to choose proper MANET routing protocols for particular MANET applications. Our study shows that reactive protocols do not always have low control overhead, as people tend to think. The control overhead for reactive protocols is more sensitive to the traffic load, in terms of the number of traffic flows, and mobility, in terms of link connectivity change rates, than other protocols. Therefore, reactive protocols may only be suitable for MANETs with small number of traffic loads and small link connectivity change rates. We also demonstrated that it is feasible to maintain full network topology in a MANET with low control overhead. This dissertation summarizes all the aforementioned methodologies and corresponding applications we developed concerning MANET routing protocols. / Ph. D.

Dans le cadre des réseaux ad hoc à grande envergure, le concept de clusterisation peut être mis à profit afin de faire face aux problèmes de passage à l'échelle et d'accroître les performances du système. Tout d’abord, cette thèse présente notre algorithme de clusterisation TBCA ‘Tiered based Clustering algorithm’, ayant pour objectif d’organiser le processus de clusterisation en couches et de réduire au maximum le trafic de contrôle associé à la phase d’établissement et de maintenance de l’infrastructure virtuelle générée. La formation et la maintenance d’une infrastructure virtuelle ne sont pas une fin en soi. Dans cet axe, on a exploité les apports de notre mécanisme de clusterisation conjointement avec le mode veille, à travers la proposition de l’approche de conservation d’énergie baptisée CPPCM ‘Cluster based Prioritized Power Conservation Mechanism’ avec deux variantes. Notre objectif principal est de réduire la consommation d’énergie tout en assurant l’acheminement des paquets de données sans endurer des temps d’attente importants aux niveaux des files d’attente des nœuds impliqués dans le transfert. Nous avons proposé aussi un algorithme de routage LCR ‘Layered Cluster based Routing’ se basant sur l’existence d’une infrastructure virtuelle. L’exploitation des apports de notre mécanisme TBCA et la limitation des tâches de routage additionnelles à un sous ensemble de nœuds sont des atouts pour assurer le passage à l’échelle de notre algorithme. / Relying on a virtual infrastructure seems a promising approach to overcome the scalability problem in large scale ad hoc networks. First, we propose a clustering mechanism, TBCA ‘Tiered based Clustering algorithm’, operating in a layered manner and exploiting the eventual collision to accelerate the clustering process. Our mechanism does not necessitate any type of neighbourhood knowledge, trying to alleviate the network from some control messages exchanged during the clustering and maintenance process. Since the energy consumption is still a critical issue, we combining a clustering technique and the power saving mode in order to conserve energy without affecting network performance. The main contribution of our power saving approach lies on the differentiation among packets based on the amount of network resources they have been so far consumed. Besides, the proposed structure of the beacon interval can be adjusted dynamically and locally by each node according to its own specific requirements. We propose also a routing algorithm, LCR ‘Layered Cluster based Routing’. The basic idea consists on assigning additional tasks to a limited set of dominating nodes, satisfying specific requirements while exploiting the benefits of our clustering algorithm TBCA.

A mobile ad-hoc network (MANET) is a collection of wireless mobile nodes forming a temporary network without the support of any established infrastructure or centralized administration. There are many potential applications based the techniques of MANETs, such as disaster rescue, personal area networking, wireless conference, military applications, etc. MANETs face a number of challenges for designing a scalable routing protocol due to their natural characteristics. Guaranteeing delivery and the capability to handle dynamic connectivity are the most important issues for routing protocols in MANETs. In this dissertation, we will propose four algorithms that address different aspects of routing problems in MANETs. Firstly, in position based routing protocols to design a scalable location management scheme is inherently difficult. Enhanced Scalable Location management Service (EnSLS) is proposed to improve the scalability of existing location management services, and a mathematical model is proposed to compare the performance of the classical location service, GLS, and our protocol, EnSLS. The analytical model shows that EnSLS has better scalability compared with that of GLS. Secondly, virtual backbone routing can reduce communication overhead and speedup the routing process compared with many existing on-demand routing protocols for routing detection. In many studies, Minimum Connected Dominating Set (MCDS) is used to approximate virtual backbones in a unit-disk graph. However finding a MCDS is an NP-hard problem. In the dissertation, we develop two new pure localized protocols for calculating the CDS. One emphasizes forming a small size initial near-optimal CDS via marking process, and the other uses an iterative synchronized method to avoid illegal simultaneously removal of dominating nodes. Our new protocols largely reduce the number of nodes in CDS compared with existing methods. We show the efficiency of our approach through both theoretical analysis and simulation experiments. Finally, using multiple redundant paths for routing is a promising solution. However, selecting an optimal path set is an NP hard problem. We propose the Genetic Fuzzy Multi-path Routing Protocol (GFMRP), which is a multi-path routing protocol based on fuzzy set theory and evolutionary computing.

This thesis has two parts. In the first part, we study some graph covering problems with a non-local covering rule that allows a "remote" node to be covered by repeatedly applying the covering rule. In the second part, we provide some results on the packing of Steiner trees. In the Propagation problem we are given a graph $G$ and the goal is to find a minimum-sized set of nodes $S$ that covers all of the nodes, where a node $v$ is covered if (1) $v$ is in $S$, or (2) $v$ has a neighbor $u$ such that $u$ and all of its neighbors except $v$ are covered. Rule (2) is called the propagation rule, and it is applied iteratively. Throughout, we use $n$ to denote the number of nodes in the input graph. We prove that the path-width parameter is a lower bound for the optimal value. We show that the Propagation problem is NP-hard in planar weighted graphs. We prove that it is NP-hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in weighted (general) graphs. The second problem that we study is the Power Dominating Set problem. This problem has two covering rules. The first rule is the same as the domination rule as in the Dominating Set problem, and the second rule is the same propagation rule as in the Propagation problem. We show that it is hard to approximate the optimal value to within a factor of $2^{\log^{1-\epsilon}{n}}$ in general graphs. We design and analyze an approximation algorithm with a performance guarantee of $O(\sqrt{n})$ on planar graphs. We formulate a common generalization of the above two problems called the General Propagation problem. We reformulate this general problem as an orientation problem, and based on this reformulation we design a dynamic programming algorithm. The algorithm runs in linear time when the graph has tree-width $O(1)$. Motivated by applications, we introduce a restricted version of the problem that we call the $\ell$-round General Propagation problem. We give a PTAS for the $\ell$-round General Propagation problem on planar graphs, for small values of $\ell$. Our dynamic programming algorithms and the PTAS can be extended to other problems in networks with similar propagation rules. As an example we discuss the extension of our results to the Target Set Selection problem in the threshold model of the diffusion processes. In the second part of the thesis, we focus on the Steiner Tree Packing problem. In this problem, we are given a graph $G$ and a subset of terminal nodes $R\subseteq V(G)$. The goal in this problem is to find a maximum cardinality set of disjoint trees that each spans $R$, that is, each of the trees should contain all terminal nodes. In the edge-disjoint version of this problem, the trees have to be edge disjoint. In the element-disjoint version, the trees have to be node disjoint on non-terminal nodes and edge-disjoint on edges adjacent to terminals. We show that both problems are NP-hard when there are only $3$ terminals. Our main focus is on planar instances of these problems. We show that the edge-disjoint version of the problem is NP-hard even in planar graphs with $3$ terminals on the same face of the embedding. Next, we design an algorithm that achieves an approximation guarantee of $\frac{1}{2}-\frac{1}{k}$, given a planar graph that is $k$ element-connected on the terminals; in fact, given such a graph the algorithm returns $k/2-1$ element-disjoint Steiner trees. Using this algorithm we get an approximation algorithm with guarantee of (almost) $4$ for the edge-disjoint version of the problem in planar graphs. We also show that the natural LP relaxation of the edge-disjoint Steiner Tree Packing problem has an integrality ratio of $2-\epsilon$ in planar graphs.