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.

A broadcast is a function $f:V \rightarrow { 0,...,diam(G)}$ that assigns an integer value to each vertex such that, for each $v\in V$, $f(v)\leq e(v)$, the eccentricity of $v$. The broadcast number of a graph is the minimum value of $\sum_{v\in V}f(v)$ among all broadcasts $f$ for which each vertex of the graph is within distance $f(v)$ from some vertex $v$ having $f(v)\geq1$. This number is bounded above by the radius of the graph, as well as by its domination number. Graphs for which the broadcast number is equal to the radius are called radial. We prove a new upper bound on the broadcast number of a graph and motivate the study of radial trees by proving a relationship between the broadcast number of a graph and those of its spanning subtrees. We describe some classes of radial trees and then provide a characterization of radial trees, as well as a geometric interpretation of our characterization.

The objective of this dissertation is to study commonly occurring location and clustering problems on graphs. The dissertation is presented as a collection of results in topics including finding maximum cliques in large graphs, graph clustering in large scale graphs, determining location of facilities for pre-positioning emergency relief supplies, and selecting nodes to form a virtual backbone in a wireless sensor network. To begin with, a new clique relaxation called a k-community is defined as a connected subgraph such that endpoints of every edge have at least k common neighbors within the subgraph. It is used to develop scale reduction techniques to obtain the maximum clique on very large scale real life networks. Analytically, the technique is been shown to be very effective on power-law random graphs. Experimental results on real life graph instances (Collaboration networks, P2P networks, Social networks, etc.) show our procedure to be much more effective than a regular k-core peeling approach. Next, a general purpose network clustering algorithm based on the clique relaxation concept of k-community is presented. A salient feature of this approach is that it does not use any prior information about the structure of the network. By defining a cluster as a k-community, the proposed algorithm aims to provide a clustering of a network into k-communities with varying values of k. Even though the algorithm is not designed to optimize any particular performance measure, the computational results suggest that it performs well on a number of criteria that are used in literature to evaluate the quality of a clustering. The third topic deals with choosing the locations of disaster response facilities for the storage of emergency supplies, which is critical to the quality of service provided in a large scale emergency like an earthquake. In the existing literature, large scale emergency facility location models have either assumed that disaster response facilities will always be functioning and available when required, or that the functioning of a facility is independent of a particular disaster scenario. In this paper new location models are presented that explicitly take into consideration the stochastic nature of the impact a disaster can have on the disaster response facilities and the population centers in surrounding areas. A comparison of the results obtained using our models with those from models available in literature using a case study suggests that the locations suggested by the model in this paper significantly reduce the expected cost of transportation of supplies when we consider the damage a disaster causes to the disaster response facilities and areas near it. Lastly, a distributed approximate algorithm for forming the communication backbone in wireless sensor networks is presented. Some of the most popular routing protocols for wireless sensor networks require a virtual backbone for efficient communication be- tween the sensors. Connected Dominating Sets (CDS) have been studied as a method of choosing nodes to be in the backbone. The traditional approach is to assume that the transmission range of each node is given and then minimize the number of nodes in the CDS representing the backbone. A recently introduced alternative strategy is based on the concept of k-bottleneck connected dominating set (k-BCDS), which, given a positive integer k, minimizes the transmission range of the nodes that ensures a CDS of size k exists in the network. This paper provides a 6-approximate distributed algorithm for the k-BCDS problem. The results of empirical evaluation of the proposed algorithm are also included.

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.

For a graph G, the minimum dominating set problem is to find a minimum size set S of vertices of G such that every vertex is either in S or adjacent to a vertex in the set. The decision version of this problem, which asks whether G has a dominating set of a particular size k, is known to be NP-complete, and no polynomial time algorithm to solve the problem is currently known to exist. The queen domination problem is to find the minimum number of queens which, collectively, can attack every square on an n by n chess board. The related border queen problem is to find such a collection of queens with the added restriction that all queens lie on the outer border of the board. This thesis studies practical exponential time algorithms for solving domination problems, and presents an experimental comparison of several different algorithms, with the goal of producing a broadly effective general domination solver for use by future researchers. The developed algorithms are then used to solve several open problems, including cases of the queen domination problem and the border queen problem. In addition, new theoretical upper bounds are presented for the border queen problem for some families of queen graphs. / Graduate

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.

A total dominating set in a graph G is a set S of vertices of G such that every vertex in G is adjacent to a vertex of S. We study graphs whose vertex set can be partitioned into two total dominating sets. In particular, we develop several sufficient conditions for a graph to have a vertex partition into two total dominating sets. We also show that with the exception of the cycle on five vertices, every selfcomplementary graph with minimum degree at least two has such a partition.

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 growing number of organizations developing complex innovation projects face the dilemma that complexity demands more stability, resulting in increased sensemaking efforts, whilst innovation requires instability which encourages the organization to breakdown meaning. The aim of this study is to shed light on the simultaneous need for a strong sensemaking process and a breaking down in meaning in complex innovation projects by analyzing the collective sensemaking process through a storytelling lens. By conducting an explorative case study, we investigated a long-term development project in the Swedish defense industry. The qualitative study is based upon ten in-depth interviews with technical experts occupying key positions in the investigated project (JAS39 Gripen E-series development). We followed a process study approach to investigate the dynamic attributes and effects of a changing dominant story on the sensemaking process of project sub-teams. We propose that the dual attributes of arising dominating narratives allow sub- collectives to "escape" the dangerous downward spiral of a collapsing sensemaking process, as they enable individuals with cause maps contradictory to an organization's dominant story to remain in action. The acceptance of temporal relaxed stability can therefore be seen as an important step in the creation of radical innovation.

An Open Locating-Dominating Set (OLD set) is a subset of vertices in a graph such that every vertex in the graph has a neighbor in the OLD set and every vertex has a unique set of neighbors in the OLD set. This can also represent where sensors, capable of detecting an event occurrence at an adjacent vertex, could be placed such that one could always identify the location of an event by the specific vertices that indicated an event occurred in their neighborhood. By the open neighborhood construct, which differentiates OLD sets from identifying codes, a vertex is not able to report if it is the location of the event. This construct provides a robustness over identifying codes and opens new applications such as disease carrier and dark actor identification in networks. This work explores various aspects of OLD sets, beginning with an Integer Linear Program for quickly identifying the optimal OLD set on a graph. As many graphs do not admit OLD sets, or there may be times when the total size of the set is limited by an external factor, a concept called maximum covering OLD sets is developed and explored. The coverage radius of the sensors is then expanded in a presentation of Mixed-Weight OLD sets where sensors can cover more than just adjacent vertices. Finally, an application is presented to optimally monitor criminal and terrorist networks using OLD sets and related concepts to identify the optimal set of surveillance targets.