In this thesis, we do an algorithmic study of optimization problems in budgeted auctions, and some well known covering problems in the multi-agent setting. We give new results for the design of approximation algorithms, online algorithms and hardness of approximation for these problems. Along the way we give new insights for many other related problems. Budgeted Auction. We study the following allocation problem which arises in budgeted auctions (such as advertisement auctions run by Google, Microsoft, Yahoo! etc.) : Given a set of m indivisible items and n agents; agent i is willing to pay b[subscript ij] for item j and has an overall budget of B[subscript i] (i.e. the maximum total amount he is willing to pay). The goal is to allocate items to the agents so as to maximize the total revenue obtained. We study the computation complexity of the above allocation problem, and give improved results for the approximation and the hardness of approximation. We also study the above allocation problem in an online setting. Online version of the problem has motivation in the sponsored search auctions which are run by search engines. Lastly, we propose a new bidding language for the budgeted auctions: decreasing bid curves with budget constraints. We make a case for why this language is better both for the sellers and for the buyers. Multi-agent Covering Problems. To motivate this class of problems, consider the network design problem of constructing a spanning tree of a graph, assuming there are many agents willing to construct different parts of the tree. The cost of each agent for constructing a particular set of edges could be a complex function. For instance, some agents might provide discounts depending on how many edges they construct. The algorithmic question that one would be interested in is: Can we find a spanning tree of minimum cost in polynomial time in these complex settings? Note that such an algorithm will have to find a spanning tree, and partition its edges among the agents. Above are the type of questions that we are trying to answer for various combinatorial problems. We look at the case when the agents' cost functions are submodular. These functions form a rich class and capture the natural properties of economies of scale or the law of diminishing returns.We study the following fundamental problems in this setting- Vertex Cover, Spanning Tree, Perfect Matching, Reverse Auctions. We look at both the single agent and the multi-agent case, and study the approximability of each of these problems.

Since the early 1960s location problems have been used throughout various industries and in various countries. During recent years the field of location problems has become increasingly popular due to the fact that it is applicable in real life situations – especially in emergency services such as hospital, police station and ambulance locations to name a few. Despite the fact that location problems are so widely used with great success, it is still not being used to full potential in industries where it can have a major impact. One of these industries is the private security industry in South Africa. This dissertation addresses various mathematical models that can assist the management of privately owned security companies to determine strategic locations for their reaction vehicles, these locations will increase both resource utilization and improve the level of service they provide to customers. These models are used in different scenarios to see how the models adapt to input changes. / Dissertation (MSc)--University of Pretoria, 2014. / Industrial and Systems Engineering / MSc / Restricted

Un système distribué est un système composé d'éléments de calcul autonomes dotés de capacité de communication. Il s'agit d'un modèle commun pour l'étude des réseaux. L'évolution rapide des réseaux sans fils et/ou mobiles aussi bien dans la vie quotidienne que dans la recherche amène progressivement à intégrer la dynamique (i.e. l'évolution dans le temps de la connectivité) dans les systèmes distribués. Concrètement, cela revient à ajouter l'hypothèse que les capacités de communication des éléments du système peuvent varier dans le temps. De nombreux modèles considèrent ainsi la dynamique comme composante à part entière du système (et non pas comme une faute). De manière récente, une nouvelle approche, appelée graphe variant dans le temps, tente d'unifier tous ces modèles dans un formalisme commun qui permet de classifier les systèmes en fonction de leurs propriétés de connexité temporelle. Dans cette thèse, nous nous intéressons à des systèmes distribués hautement dynamiques dans lesquels les hypothèses de connexité sont minimalistes. Plus précisément, nous concentrons nos efforts sur les systèmes connexes à travers le temps dans lesquels la seule garantie est que tout élément du système peut infiniment souvent envoyer un message à tout autre (sans garantie sur la pérennité de la route utilisée ni sur le délai de communication). Nous nous intéressons plus particulièrement aux problèmes de couverture (par exemple, ensemble dominant minimal, couplage maximal, ensemble indépendant maximal, ...) dans ces systèmes distribués hautement dynamiques. Les contributions de cette thèse dans ce contexte sont les suivantes. Nous proposons tout d'abord une nouvelle définition pour les problèmes de couverture qui est plus adaptée aux systèmes distribués hautement dynamiques que les définitions existantes. Dans un deuxième temps, nous fournissons un outil générique qui permet de faciliter les preuves de résultats d'impossibilité dans les systèmes distribués dynamiques. Nous appliquons cet outil pour prouver plusieurs résultats d'impossibilité à propos de problèmes de couverture. Ensuite, nous proposons une nouvelle mesure de complexité en temps qui permet de comparer équitablement les performances de protocoles dans les systèmes distribués dynamiques. Enfin, nous donnons un algorithme de construction d'un ensemble dominant minimal dans les systèmes distribués hautement dynamiques. / A distributed system is a system of autonomous computing components endowed with communication abilities. This is a common model for the study of networks. The quick evolution of wireless and mobile network both in everyday life and in research gradually leads to take in account the dynamics (i.e. the evolution over time) in distributed systems. Concretely, this means to add the assumption that the communication abilities of the components of the system may vary over time. Many models consider the dynamics as an integral component of the system (and not as a fault). Recently, a new approach, called time-varying graph, attempts to unify all these models in a common formalism which allows the classification systems based on their temporal connectivity properties. In this thesis, we are interested in highly dynamic distributed systems with minimal connectivity assumptions. Specifically, we focus on connected over time systems where the only guarantee is that any element of the system can infinitely often send a message to any other (no guarantee are provided on the sustainability of the used path nor on the time communication). We are particularly interested in covering problems (e.g., minimal dominanting set, maximal matching, maximal independent set, ...) in these highly dynamic distributed systems. The contributions of this thesis in this context are as follows. We first propose a new definition for the covering problems which is more suited to highly dynamic distributed systems that the existing definitions. Secondly, we provide a generic tool to simplify proof of impossibility results in dynamic distributed systems. We use this tool to prove some impossibility results of covering problems. Then, we propose a new time complexity measure to fairly compare the algorithms performance in dynamic distributed systems. Finally, we give an algorithm that compute a minimal dominating set in highly dynamic distributed systems.

<p>This thesis is devoted to introducing new problem formulations and exact solution methods for a class of continuous covering location models. The manuscript includes three self-contained studies which are organized as in the following. </p> <p> In the first study, we introduce the planar expropriation problem with non-rigid rectangular facilities which has many applications in regional planning and undesirable facility location domains. This model is proposed for determining the locations and formations of two-dimensional rectangular facilities. Based on the geometric properties of such facilities, we developed a new formulation which does not require employing distance measures. The resulting model is a mixed integer nonlinear program. For solving this new model, we derived a continuous branch-and-bound framework utilizing linear approximations for the tradeoff curve associated with the facility formation alternatives. Further, we developed new problem generation and bounding strategies suitable for this particular branch-and-bound procedure. We designed a computational study where we compared this algorithm with two well-known mixed integer nonlinear programming solvers. Computational experience showed that the branch-and-bound procedure we developed performs better than BARON and SBB both in terms of processing time and size of the branching tree.</p> <p> The second study is referred to as the planar maximal covering problem with single convex polygonal shapes and it has ample applications in transmitter location, inspection of geometric shapes and directional antenna location. In this study, we investigated maximal point containment by any convex polygonal shape in the Euclidean plane. Using a fundamental separation property of convex sets, we derived a mixed integer linear formulation for this problem. We were able to identify two types of special cuts based on the geometric properties of the shapes under study, which were later employed for developing a branch-and-cut procedure for solving this particular location model. We also evaluated the resultant bound quality after employing the afore-mentioned cuts. </p> <p> In the third study, we discuss the dynamic planar expropriation problem with single convex polygonal shapes. We showed how the basic problem formulations discussed in the first two studies extend to their diametric opposites, and further to models in higher dimensions. Subsequently, we allowed a dynamic setting where the shape under study is expected to function over a finite planning horizon and the system parameters such as the fixed point locations and expropriation costs are subject to change. The shape was permitted to relocate at the beginning of each time period according to particular relocation costs. We showed that this dynamic problem structure can be decomposed into a set of static problems under a particular vector of relocations. We discussed the solution of this model by two enumeration procedures. Subsequently, we derived an incomplete dynamic programming procedure which is suitable for this distinct problem structure. In this method, there is no need to evaluate all the branches of the branching tree and one proceeds with keeping the minimum stage cost. The evaluation of a branch is postponed until relocation takes place in the lower-level problems. With this postponing structure, the procedure turned out to be superior to the two enumeration procedures in terms of tree size. </p> / Thesis / Doctor of Philosophy (PhD)