In combinatorial optimization, covering problems are those problems where given a ground set and a family of subsets of the ground set, the objective is to find a minimum cost set of subsets whose union contains the ground set. We consider covering problems in the context of Computational Geometry, which is a subfield of Computer Science that deals with problems associated with geometric objects such as points, lines, disks, polygons etc. In particular, geometric covering is an active research area, where the ground set and the family of subsets are induced by geometric objects. Covering problems in combinatorial optimizations often have a geometric analogue that arises naturally, and though such problems remain NP-hard, it is often possible to exploit the geometric properties of the set system to obtain better approximation algorithms. In this work, the fundamental problem that we consider is a generalization of geometric covering, where each element in the ground set may need to covered by more than one subset. To be precise, the problem is defined as follows: given two sets of points X, Y and a coverage function κ : X → Z+ ∪ {0}, construct balls centered on the points in Y such that each point in X is covered by at least κ(x) distinct balls. The objective in this problem is to minimize the total cost, which is a function of the radii of the balls. This problem is termed as the metric multi-cover (MMC) problem. We first consider version of the MMC problem where κ(x) = 1 for all clients, i.e. the 1-covering case. The known results that give a constant factor approximation for this problem are variations of LP-based primal-dual algorithm. We use a modified local search technique, motivated by geometric idea, to derive a simple, constant- factor approximation for this problem in Chapter 2. We then look at the MMC problem where the point sets X,Y are in the Euclidean plane, and each client x ∈ X needs to be covered by at least κ(x) distinct disks centered on the points in Y . In Chapter 4, we give the first polynomial time constant factor approximation for this problem, in which the constant is independent of the coverage function κ. Our solution also has an incremental property, which allows the algorithm to handle increases in the coverage requirement by increasing the radii of the current server disks, without affecting the approximation factor. In the next problem, we move from the Euclidean plane to arbitrary metric spaces where we consider the uniform MMC problem. In this problem, each client x has the demand κ(x) = k, where k > 0 is an integer. We give the first constant factor approximation (independent of k) for this problem. The key contribution that led to this result is the formulation of a partitioning scheme of the servers in the uniform MMC problem, that reduces the uniform MMC problem to k instances of 1-covering problem, while preserving the optimality of the solution to a constant multiplicative factor. We present the partitioning scheme as an independent result in Chapter 5, which we then use to solve the uniform MMC problem in Chapter 6. The MMC problem with arbitrary coverage function κ is then considered in Chapter 7. The key challenge that the non-uniform version presents is that the symmetry of the server partitioning scheme breaks down as the coverage demands of clients are independent of each other. We present a constant factor algorithm for this problem in Chapter 7. The last problem that we consider is the t-MMC problem, which is a restricted version of the uniform MMC problem. The objective is to compute a cover in which each client is covered by at least k distinct server disks, using atmost t server disks in total. This problem is a generalization of the clustering problem (where k = 1), and to our knowledge this is the first time this generalization has been considered. We give a constant factor approximation for this problem in Chapter 8.

Neste trabalho, objetivamos apresentar o Teorema de Alon e Naor, o qual afirma que existe um algoritmo de aproximação para a norma de corte de uma matriz qualquer, sendo que a garantia de desempenho desse algoritmo é a inversa da constante de Grothendieck. Introduzimos a norma de corte de uma matriz e exibimos algumas de suas propriedades. Uma delas é que a norma de corte é equivalente a uma outra norma, que é valor ótimo de um programa inteiro quadrático que pode ser relaxado por um programa semidefinido. Além do Teorema de Alon e Naor, construímos mais dois algoritmos de aproximação para a norma de corte. Ambos possuem garantia de desempenho inferior que a do Teorema de Alon e Naor, porém as técnicas que foram utilizadas para obter tais algoritmos são interessantes. Enunciamos a Desigualdade e Grothendieck reformulada por Lindenstrauss e Pelcýnski e mostramos uma cota superior para a constante de Grothendieck que se baseia no Argumento de Krivine. Finalmente, apresentamos três aplicações do Teorema de Alon e Naor: em corte máximo de um grafo; na versão algorítmica do Lema da Regularidade de Szemerédi; e no Teorema de Frieze e Kannan. / In this work, our objective is to present Alon and Naor\'s Theorem, which states that there exists an approximation algorithm for cut-norm of any matrix and that the approximations guarantee of the algorithm is the inverse of the Grothendieck\'s constant. We introduce the cut-norm of a matrix and we show some of its properties. One is that the cut-norm is equivalent of some other norm which is the optimum value of quadratic integer program which can be relaxed for a semidefinite program. Beyond Alon and Naor\'s Theorem, we construct two more approximation algorithm for cut-norm. The approximation guarantee of both is inferior to the Alon and Naor\'s Theorem, but the techniques for obtaining such algorithms is interesting. We show Grothendieck\'s Inequality reformulated by Lindenstrauss e Pelcýnski and we show an upper bound for the Grothendieck\'s constant which is based on Krivine\'s Argument. Furthermore, we show three applications of Alon and Naor\'s Theorem: Maximum cut of a graph, an algorithmic version of Szemerédi Regularity Lemma, and Frieze and Kannan\'s Theorem.

We present a set of approximation results for several covering problems in dense graphs. These results show that for several problems, classical algorithms with constant approximation ratios can be analyzed in a finer way, and provide better constant approximation ratios under some density constraints. In particular, we show that the maximal matching heuristic approximates VERTEX COVER (VC) and MINIMUM MAXIMAL MATCHING (MMM) with a constant ratio strictly smaller than 2 when the proportion of edges present in the graph (weak density) is at least 3/4, or when the normalized minimum degree (strong density) is at least 1/2. We also show that this result can be improved by a greedy algorithm which provides a constant ratio smaller than 2 when the weak density is at least 1/2. We also provide tight families of graphs for all these approximation ratios. We then looked at several algorithms from the literature for VC and SET COVER (SC). We present a unified and critical approach to the Karpinski/Zelikovsky, Imamura/Iwama and Bar-Yehuda/Kehat algorithms, identifying the general the general scheme underlying these algorithms. Finally, we look at the CONNECTED VERTEX COVER (CVC) problem,for which we proposed new approximation results in dense graphs. We first analyze Carla Savage's algorithm, then a new variant of the Karpinski-Zelikovsky algorithm. Our results show that these algorithms provide the same approximation ratios for CVC as the maximal matching heuristic and the Karpinski-Zelikovsky algorithm did for VC. We provide tight examples for the ratios guaranteed by both algorithms. We also introduce a new invariant, the "price of connectivity of VC", defined as the ratio between the optimal solutions of CVC and VC, and showed a nearly tight upper bound on its value as a function of the weak density. Our last chapter discusses software aspects, and presents the use of the GRAPHEDRON software in the framework of approximation algorithms, as well as our contributions to the development of this system. / Nous présentons un ensemble de résultats d'approximation pour plusieurs problèmes de couverture dans les graphes denses. Ces résultats montrent que pour plusieurs problèmes, des algorithmes classiques à facteur d'approximation constant peuvent être analysés de manière plus fine, et garantissent de meilleurs facteurs d'aproximation constants sous certaines contraintes de densité. Nous montrons en particulier que l'heuristique du matching maximal approxime les problèmes VERTEX COVER (VC) et MINIMUM MAXIMAL MATCHING (MMM) avec un facteur constant inférieur à 2 quand la proportion d'arêtes présentes dans le graphe (densité faible) est supérieure à 3/4 ou quand le degré minimum normalisé (densité forte) est supérieur à 1/2. Nous montrons également que ce résultat peut être amélioré par un algorithme de type GREEDY, qui fournit un facteur constant inférieur à 2 pour des densités faibles supérieures à 1/2. Nous donnons également des familles de graphes extrémaux pour nos facteurs d'approximation. Nous nous somme ensuite intéressés à plusieurs algorithmes de la littérature pour les problèmes VC et SET COVER (SC). Nous avons présenté une approche unifiée et critique des algorithmes de Karpinski-Zelikovsky, Imamura-Iwama, et Bar-Yehuda-Kehat, identifiant un schéma général dans lequel s'intègrent ces algorithmes. Nous nous sommes finalement intéressés au problème CONNECTED VERTEX COVER (CVC), pour lequel nous avons proposé de nouveaux résultats d'approximation dans les graphes denses, au travers de l'algorithme de Carla Savage d'une part, et d'une nouvelle variante de l'algorithme de Karpinski-Zelikovsky d'autre part. Ces résultats montrent que nous pouvons obtenir pour CVC les mêmes facteurs d'approximation que ceux obtenus pour VC à l'aide de l'heuristique du matching maximal et de l'algorithme de Karpinski-Zelikovsky. Nous montrons également des familles de graphes extrémaux pour les ratios garantis par ces deux algorithmes. Nous avons également étudié un nouvel invariant, le coût de connectivité de VC, défini comme le rapport entre les solutions optimales de CVC et de VC, et montré une borne supérieure sur sa valeur en fonction de la densité faible. Notre dernier chapitre discute d'aspects logiciels, et présente l'utilisation du logiciel GRAPHEDRON dans le cadre des algorithmes d'approximation, ainsi que nos contributions au développement du logiciel.

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We study the problem of finding the smallest power of an input graph that has <em>k</em> disjoint dominating sets, where the <em>i</em>th power of an input graph <em>G</em> is constructed by adding edges between pairs of vertices in <em>G</em> at distance <em>i</em> or less, and a subset of vertices in a graph <em>G</em> is a dominating set if and only if every vertex in <em>G</em> is adjacent to a vertex in this subset.   The problem is a different view of the <em>d</em>-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the <em>d</em>th power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are <em>k</em> types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance <em>d</em>".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the <em>d</em>-domatic number problem. We show that the problem is NP-complete for any fixed <em>k</em> greater than two even when the input graph is restricted to split graphs, <em>2</em>-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering <em>k</em> as a constant, and for strongly chordal graphs, for any <em>k</em>. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time <em>O</em>(2. 73^<em>n</em>). Moreover, we prove that the problem cannot be approximated within ratio smaller than <em>2</em> even for split graphs, <em>2</em>-connected graphs, and planar bipartite graphs of degree four. We propose a greedy <em>3</em>-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work.

Piercing problems arise often in facility location, which is a well-studied area of computational geometry. The general form of the piercing problem discussed in this dissertation asks for the minimum number of facilities for a set of given rectangular demand regions such that each region has at least one facility located within it. It has been shown that even if all regions are uniform sized squares, the problem is NP-hard. Therefore we concentrate on approximation algorithms for the problem. As the known approximation ratio for arbitrarily sized rectangles is poor, we restrict our effort to designing approximation algorithms for unit-height rectangles. Our e-approximation scheme requires <I>n</I>^{<I>O</I>(1/&epsilon;²)} time. We also consider the problem with restrictions like bounding the depth of a point and the width of the rectangles. The approximation schemes for these two cases take <I>n</I>^{<I>O</I>(1/&epsilon;)} time. We also show how to maintain a factor 2 approximation of the piercing set in <I>O</I>(log <I>n</I>) amortized time in an insertion-only scenario.

We study the problem of finding the smallest power of an input graph that has <em>k</em> disjoint dominating sets, where the <em>i</em>th power of an input graph <em>G</em> is constructed by adding edges between pairs of vertices in <em>G</em> at distance <em>i</em> or less, and a subset of vertices in a graph <em>G</em> is a dominating set if and only if every vertex in <em>G</em> is adjacent to a vertex in this subset.   The problem is a different view of the <em>d</em>-domatic number problem in which the goal is to find the maximum number of disjoint dominating sets in the <em>d</em>th power of the input graph.   This problem is motivated by applications in multi-facility location and distributed networks. In the facility location framework, for instance, there are <em>k</em> types of services that all clients in different regions of a city should receive. A graph representing the map of regions in the city is given where the nodes of the graph represent regions and neighboring regions are connected by edges. The problem is how to establish facility servers in the city (each region can host at most one server) such that every client in the city can access a facility server in its region or in a region in the neighborhood. Since it may not be possible to find a facility location satisfying this condition, "a region in the neighborhood" required in the question is modified to "a region at the minimum possible distance <em>d</em>".   In this thesis, we study the connection of the above-mentioned problem with similar problems including the domatic number problem and the <em>d</em>-domatic number problem. We show that the problem is NP-complete for any fixed <em>k</em> greater than two even when the input graph is restricted to split graphs, <em>2</em>-connected graphs, or planar bipartite graphs of degree four. In addition, the problem is in P for bounded tree-width graphs, when considering <em>k</em> as a constant, and for strongly chordal graphs, for any <em>k</em>. Then, we provide a slightly simpler proof for a known upper bound for the problem. We also develop an exact (exponential) algorithm for the problem, running in time <em>O</em>(2. 73^<em>n</em>). Moreover, we prove that the problem cannot be approximated within ratio smaller than <em>2</em> even for split graphs, <em>2</em>-connected graphs, and planar bipartite graphs of degree four. We propose a greedy <em>3</em>-approximation algorithm for the problem in the general case, and other approximation ratios for permutation graphs, distance-hereditary graphs, cocomparability graphs, dually chordal graphs, and chordal graphs. Finally, we list some directions for future work.

In this dissertation, we study the single machine scheduling problem with an objective of minimizing the total completion time subject to release dates. The problem, denoted 1|rj £UCj ,was known to be strongly NP-hard and both theoretically and practically important. The focus of the research in this dissertation is to develop the efficient algorithms for solving the 1|rj|£UCj problem. This thesis contains two parts. In the first part, the theme concerns the approximation approach. We derive a necessary and sufficient condition for local optimality, which can be implemented as a priority rule and be used to construct three heuristic algorithms with running times of O(n log n). By ¡¨local optimality¡¨, we mean the optimality of all candidates whenever a job is selected in a schedule, without considering the other jobs preceding or following. This is the most broadly considered concepts of locally optimal rule. We also identify a dominant subset which is strictly contained in each of all known dominant subsets, where a dominant subset is a set of solutions containing all optimal schedules. In the second part, we develop our optimality algorithms for the 1|rj |£UCj problem. First, we present a lemma for estimating the sum of delay times of the rest jobs, if the starting time is delayed a period of time in a schedule. Then, using the lemma, partially, we proceed to develop a new partition property and three dominance theorems, that will be used and have improved the branch-and-bound algorithms for our optimization approach. By exploiting the insights gained from our heuristics as a branching scheme and by exploiting our heuristics as an upper bounding procedure, we propose three branch-and-bound algorithms. Our algorithms can optimally solve the problem up to 120 jobs, which is known to be the best till now.

The Macroscopic Fundamental Diagram (MFD) is a diagram that presents a relationship between the average flow (production) and the average density in an urban network. Ever since the existence of low scatter MFD in urban road network was verified, significant efforts have been made to describe the MFD quantitatively. Due to the complexity of the traffic environment in urban networks, an accurate and explicit expression for the MFD is not yet developed and many recent research efforts for MFD rely on computer simulations. On a single corridor, an analytical approximation model for the MFD exists. However, this thesis expanded this theory in two directions. First, we specialize the method for models with equal road length on the corridor, which greatly reduces the complexity of the method. We introduce the adoption of seven straight cuts in approximation. Computer simulations are conducted and show a high compatibility with the approximated results. However the analytical approximation can only be applied with the assumption of constant circulating vehicles in the system without turnings and endogenous traffics. Secondly, we show that turnings and endogenous traffic can bring various impact on the shape of the MFD, the capacity, the critical density, the variance in density and cause a phenomenon of clustered traffic status along the MFD curve. Furthermore, the simulation using stochastic variables reveals that the variance in turning rates and endogenous traffic don’t have significant impact on the MFD. This discovery enables studies to focus on scenarios with deterministic parameters for those factors. While traditional objective of engineering for network is to maximize capacity and widen the range for the maximum capacity, our results indicate that traffic stability at the maximum performance is poor if the system does not stay constantly in equilibrium status. This thesis provides insights into the factors that affect the shape of the MFD by analytical approximation and simulation.

Both the maximum agreement forest problem and the multicut on trees problem are NP-hard, thus cannot be solved efficiently if P /=NP. The maximum agreement forest problem was motivated in the study of evolution trees in bioinformatics, in which we are given two leaf-labeled trees and are asked to find a maximum forest that is a subgraph of both trees. The multicuton trees problem has applications in networks, in which we are given a forest and a set of pairs of termianls and are asked to find a cut that separates all pairs of terminals. We develop combinatorial and algorithmic techniques that lead to improved parameterized algorithms, approximation algorithms, and kernelization algorithms for these problems. For the maximum agreement forest problem, we proceed from the bottommost level of trees and extend solutions to whole trees. With this technique, we show that the maxi- mum agreement forest problem is fixed-parameterized tractable in general trees, resolving an open problem in this area. We also provide the first constant ratio approximation algorithm for the problem in general trees. For the multicut on trees problem, we take a new look at the problem through the eyes of vertex cover problem. This connection allows us to develop an kernelization algorithm for the problem, which gives an upper bound of O(k3) on the kernel size, significantly improving the previous best upper bound O(k6). We further exploit this connection to give a parameterized algorithm for the problem that runs in time O∗ (1.62k), thus improving the previous best algorithm of running time O∗ (2k). In the protein complex prediction problem, which comes directly from the study of bioinformatics, we are given a protein-protein interaction network, and are asked to find dense regions in this graph. We formulate this problem as a graph clustering problem and develop an algorithm to refine the results for identifying protein complexes. We test our algorithm on yeast protein- protein interaction networks, and we show that our algorithm is able to identify complexes more accurately than other existing algorithms.