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Placing security cameras in buildings, finding good locations for cameras to enforce speed limits or placing guards to defend a border are some of the problems we face everyday. A nation that wishes to defend its border with armed guards wants to be sure the entire border is secure. However, hiring more guards than necessary can be costly. A start-up company moving into a new building wants to be sure every room in the building is seen by some security camera. Cameras are expensive and the company wants to install the smallest number of cameras; at the same time the company wants to be sure the building is secure. These problems, and many other visibility type problems, are not easy to solve in general. In some specific cases, optimal solutions can be obtained quickly. In general, finding an optimal solution may take a very long time. The original results of this thesis address some of these problems. We show some positive results for solving some of these visibility problems. We also give some negative results for some of these problems. These negative results are useful because they tell us that we are unlikely to find a fast algorithm to solve a particular problem optimally.

In this thesis, I survey 11 approximation algorithms for unweighted set cover problem. I have also implemented the three algorithms and created a software library that stores the code I have written. The algorithms I survey are: 1. Johnson's standard greedy; 2. f-frequency greedy; 3. Goldsmidt, Hochbaum and Yu's modified greedy; 4. Halldorsson's local optimization; 5. Dur and Furer semi local optimization; 6. Asaf Levin's improvement to Dur and Furer; 7. Simple rounding; 8. Randomized rounding; 9. LP duality; 10. Primal-dual schema; and 11. Network flow technique. Most of the algorithms surveyed are refinements of standard greedy algorithm.

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Satellite constellation design has been a well-studied problem since the beginning of the space age. In recent years new concepts and approaches tried to solve it with fewer satellites whilst guaranteeing coverage to the areas of interest, whether globally or regionally. This thesis introduces a novel approach based on the repeating ground track concept. It then links and converts the constellation design problem to a Set Cover problem. Although it is NP-hard, the Greedy Algorithm is capable to approximate the solution in a polynomial time with a logarithm ratio. An application of the non-symmetric strategy is illustrated with in 36 different scenarios, where altitude, sensor swath and time requirement are varied. In addition to that, a comparison with the Walker constellation on 6 scenarios is analyzed and discussed. In most cases the non-symmetric strategy produces constellations with significantly less satellites required. / Satellitkonstellationsdesign har varit ett väl studerat problem sedan början av rymdåldern. Under de senaste åren har nya koncept och tillvägagångssätt försökt lösa det med färre satelliter samtidigt som de garanterar täckning till intresseområdena, globalt eller regionalt. Detta examensarbete introducerar ett nytt tillvägagångssätt baserat på konceptet med återkommande markspår. Den länkar sedan och konverterar konstellationsdesignproblemet till ett Set Cover-problem. Även om problemet är NP-hårt, är den giriga algoritmen kapabel att approximera lösningen under polynomtid med ett logaritmförhållande. En tillämpning av den icke-symmetriska strategin illustreras med i 36 olika scenarier, där höjd, sensorsvep och tidsbehov varieras. Utöver det analyseras och diskuteras en jämförelse med Walker-konstellationen på 6 scenarier. I de allra flesta fall producerar den icke-symmetriska strategin konstellationer med betydligt färre satelliter.

Un stable (ou ensemble indépendant) est un ensemble de sommets qui sont deux à deux non adjacents. De nombreux résultats classiques en optimisation combinatoire portent sur le nombre de stabilité (défini comme la plus grande taille d'un stable), et les stables se classent certainement parmi les structures les plus simples et fondamentales en théorie des graphes. La thèse est divisée en deux parties, toutes deux liées à la notion de stables dans un graphe. Dans la première partie, nous étudions un problème de coloration de graphes, c'est à dire de partition en stables, où le but est de minimiser l'entropie de la partition. C'est une variante du problème classique de minimiser le nombre de couleurs utilisées. Nous considérons aussi une généralisation du problème aux couvertures d'ensembles. Ces deux problèmes sont appelés respectivement minimum entropy coloring et minimum entropy set cover, et sont motivés par diverses applications en théorie de l'information et en bioinformatique. Nous obtenons entre autres une caractérisation précise de la complexité de minimum entropy set cover : le problème peut être approximé à une constante lg e (environ 1.44) près, et il est NP-difficile de faire strictement mieux. Des résultats analogues sont prouvés concernant la complexité de minimum entropy coloring. Dans la deuxième partie de la thèse, nous considérons les graphes dont le nombre de stabilité augmente dès qu'une arête est enlevée. Ces graphes sont dit être "alpha-critiques", et jouent un rôle important dans de nombreux domaines, comme la théorie extrémale des graphes ou la combinatoire polyédrique. Nous revisitons d'une part la théorie des graphes alpha-critiques, donnant à cette occasion de nouvelles démonstrations plus simples pour certains théorèmes centraux. D'autre part, nous étudions certaines facettes du polytope des ordres totaux qui peuvent être vues comme une généralisation de la notion de graphe alpha-critique. Nous étendons de nombreux résultats de la théorie des graphes alpha-critiques à cette famille de facettes.

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.

The set cover problem is a well studied problem in computer science. The problem cannot be approximated to better than an log n-factor in polynomial time unless P = NP and has an O(log n)-factor approximation algorithm. We consider several special cases of the set cover problem in which geometry plays a key role. With geometric structure introduced to the problem, it may be possible to construct approximation algorithms with approximation ratios asymptotically better than log n. The first problem we consider is the decomposing coverings problem. Here, we consider a combinatorial problem: given a collection of points in the plane and a collection of objects in the plane such that each point is contained in at least k objects, partition the objects into as many sets as possible so that each set covers all of the points. We show that if the objects are translates of a convex polygon, then it is possible to partition the translates into Ω(k) covers. The second problem we consider is the planar sensor cover problem. This problem is a generalization of the decomposing coverings problem. We are given a collection of points in the plane and a collection of objects in the plane. Each of the objects can be thought of as a sensor. The sensors have a duration which can be thought of as the battery life of the sensor. The planar sensor cover problem is to schedule a start time to each of the sensors so that the points are covered by a sensor for as long as possible. We give a constant factor approximation for this problem. The key contribution to this result is a constant factor approximation to a one-dimensional version of the problem called the restricted strip cover (RSC) problem. Our result for RSC improves upon the previous best O(log log log n)-approximation, and our result for the planar sensor cover problem improves upon the previous best O(log n)-approximation. The next problem we consider is the metric clustering to minimize the sum of radii problem. Here, we are given an n-point metric (P,d) and an integer k > 0. We are interested in covering the points in P with at most k balls so that the sum of the radii of the balls is minimized. We give a randomized algorithm which solves the problem exactly in nO(log n log Δ) time, where Δ is the ratio of the maximum interpoint distance to the minimum interpoint distance. We also show that the problem is NP-hard, even in metrics induced by weighted planar graphs and when the metric has constant doubling dimension. The last problem we consider is the minimum dominating set problem for disk graphs. In this problem, we are given a set of disks in the plane, and we want to choose a minimum-cardinality subset of disks such that every disk is either in the set or intersects a disk in the set. For any ε > 0, we show that a simple local search algorithm is a (1+ ε)-approximation for the problem which improves upon the previous best O(log n)-approximation algorithm.

In this Master's thesis I am focusing on the creation of a series of exercices for the libero player in volleyball. Exercises focus on the training of specific activities that are mandatory while participating in the game: passing, digging, setting and covering. In the description of individual game activities, always emphasize the key moments of training. The first part focuses on introducing readers to the theoretical basics of volleyball. The main focus is on the detailed description and analysis of the key gaming activities of libero such as mentioned before. The second part of the Master's thesis consists of the exercises I have created for libero player specialization. Exercises are divided into four parts, according to individual game activities. Exercises shown in kinograms (phased footage from the video that is verbally described). Attached to the Master's thesis is a video with a complete series of skill developing exercises with comments. Key words: libero, methodical series, pass, set, dig, cover

Background. With the increasing demand for cybersecurity, there is a growing interest in understanding cyber-attack surfaces and vectors. Security Operation Centers (SOCs) play a crucial role in defensive cybersecurity, and Security Informationand Event Management (SIEM) systems are used to monitor and analyze the security status of computer systems. However, SIEM systems face challenges such asdata overload and the need for effective data selection.Objectives. This research aims to develop a method for reducing the number ofsets of Indicators of Compromise (IOCs) processed by SIEM systems while maintaining maximum coverage. The objectives include conducting a literature review onIOCs processing and data reduction, preparing data from the Open Threat Exchange(OTX) platform, developing a method for minimizing IOCs sets, and evaluating theeffectiveness of the proposed solution.Methods. The evaluation of the methods is performed numerically using a FuzzyTable. The research also involves developing a mathematical model that describesthe relationships between different types of IOCs and the possibility of various representations for the same object. The model takes into account weight assignmentto each indicator. Software implementation is carried out. The effectiveness of thedeveloped method is evaluated using metrics such as the coverage of the initial setof IOCs and the data reduction rateResults. Unfortunately, none of the methods fully met all the criteria. Fuzzy logicwas selected as the decision-making approach. A mathematical data model was developed to represent IOCs and associated pulses as sets. Dependencies were described tofilter out duplicate indicators. Implementation was done using the Python programming language. Three algorithms were implemented: Set cover problem, Weightedcoverage maximization, and Budget cover problem. Tests were conducted on theentire data set and subsets to analyze performance. The number of IOCs decreasedfrom 4115 to 3341, representing a reduction of 25.5% to 93% according to the Totaldata reduction metric. Conclusions. Overall, the developed method reduced information and minimizedindicator sources, offering a valuable approach for reducing data in IOC processing.