Neogi, Rian2021-09-032021-09-032021-09-032021-08-31http://hdl.handle.net/10012/17338In this thesis, we study approximation algorithms for Connectivity Augmentation and related problems. In the Connectivity Augmentation problem, one is given a base graph G=(V,E) that is k-edge-connected, and an additional set of edges $L \subseteq V\times V$ that we refer to as links. The task is to find a minimum cost subset of links $F \subseteq L$ such that adding F to G makes the graph (k+1)-edge-connected. We first study a special case when k=1, which is equivalent to the Tree Augmentation problem. We present a breakthrough result by Adjiashvili that gives an approximation algorithm for Tree Augmentation with approximation guarantee below 2, under the assumption that the cost of every link $\ell \in L$ is bounded by a constant. The algorithm is based on an elegant decomposition based method and uses a novel linear programming relaxation called the $\gamma $-bundle LP. We then present a subsequent result by Fiorini, Gross, Konemann and Sanita who give a $3/2+\epsilon$ approximation algorithm for the same problem. This result uses what are known as Chvatal-Gomory cuts to strengthen the linear programming relaxation used by Adjiashvili, and uses results from the theory of binet matrices to give an improved algorithm that is able to attain a significantly better approximation ratio. Next, we look at the special case when k=2. This case is equivalent to what is known as the Cactus Augmentation problem. A recent result by Cecchetto, Traub and Zenklusen give a 1.393-approximation algorithm for this problem using the same decomposition based algorithmic framework given by Adjiashvili. We present a slightly weaker result that uses the same ideas and obtains a $3/2+\epsilon $ approximation ratio for the Cactus Augmentation problem. Next, we take a look at the integrality ratio of the natural linear programming relaxation for Tree Augmentation, and present a result by Nutov that bounds this integrality gap by 28/15. Finally, we study the related Forest Augmentation problem that is a generalization of Tree Augmentation. There is no approximation algorithm for Forest Augmentation known that obtains an approximation ratio below 2. We show that we can obtain a 29/15-approximation algorithm for Forest Augmentation under the assumption that the LP solution is half-integral via a reduction to Tree Augmentation. We also study the structure of extreme points of the natural linear programming relaxation for Forest Augmentation and prove several properties that these extreme points satisfy.enApproximation AlgorithmsCombinatorial OptimizationConnectivity AugmentationMaster Thesis