Many real world applications which involve computational steps are closely tied to theoretical computer science. In order for these systems to be efficiently deployed and used, a thorough analysis is required in advance. This dissertation deals with several real world problems related to the field of Robotics, which can be mathematically modeled and analyzed. One of these problems is known as the pursuit evasion problem and involves the use of independent automated robots to capture a fugitive hiding in a building or a cave system. This is an extensively studied game theory and combinatorics problem which has multiple variations. It can be modeled as a graph and the goal is to minimize the cost of capturing the evader. We deal with two completely different variations of this problem: a vision based variant, in which the robots have limited vision and thus can react when the fugitive is in line of sight; and a no-vision variant, in which the robots do not have any knowledge about the fugitive. Another problem we deal with is the problem of neighbor discovery in wireless networks using directional antennas. This is another problem which received a growing interest in the last years. Our approach to solving this problem, as well as the model, is different from the other results that have been previously published in the literature. Besides modeling and formally analyzing these problems, our focus in this dissertation is to design efficient algorithms that solve them either completely or partially.