Given a configuration of pebbles on the vertices of a graph, a pebbling move is defined by removing two pebbles from some vertex and placing one pebble on an adjacent vertex. The cover pebbling number of a graph, γ (G), is the smallest number of pebbles such that through a sequence of pebbling moves, a pebble can eventually be placed on every vertex simultaneously, no matter how the pebbles are initially distributed. We determine Bose-Einstein and Maxwell-Boltzmann cover pebbling thresholds for the complete graph. Also, we show that the cover pebbling decision problem is NP-complete.

We obtain first-order cover pebbling thresholds of the complete graph for Maxwell Boltzmann and Bose Einstein configurations.

Machine Learning Techniques have been used on a wide array of input types: images, sound waves, text, and so forth. In articulating these input types to the almighty machine, there have been all sorts of amazing problems that have been solved for many practical purposes. Nevertheless, there are some input types which don’t lend themselves nicely to the standard set of machine learning tools we have. Moreover, there are some provably difficult problems which are abysmally hard to solve within a reasonable time frame. This thesis addresses several of these difficult problems. It frames these problems such that we can then attempt to marry the allegedly powerful utility of existing machine learning techniques to the practical solvability of said problems.

Complete coverage relies on a path planning algorithm that will move one or more robots, including the actuator, sensor, or body of the robot, over the entire environment. Complete coverage of an unknown environment is used in applications like automated vacuum cleaning, carpet cleaning, lawn mowing, chemical or radioactive spill detection and cleanup, and humanitarian de-mining. The environment is typically decomposed into smaller areas and then assigned to individual robots to cover. The robots typically use the Boustrophedon motion to cover the cells. The location and size of obstacles in the environment are unknown beforehand. An online algorithm using sensor-based coverage with unlimited communication is typically used to plan the path for the robots. For certain applications, like robotic lawn mowing, a pattern might be desirable over a random irregular pattern for the coverage operation. Assigning directional constraints to the cells can help achieve the desired pattern if the path planning part of the algorithm takes the directional constraints into account. The goal of this dissertation is to adapt the distributed coverage algorithm with unrestricted communication developed by Rekleitis et al. (2008) so that it can be used to solve the complete coverage problem with directional constraints in unknown environments while minimizing repeat coverage. It is a sensor-based approach that constructs a cellular decomposition while covering the unknown environment. The new algorithm takes directional constraints into account during the path planning phase. An implementation of the algorithm was evaluated in simulation software and the results from these experiments were compared against experiments conducted by Rekleitis et al. (2008) and with an implementation of their distributed coverage algorithm. The results of this study confirm that directional constraints can be added to the complete coverage algorithm using multiple robots without any significant impact on performance. The high-level goals of complete coverage were still achieved. The work was evenly distributed between the robots to reduce the time required to cover the cells.

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