Combinatorial Optimization plays a significant role in applied mathematics, supplying solutions to many scientific problems in a variety of fields, including computer science and computational networks. This dissertation first reviews a number of problems from combinatorial optimization and the algorithms used to solve them.; The author then presents original solutions to the domino portrait problem, which involves arranging complete sets of dominos to resemble photographic portraits when seen from a distance. The first approach makes use of a greedy algorithm. Because the greedy algorithm often encounters blockages, a new technique was developed to avoid these blockages. Next, a local search algorithm was used to solve the problem. In both new solutions, the cost function was modified so that important positions in the portrait such as facial features were emphasized, thus improving the results. A singular value decomposition (SVD) was used to construct a "support matrix" necessary for this new cost function. Algorithms used in computing the SVD include the Householder method and the QR method.; The second problem dealt with is the maximum clique problem and its application of finding ovoids in finite polar spaces. Again, local search provides an efficient way to search for maximum cliques in graphs and hence for finding ovoids in finite polar spaces.
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