The first part of this dissertation is on the topic of self-polar polytopes, which are polytopes that are orthogonal transformations of their polar sets. These polytopes were first studied by Lovasz as a means of establishing the chromatic number of distance graphs on spheres, and they can also be used to construct triangle-free graphs with arbitrarily high chromatic number. We investigate the existence, construction, facial structure, and practical applications of self-polar polytopes, as well as the place of these polytopes within the broader set of self-dual polytopes.

The second part of this dissertation is on the topic of sequential importance sampling algorithms. A great many varied problems in the computational sciences can be solved by counting the leaves of a decision tree, or, more generally, summing a cost function over the nodes of the tree. The main two classes of algorithms for summing a cost function over a tree are Markov chain Monte Carlo methods and sequential importance sampling. We explore in depth the application of our algorithm to counting linear extensions of a partially ordered set. We also provide a generalization of one of the newest sequential importance sampling algorithms, stochastic enumeration, and conduct rigorous analysis, both theoretical and numerical, to the resulting algorithm.