TWO APPLICATIONS OF COMBINATORIAL OPTIMIZATION

Abstract

OF THE DISSERTATION Two applications of combinatorial optimization by Matthew R. Oster Dissertation Director: Professor Jonathan Eckstein This thesis presents two applications of combinatorial optimization. The first part contains a detailed description of a conference scheduling problem. We model the problem as a symmetric clustering problem, or a variant of minimum k-partition we call capacitated k-partition. This problem is proved to be NP-hard to solve to optimality, and further, unless P = NP, no constant factor polynomial-time approximation algorithm exists. We also propose a branch-and-cut algorithm with semidefinite programming relaxations enhanced with polyhedral cuts found at each tree node. Many cutting planes are demonstrated to be satisfied, or provably close to being satisfied, by semidefinite matrices in the variable space [−1/(k − 1), 1], which is in contrast to basic linear programming relaxations. Our algorithm also relies on a novel heuristic strategy when attempting to generate feasible solutions at every tree node. We test an implementation of our algorithm on random k-partition instances as well as a particular conference data set which comes from the 13th Annual INFORMS Computing Society Conference and was solved to within 0.85% of optimum in under 4 hours. The results here are promising and provide a starting point for future projects. In the second part, we describe a project called the Boat Allocation Module, where a team comprised of United States Coast Guard (USCG) analysts, and Command, Control, and Interoperability Center for Advanced Data Analysis researchers worked