Applications of directed algebraic topology in optimization theory

Abstract

In this thesis, we consider applications of directed algebraic topology in optimization theory, by representing directed graphs as directed topological spaces. We review the classical max-flow min-cut theorem and a generalization of the theorem from numerical to semimodule-valued edge weights, which we use to develop a generalization of the linear programming duality theorem from numerical to semimodule-valued variables for linear programs that correspond to max-flow and min-cut problems.