Sammendrag
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.