Rees algebras of filtrations of covering polyhedra and integral closure of powers of monomial ideals

The aims of this work are to study Rees algebras of filtrations of monomial ideals associated to covering polyhedra of rational matrices with non-negative entries and non-zero columns using combinatorial optimization and integer programming, and to study powers of monomial ideals and their integral closures using irreducible representations and polyhedral geometry. We study the Waldschmidt constant and the ic-resurgence of the filtration associated to a covering polyhedron and show how to compute these constants using linear programming. Then we show lower bounds for the ic-resurgence of the ideal of covers of a graph and prove that the lower bound is attained when the graph is perfect.