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 decompositions 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 a lower bound 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. We also show lower bounds for the ic-resurgence of the edge ideal of a graph and give an algorithm to compute the asymptotic resurgence of squarefree monomial ideals. A classification of when Newton's polyhedron is the irreducible polyhedron is presented using integral closure.