  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.
