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The aims of this work are to study Rees algebras of filtrations of monomialideals associated to covering polyhedra of rational matrices with non-negativeentries and non-zero columns using combinatorial optimization and integerprogramming, and to study powers of monomial ideals and their integral closuresusing irreducible decompositions and polyhedral geometry. We study theWaldschmidt constant and the ic-resurgence of the filtration associated to acovering polyhedron and show how to compute these constants using linearprogramming. Then we show a lower bound for the ic-resurgence of the ideal ofcovers of a graph and prove that the lower bound is attained when the graph isperfect. We also show lower bounds for the ic-resurgence of the edge ideal of agraph and give an algorithm to compute the asymptotic resurgence of squarefreemonomial ideals. A classification of when Newton's polyhedron is theirreducible polyhedron is presented using integral closure.

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