ABSTRACT
In this chapter we relate commutative algebra and blowup algebras with combinatorial optimization to gain insight on these research areas. A main goal here is to connect algebraic properties of blowup algebras associated to edge ideals with combinatorial and optimization properties of clutters and polyhedra. A conjecture of Conforti and Cornue´jols about packing problems is examined from an algebraic point of view. We study max-flow min-cut problems of clutters, packing problems, and integer rounding properties of systems of linear inequalities-and their underlying polyhedra-to analyze algebraic properties of blowup algebras and edge ideals (e.g., normality, normally torsion freeness, Cohen-Macaulayness, unmixedness). Systems with integer rounding properties and clutters with the max-flow min-cut property come from linear optimization problems [372, 373].
