A note on Rees algebras and the MFMC property

TL;DR

This paper explores the relationship between Rees algebras and the max-flow min-cut property of clutters, providing algebraic characterizations and practical methods for verification in combinatorial optimization.

Contribution

It characterizes the max-flow min-cut property via the normality of Rees algebras and introduces an effective computational approach using Normaliz.

Findings

C has the max-flow min-cut property iff I is normally torsion free.

Provides an algebraic criterion for the max-flow min-cut property.

Offers a practical method to check the property using Normaliz.

Abstract

We study irreducible representations of Rees cones and characterize the max-flow min-cut property of clutters in terms of the normality of Rees algebras and the integrality of certain polyhedra. Then we present some applications to combinatorial optimization and commutative algebra. As a byproduct we obtain an "effective" method, based on the program "Normaliz", to determine whether a given clutter satisfies the max-flow min-cut property. Let C be a clutter and let I be its edge ideal. We prove that C has the max-flow min-cut property if and only if I is normally torsion free, that is, I^i=I^{(i)} for all i>=1, where I^{(i)} is the ith symbolic power of I.