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A note on Rees algebras and the MFMC property

I. Gitler·C. E. Valencia·R. H. Villarreal
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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.

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I. Gitler, C. E. Valencia, R. H. Villarreal. 2005-11-11. A note on Rees algebras and the MFMC property. https://arxiv.org/abs/math/0511307

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