Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems

TL;DR

This paper explores the algebraic properties of blowup algebras of square-free monomial ideals and links them to combinatorial optimization problems, providing new insights and applications in algebra and combinatorics.

Contribution

It establishes connections between algebraic properties of graded algebras and combinatorial optimization of associated polyhedra and clutters, including applications to Rees algebras.

Findings

Algebraic properties relate to combinatorial optimization characteristics.

Applications to Rees algebras demonstrate practical relevance.

Provides an algebraic approach to a conjecture by Conforti and Cornuéjols.

Abstract

Let I=(x^{v_1},...,x^{v_q} be a square-free monomial ideal of a polynomial ring K[x_1,...,x_n] over an arbitrary field K and let A be the incidence matrix with column vectors {v_1},...,{v_q}. We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedrons and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornu\'ejols using an algebraic approach.