Blowup Algebras of Square-Free Monomial Ideals and Some Links to Combinatorial Optimization Problems

Abstract

Let I = (x v1 , . . . , x vq ) 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 polyhedra 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 Cornuéjols using an algebraic approach.

Algebraic and combinatorial properties of ideals and algebras of uniform clutters of TDI systems

“…The next result was shown in [9] using commutative algebra methods. Here we give a simple combinatorial proof.…”

“…A breakthrough in the area of monomial algebras is the translation of combinatorial problems (e.g., the Conforti-Cornuéjols conjecture [5], the max-flow min-cut property, or the idealness of a clutter) into algebraic problems of monomial algebras [9,11]. A typical example is the following result that describes the max-flow min-cut property in algebraic and optimization terms.…”

“…We set d = α 0 (G). Clearly C is a d-uniform clutter because all minimal vertex covers of G have size d. By [19,Theorem 78.1] the polyhedron Q(A) is integral because G is a bipartite graph and bipartite graphs have integral set covering polyhedron [9,Proposition 4.27]. Any minimal vertex cover of C is of the form {x i , x j } for some edge {x i , x j } of G. Therefore by Proposition 2.2 the clutter C is 2-partitionable.…”

“…We begin in Section 2 by introducing various combinatorial properties of clutters. We then give a simple combinatorial proof of the following result of [9]: Lemma 2.1, Proposition 2.2. If C is a d-uniform clutter whose set covering polyhedron Q(A) is integral, then there are X 1 , .…”

Systems with the integer rounding property in normal monomial subrings

“…If we work in the more general context of clutters, none of the conditions (a) to (e) are equivalent. Some of these conditions are equivalent under certain assumptions (Gitler et al 2009). …”