[Paper Review] Blowup algebras of square-free monomial ideals and some links to combinatorial optimization problems

Isidoro Gitler, Enrique Reyes+1 more|arXiv (Cornell University)|Sep 21, 2006

Commutative Algebra and Its Applications33 references3 citations

TL;DR

This paper establishes algebraic-geometric connections between blowup algebras of square-free monomial ideals and combinatorial optimization via incidence matrices and polyhedra. It proves that the normality of the Rees algebra and reducedness of the associated graded ring are equivalent to total dual integrality of the system defining the set covering polyhedron, supporting a conjecture of Conforti and Cornuéjols on the König property and integrality of polyhedra.

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 Cornuéjols using an algebraic approach.

Motivation & Objective

• To establish algebraic-geometric connections between blowup algebras of square-free monomial ideals and combinatorial optimization problems.
• To investigate the normality of Rees algebras and reducedness of associated graded rings in terms of polyhedral and clutter-theoretic properties.
• To provide algebraic evidence for the Conforti–Cornuéjols conjecture on the equivalence between integrality of the set covering polyhedron and the König property of the clutter.
• To characterize minimal vertex covers of clutters via integral vertices of the set covering polyhedron and minimal primes of edge ideals.
• To explore the implications of these algebraic conditions on the structure of Rees algebras and their canonical modules.

Proposed method

• Use the incidence matrix $A$ of a square-free monomial ideal $I$ to define the set covering polyhedron $Q(A) = \{x \in \mathbb{R}^n \mid x \geq 0, xA \geq \mathbf{1}\}$.
• Apply total dual integrality (TDI) theory to link integrality of $Q(A)$ with algebraic properties of $R[It]$ and $\mathrm{gr}_I(R)$.
• Characterize minimal vertex covers of the clutter $\mathcal{C}$ associated with $I$ as integral vertices of $Q(A)$ and minimal primes of $I=I(\mathcal{C})$, using support and linear independence arguments.
• Use canonical module theory and determinantal representations to analyze Gorenstein and normal properties of Rees algebras.
• Apply induction and monomial substitution to show existence of disjoint minimal vertex covers when $\overline{R[It]} = R_s(I)$, the integral closure of the Rees algebra.
• Leverage Alexander duality and the structure of associated primes to relate the König property of $\mathcal{C}$ and its dual clutter $\mathcal{D}$.

Experimental results

Research questions

• RQ1 Under what conditions is the Rees algebra $R[It]$ normal, and how does this relate to the integrality of the set covering polyhedron $Q(A)$?
• RQ2 When is the associated graded ring $\mathrm{gr}_I(R)$ reduced, and how does this connect to the TDI property of the system $xA \geq \mathbf{1}, x \geq 0$?
• RQ3 How do the existence of disjoint minimal vertex covers in a clutter $\mathcal{C}$ reflect on the algebraic structure of $R[It]$?
• RQ4 What is the relationship between the Gorenstein property of $R[It]$ and the height of the ideal $I$?
• RQ5 To what extent does the condition $\overline{R[It]} = R_s(I)$ imply the König property for the clutter $\mathcal{C}$ and its dual?

Key findings

• The Rees algebra $R[It]$ is normal if and only if the system $xA \geq \mathbf{1}, x \geq 0$ is totally dual integral, which implies that $Q(A)$ has only integral vertices.
• The associated graded ring $\mathrm{gr}_I(R)$ is reduced if and only if $I_i = I \cap K[X \setminus \{x_i\}]$ is normal for all $i=1,\ldots,n$, provided $I$ is unmixed and $Q(A)$ is integral.
• If $\overline{R[It]} = R_s(I)$, then there exist $d$ mutually disjoint minimal vertex covers $C_1, \ldots, C_d$ such that each edge of $\mathcal{C}$ intersects each $C_k$ in exactly one vertex.
• The clutter $\mathcal{C}$ has the König property if and only if its Alexander dual $I_c(\mathcal{C})$ is unmixed and $\overline{R[I_c(\mathcal{C})t]} = R_s(I_c(\mathcal{C}))$, which holds when $\overline{R[It]} = R_s(I)$.
• The Rees algebra $R[It]$ is Gorenstein only if the height $g$ of $I$ is 2, as higher heights lead to non-Gorenstein determinantal structures.
• The canonical module of $R[It]$ is isomorphic to $\omega_R(1,t)^{g-2}$ when $I$ is a complete intersection locally, and $R[It]$ is Gorenstein only when $g=2$.