Ehrhart clutters: Regularity and Max-Flow Min-Cut

TL;DR

This paper explores Ehrhart clutters, establishing their properties, providing bounds on algebraic invariants, and connecting them to graph theory and packing conjectures, with proofs for specific graph classes.

Contribution

It introduces Ehrhart clutters, links them to the MFMC property, and proves the conjecture for Meyniel and perfect graphs, advancing understanding of packing problems.

Findings

Ehrhart clutters are linked to the MFMC property.

Bounds on Castelnuovo-Mumford regularity are established.

Conjecture proven for Meyniel and perfect graphs.

Abstract

If C is a clutter with n vertices and q edges whose clutter matrix has column vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide sharp bounds on the Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC property. We prove this conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when C is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.