On ideal minimally non-packing clutters

TL;DR

This paper investigates the structure of ideal minimally non-packing clutters, proposing necessary conditions and analyzing specific cases like affine planes to understand their properties and the conjecture related to transversals.

Contribution

It introduces a two-step approach to analyze ideal minimally non-packing clutters and examines the case of affine plane clutters to verify the conjecture's conditions.

Findings

Affine plane clutters satisfy the necessary conditions for ideal minimally non-packing clutters.

Affine plane clutters do not have ideal minimally non-packing clutters with blocking number ≥ 3.

Abstract

We consider the following conjecture proposed by Cornu\'ejols, Guenin and Margot: every ideal minimally non-packing clutter has a transversal of size 2. For a clutter C, the tilde clutter is the set of hyperedges of C which intersect any minimum transversal in exactly one element. We divide the (non-)existence problem of an ideal minimally non-packing clutter D into two steps. In the first step, we give necessary conditions for C = the tilde clutter of D when a clutter D is an ideal minimally non-packing clutter. In the second step, for a clutter C satisfying the conditions in the first step, we consider whether C has an ideal minimally non-packing clutter D with C= the tilde clutter of D. We show that the clutter of a combinatorial affine plane satisfies the conditions in the first step. Moreover, we show that the clutter of a combinatorial affine plane does not have any ideal minimally non-packing clutter of blocking number at least 3.