Take a prime power q, an integer nβ₯2, and a coordinate subspace
SβGF(q)n over the Galois field GF(q). One can associate with S
an n-partite n-uniform clutter C, where every part has size q
and there is a bijection between the vectors in S and the members of
C.
In this paper, we determine when the clutter C is ideal, a
property developed in connection to Packing and Covering problems in the areas
of Integer Programming and Combinatorial Optimization. Interestingly, the
characterization differs depending on whether q is 2,4, a higher power of
2, or otherwise. Each characterization uses crucially that idealness is a
minor-closed property: first the list of excluded minors is identified, and
only then is the global structure determined. A key insight is that idealness
of C depends solely on the underlying matroid of S.
Our theorems also extend from idealness to the stronger max-flow min-cut
property. As a consequence, we prove the Replication and Ο=2 Conjectures
for this class of clutters.Comment: 32 pages, 6 figure