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Let ''<math>{\mathcal C'' }</math> be a clutter on ground set ''V''. We investigate the following linear system ''<math>LP({\mathcal C})'' </math> associated with ''<math>{\mathcal C''}</math>:

x(A) & \geq 1 & \text{ for each } A \in {\mathcal C}\\

* A clutter ''<math>{\mathcal C'' }</math> '''packs''' if both ''<math>LP({\mathcal C})'' </math> and its dual have integral optimal solutions for ''<math>w''='''{\mathbf 1'''}</math>.* A clutter ''<math>{\mathcal C'' }</math> has the '''packing property''' if both ''<math>LP({\mathcal C})'' </math> and its dual have integral optimal solutions if <math>w \in \{0,1, +\infty\}^V</math>.* A clutter ''<math>{\mathcal C'' }</math> has the '''max flow min cut property''' ('''MFMC property''' for short) if both ''<math>LP({\mathcal C})'' </math> and its dual have integral optimal solutions if <math>w \in \mathbb{Z}_+^V</math>. Equivalently, if ''<math>LP({\mathcal C})'' </math> is [[Total dual integrality|TDI]].* A clutter ''<math>{\mathcal C'' }</math> is '''ideal''' if ''<math>LP({\mathcal C})'' </math> has an integral optimal solution if <math>w \in \mathbb{Z}_+^V</math>. Equivalently, if ''<math>LP({\mathcal C})'' </math> describes an integer polyhedron.

The '''blocker''' of a clutter is the clutter of inclusionwise minimal transversals. Given a clutter ''<math>{\mathcal C'' }</math> and an element <math>v \in V</math>, the '''deletion''' of ''v'' means the restriction of the clutter to <math>V-v</math>, while the '''contraction''' of ''v'' is the clutter on ground set <math>V-v</math> formed by the inclusionwise minimal elements of <math>\{X-v: X \in {\mathcal C}\}</math>. A '''minor''' of ''<math>{\mathcal C'' }</math> is a clutter obtained by a sequence of deletions and contractions.