Light on the infinite group relaxation II: sufficient conditions for extremality, sequences, and algorithms

Minimal cut-generating functions are nearly extreme

“…Since π is subadditive and symmetric by Theorems 1.1 and 2.1, π| 1 q Z is subadditive and symmetric for all rational numbers in 1 q Z. It is then easy to see that π pwl is also subadditive and symmetric and therefore satisfies conditions (ii) and (iii) in Theorems 1.1 and 2.1 (see also [6,Theorem 8.3]). Also, since all integers are in 1 q Z, condition (i) in Theorems 1.1 and 2.1 is easily verified for π pwl .…”

“…We say that a CGF π is extreme if there do not exist distinct CGFs π 1 and π 2 such that π = π 1 +π 2 2 . This is a subset of strongly minimal functions [18] that corresponds to a notion of "facets" in the context of CGFs (see also [5,6] for other notions of "facet" for CGFs). Because of the importance of facet-defining cuts in Integer Programming, there has been substantial interest in obtaining and understanding extreme functions (see [5,6] for a survey).…”

“…This is a subset of strongly minimal functions [18] that corresponds to a notion of "facets" in the context of CGFs (see also [5,6] for other notions of "facet" for CGFs). Because of the importance of facet-defining cuts in Integer Programming, there has been substantial interest in obtaining and understanding extreme functions (see [5,6] for a survey). For example, a celebrated result is Gomory and Johnson's 2-Slope Theorem (Theorem 2.4 below) that gives a sufficient condition for a CGF to be extreme (in the affine lattice setting with n = 1; see [7,10,18] for generalizations).…”

“…For example, even verifying the extremality of a function is not completely understood (see [5,6] for preliminary steps in this direction); a simple characterization like Theorem 1.1 seems all the more unlikely.…”

“…Cut-generating functions have received significant attention in the literature (see the surveys [3,5,6] and the references therein). One important feature is that cut-generating functions capture known general purpose cuts, for example the prominent GMI cuts and more generally split cuts (when S is an affine lattice) and the lopsided cuts of Balas and Qualizza [1] (when S is a truncated affine lattice).…”