On the Notions of Facets, Weak Facets, and Extreme Functions of the Gomory–Johnson Infinite Group Problem

Abstract

Abstract. We investigate three competing notions that generalize the notion of a facet of finite-dimensional polyhedra to the infinite-dimensional Gomory-Johnson model. These notions were known to coincide for continuous piecewise linear functions with rational breakpoints. We show that two of the notions, extreme functions and facets, coincide for the case of continuous piecewise linear functions, removing the hypothesis regarding rational breakpoints. We then separate the three notions using discontinuous examples.

Equivariant perturbation in Gomory and Johnson’s infinite group problem. VI. The curious case of two-sided discontinuous minimal valid functions

“…Thus, we give a negative answer to Question 1.3 (b): Piecewise linear effective perturbations do not suffice to certify nonextremality of piecewise linear functions. (Moreover, in the authors' IPCO 2017 paper [16], building upon the present paper, the function π and a certain perturbation of it are instrumental in separating the classes of extreme functions, facets, and so-called weak facets, thereby solving the long-standing open question [4, Open question 2.9] regarding the relations of these notions. )…”

“…This was a vast generalization of the extremality of the Gomory mixed-integer cut. In parts of the later literature, the related notion of facets, instead of extreme functions, was considered; see [16].…”

All Cyclic Group Facets Inject

“…Minimal functions form a convex set in the space of functions π : G → R, again characterized by (3). For the infinite group problem, there is a subtle difference between the notions of extreme functions, defined by (4), and facets; see [18]. However, for the important case of continuous piecewise linear functions of R/Z, both notions agree (see [3,Proposition 2.8] and [18,Theorem 1.2]), and we will use them interchangably.…”

An extreme function which is nonnegative and discontinuous everywhere

“…So it is discontinuous everywhere and the left and right limits do not exist at any point. The function in [14] also has no left and right limits at the points of discontinuity. The graph of this function is not dense in R × [0, 1]; in fact, the closure of this graph is the union of finitely many piecewise linear curves in R 2 .…”