Minimal cut-generating functions are nearly extreme

Abstract

We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory-Johnson setting as well as a recent extension by Cornuéjols and Yıldız. We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are "dense" in the set of continuous (strongly) minimal functions.

Approximation of Minimal Functions by Extreme Functions

“…In our opinion, this result alone is likely to be a very useful tool for subsequent research in the area. Similar non-trivial departures from the techniques of [7] are required to obtain our main result.…”

“…In [7], Basu, Hildebrand, and Molinaro observed a curious property for the convex set of continuous minimal functions π : R → R, i.e., with n = 1. Moreover, they demonstrated that the extreme points of this convex set are dense in this set (under the topology of pointwise convergence).…”

“…This observation is clearly an infinite-dimensional phenomenon; compact convex sets in finite dimensions do not possess dense subsets of extreme points. The question as to whether the "density" result of [7] continues to hold for general integers n ≥ 2 was unresolved. In fact, for a related family of functions -the so-called cut generating functions for the continuous model -an analogous result was established to be true only for n = 1, 2, and false for n ≥ 3.…”

“…This paper resolves the above "density" question and shows that for all n ≥ 1, continuous extreme functions form a dense subset of the set of continuous minimal functions. While the proof strategy in this paper borrows ideas from the 1-dimensional proof in [7], there are significant technical barriers that must be surmounted in order to obtain the general n-dimensional result. For example, the fact that any continuous minimal function can be arbitrarily well-approximated by a piecewise linear minimal function can be derived easily via interpolation for n = 1, and is the first step towards the 1-dimensional proof in [7].…”

“…While the proof strategy in this paper borrows ideas from the 1-dimensional proof in [7], there are significant technical barriers that must be surmounted in order to obtain the general n-dimensional result. For example, the fact that any continuous minimal function can be arbitrarily well-approximated by a piecewise linear minimal function can be derived easily via interpolation for n = 1, and is the first step towards the 1-dimensional proof in [7]. The absence of such a result for n ≥ 2 is a major obstacle in extending the density result to higher dimensions.…”