Equivariant Perturbation in Gomory and Johnson's Infinite Group Problem. I. The One-Dimensional Case

Abstract

We give an algorithm for testing the extremality of minimal valid functions for Gomory and Johnson's infinite group problem that are piecewise linear (possibly discontinuous) with rational breakpoints. This is the first set of necessary and sufficient conditions that can be tested algorithmically for deciding extremality in this important class of minimal valid functions. We also present an extreme function that is a piecewise linear function with some irrational breakpoints, whose extremality follows from a new principle.

Equivariant perturbation in Gomory and Johnson’s infinite group problem—III: foundations for the k-dimensional case with applications to $$k=2$$ k = 2

“…We will use three important properties of π 1 , π 2 in our proofs, which are summarized in the following lemma. These facts for the one-dimensional case can be found, for instance, in [5], and are easily extended to the general k-dimensional case. Lemma 1.4.…”

“…Combinatorial representation of additivity domain through P. Our second main contribution in the present paper is a detailed study of the discrete geometry of the additivity domain E(π), as defined in (2), of a function π that is continuous piecewise linear on a polyhedral complex P. This is missing from the previous literature on R f (R k , Z k ) for k ≥ 2 and extends the discussion in the one-dimensional case in [5]. In section 3.2, we show that the subadditivity slack function ∆π (as defined in (2)) is continuous piecewise linear over a polyhedral complex in R k × R k that we call ∆P.…”

“…This addressed an inherent previously unknown arithmetic (number-theoretic) aspect of the problem and allowed the authors to give an algorithm that tests extremality of piecewise linear functions with rational breakpoints and relate extremality to a finite-dimensional problem. Theorem 1.3 (Theorems 1.3 and 1.5 in [5]). Consider the following problem.…”

“…Various sufficient conditions for extremality have been proved in the previous literature [7, 9, 11-13, 19, 24, 26]. In part I [5] of the present series of papers, the authors initiated the study of perturbation functions that are equivariant with respect to certain finitely generated reflection groups. This addressed an inherent previously unknown arithmetic (number-theoretic) aspect of the problem and allowed the authors to give an algorithm that tests extremality of piecewise linear functions with rational breakpoints and relate extremality to a finite-dimensional problem.…”

“…Contributions, techniques, and outline of this paper. In the present paper, we continue the program of [5] of algorithmically studying the extemality of piecewise linear functions. We prove several general results that hold for arbitrary dimension k and then apply them to give an algorithm that tests the extremality of a large class of functions for the case k = 2.…”