Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. VII. Inverse semigroup theory, closures, decomposition of perturbations

In this self-contained paper, we present a theory of the piecewise linear minimal valid functions for the 1-row Gomory-Johnson infinite group problem. The non-extreme minimal valid functions are those that admit effective perturbations. We give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The … Read more

All Cyclic Group Facets Inject

We give a variant of Basu–Hildebrand–Molinaro’s approximation theorem for continuous minimal valid functions for Gomory–Johnson’s infinite group problem by piecewise linear two-slope extreme functions [Minimal cut-generating functions are nearly extreme, IPCO 2016]. Our theorem is for piecewise linear minimal valid functions that have only rational breakpoints (in 1/q Z for some q ∈ N) and … Read more

New computer-based search strategies for extreme functions of the Gomory–Johnson infinite group problem

We describe new computer-based search strategies for extreme functions for the Gomory–Johnson infinite group problem. They lead to the discovery of new extreme functions, whose existence settles several open questions. Article Download View New computer-based search strategies for extreme functions of the Gomory–Johnson infinite group problem

Equivariant Perturbation in Gomory and Johnson’s Infinite Group Problem. III. Foundations for the k-Dimensional Case with Applications to k=2

We develop foundational tools for classifying the extreme valid functions for the k-dimensional infinite group problem. In particular, (1) we present the general regular solution to Cauchy’s additive functional equation on bounded convex domains. This provides a k-dimensional generalization of the so-called interval lemma, allowing us to deduce affine properties of the function from certain … Read more

Constrained Infinite Group Relaxations of MIPs

Recently minimal and extreme inequalities for continuous group relaxations of general mixed integer sets have been characterized. In this paper, we consider a stronger relaxation of general mixed integer sets by allowing constraints, such as bounds, on the free integer variables in the continuous group relaxation. We generalize a number of results for the continuous … Read more