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, we present the general regular solution to Cauchy’s additive functional equation on restricted lower-dimensional 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 additivity relations. Next, we study the discrete geometry of additivity domains of piecewise linear functions, providing a framework for finite tests of minimality and extremality. We then give a theory of non-extremality certificates in the form of perturbation functions. We apply these tools in the context of minimal valid functions for the two-dimensional infinite group problem that are piecewise linear on a standard triangulation of the plane, under a regularity condition called diagonal constrainedness. We show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite two-dimensional group problem. This gives an algorithm for testing the extremality of a given minimal valid function. Date: Revision: 2090 − Date: 2016-07-28 13:56:14 -0700 (Thu, 28 Jul 2016). An extended abstract with some of the results of the paper has appeared as: Equivariant perturbation in Gomory and Johnson’s infinite group problem. II. The unimodular two-dimensional case in: Michel Goemans and José Correa (eds.), Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, vol. 7801, Springer, 2013, pp. 62– 73, ISBN 978-3-642-36693-2. The authors gratefully acknowledge partial support from the National Science Foundation through grants DMS-0636297 (R. Hildebrand), DMS-0914873 (R. Hildebrand, M. Köppe), and DMS-1320051 (M. Köppe). 1 ar X iv :1 40 3. 46 28 v2 [ m at h. O C ] 2 3 A ug 2 01 6 2 AMITABH BASU, ROBERT HILDEBRAND, AND MATTHIAS KÖPPE