EQUIVARIANT PERTURBATION IN GOMORY AND JOHNSON’S INFINITE GROUP PROBLEM III. FOUNDATIONS FOR THE k-DIMENSIONAL CASE AND APPLICATIONS

We develop foundational tools for classifying extremality of minimal valid functions for the kdimensional 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 from certain additivity relations. (2) We study the discrete geometry of additivity domains of piecewise linear minimal valid functions. (3) We give a powerful necessary condition based on the existence of certain 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. We give an algorithm for testing the extremality of a minimal function, under the assumption of a regularity condition called diagonal constrainedness. We also show that the extremality of a minimal valid function is equivalent to the extremality of its restriction to a certain finite k-dimensional group problem. Date: Revision: 1332 − Date: 2014-03-18 00:24:47 -0400 (Tue, 18 Mar 2014). 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).