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.
We consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovasz characterizing maximal lattice-free convex sets in R(n)
This is a survey on the infinite group problem, an infinite-dimensional relaxation of integer linear optimization problems introduced by Ralph Gomory and Ellis Johnson in their groundbreaking papers titled Some continuous functions related to corner polyhedra I, II [Math. Programming 3 (1972), 23-85, 359-389]. The survey presents the infinite group problem in the modern context of cut generating functions. It focuses on the recent developments, such as algorithms for testing extremality and breakthroughs for the k-row problem for general k ≥ 1 that extend previous work on the single-row and two-row problems. The survey also includes some previously unpublished results; among other things, it unveils piecewise linear extreme functions with more than four different slopes. An interactive companion program, implemented in the open-source computer algebra package Sage, provides an updated compendium of known extreme functions.1 This notation for functions of finite support is used, for example, in [2]. 2 This model is called the mixed-integer infinite relaxation, for example in the survey [22], or sometimes the mixed-integer group problem, but we shall not use either of these terms in the remainder of our survey.3 This model is called the continuous infinite relaxation, for example in the survey [22], or sometimes the continuous group problem.
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