2 research outputs found
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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 additivity relations. (2) We study the
discrete geometry of additivity domains of piecewise linear functions, providing a
framework for finite tests of minimality and extremality. (3) We 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 the assumption of 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
Recommended from our members
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 additivity relations. (2) We study the
discrete geometry of additivity domains of piecewise linear functions, providing a
framework for finite tests of minimality and extremality. (3) We 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 the assumption of 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