On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup Theory and Equivariant Decomposition Theorem

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Abstract

The non-extreme minimal valid functions for the Gomory–Johnson infinite group problem are those that admit effective perturbations. For a class of piecewise linear functions for the 1-row problem we give a precise description of the space of these perturbations as a direct sum of certain finite- and infinite-dimensional subspaces. The infinite-dimensional subspaces have partial symmetries; to describe them, we develop a theory of inverse semigroups of partial bijections, interacting with the functional equations satisfied by the perturbations. Our paper provides the foundation for grid-free algorithms for testing extremality and for computing liftings of non-extreme functions. The grid-freeness makes the algorithms suitable for piecewise linear functions whose breakpoints are rational numbers with huge denominators.

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Hildebrand, R., Köppe, M., & Zhou, Y. (2019). On Perturbation Spaces of Minimal Valid Functions: Inverse Semigroup Theory and Equivariant Decomposition Theorem. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 11480 LNCS, pp. 247–260). Springer Verlag. https://doi.org/10.1007/978-3-030-17953-3_19

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