Abstract
We study continuous (strongly) minimal cut generating functions for the model where all variables are integer. We consider both the original Gomory-Johnson setting as well as a recent extension by Cornuéjols and Yıldız. We show that for any continuous minimal or strongly minimal cut generating function, there exists an extreme cut generating function that approximates the (strongly) minimal function as closely as desired. In other words, the extreme functions are “dense” in the set of continuous (strongly) minimal functions.
Amitabh Basu gratefully acknowledges partial support from NSF grant CMMI1452820.
Most of this research was conducted while Robert Hildebrand was a postdoctoral researcher at the Institute for Operations Research, Department of Mathematics, ETH Zürich.
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When S is an affine lattice this notion is equivalent to notion of minimal inequality used in the literature.
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Basu, A., Hildebrand, R., Molinaro, M. (2016). Minimal Cut-Generating Functions are Nearly Extreme. In: Louveaux, Q., Skutella, M. (eds) Integer Programming and Combinatorial Optimization. IPCO 2016. Lecture Notes in Computer Science(), vol 9682. Springer, Cham. https://doi.org/10.1007/978-3-319-33461-5_17
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DOI: https://doi.org/10.1007/978-3-319-33461-5_17
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