Sublinear Bounds for a Quantitative Doignon-Bell-Scarf Theorem

The recent paper "A quantitative Doignon-Bell-Scarf Theorem" by Aliev et al. generalizes the famous Doignon-Bell-Scarf Theorem on the existence of integer solutions to systems of linear inequalities. Their generalization examines the number of facets of a polyhedron that contains exactly k integer points in Rn . They show that there exists a number c(n, k) such that any polyhedron in Rn that contains exactly k integer points has a relaxation to at most c(n; k) of its inequalities that will define a new polyhedron with the same integer points. They prove that c(n, k) = O(k2n ). In this paper, we improve the bound asymptotically to be sublinear in k. We also provide lower bounds on c(n, k), along with other structural results. For dimension n = 2, our bounds are asymptotically tight to within a constant.

By: Stephen R. Chestnut, Robert Hildebrand, Rico Zenklusen

Published in: RC25609 in 2016


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