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Complexity of linear relaxations in integer programming
For a set of integer points in a polyhedron, the smallest number of
facets of any polyhedron whose set of integer points coincides with is
called the relaxation complexity . This parameter was
introduced by Kaibel & Weltge (2015) and captures the complexity of linear
descriptions of without using auxiliary variables.
Using tools from combinatorics, geometry of numbers, and quantifier
elimination, we make progress on several open questions regarding
and its variant , restricting the
descriptions of to rational polyhedra.
As our main results we show that when: (a) is at most four-dimensional, (b)
represents every residue class in , (c) the convex
hull of contains an interior integer point, or (d) the lattice-width of
is above a certain threshold. Additionally, can be
algorithmically computed when is at most three-dimensional, or
satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an
improved lower bound on in terms of the dimension of .Comment: 28 pages, 5 figure