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For a set X of integer points in a polyhedron, the smallest number of facets of any polyhedron whose set of integer points coincides with X is called the relaxation complexity rc(X). This parameter was introduced by Kaibel Weltge (2015) and captures the complexity of linear descriptions of X without using auxiliary variables. Using tools from combinatorics, geometry of numbers, and quantifier elimination, we make progress on several open questions regarding rc(X) and its variant rc_Q(X),<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2003.07817v1">arXiv:2003.07817v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iyjooahm4netxerxmi6vyrw3rq">fatcat:iyjooahm4netxerxmi6vyrw3rq</a> </span>
more »... ting the descriptions of X to rational polyhedra. As our main results we show that rc(X) = rc_Q(X) when: (a) X is at most four-dimensional, (b) X represents every residue class in (Z/2Z)^d, (c) the convex hull of X contains an interior integer point, or (d) the lattice-width of X is above a certain threshold. Additionally, rc(X) can be algorithmically computed when X is at most three-dimensional, or X satisfies one of the conditions (b), (c), or (d) above. Moreover, we obtain an improved lower bound on rc(X) in terms of the dimension of X.
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