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 \({{\,\mathrm{rc}\,}}(X)\). This parameter, introduced by Kaibel & Weltge (2015), 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 \({{\,\mathrm{rc}\,}}(X)\) and its variant \({{\,\mathrm{rc}\,}}_\mathbb {Q}(X)\), restricting the descriptions of X to rational polyhedra. As our main results we show that \({{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)\) when: (a) X is at most four-dimensional, (b) X represents every residue class in \((\mathbb {Z}/2\mathbb {Z})^d\), (c) the convex hull of X contains an interior integer point, or (d) the lattice-width of X is above a certain threshold. Additionally, \({{\,\mathrm{rc}\,}}(X)\) 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 \({{\,\mathrm{rc}\,}}(X)\) in terms of the dimension of X.

对于多面体中的整数点集合X ，整数点集合与X重合的任何多面体的最小面数 称为松弛复杂度 \({{\,\mathrm{rc}\,}}(X) \)。该参数由 Kaibel & Weltge (2015) 引入，在 不使用辅助变量的情况下捕获X的线性描述的复杂性。使用组合学、数字几何和量词消除工具，我们在几个关于\({{\,\mathrm{rc}\,}}(X)\)及其变体\({{\, \mathrm{rc}\,}}_\mathbb {Q}(X)\) ，将X的描述限制 为有理多面体。作为我们的主要结果，我们表明\({{\,\mathrm{rc}\,}}(X) = {{\,\mathrm{rc}\,}}_\mathbb {Q}(X)\)当： (a) X是最多四维， (b) X表示\((\mathbb {Z}/2\mathbb {Z})^d\)中的每个残差类， (c) X的凸包 包含一个内部整数点，或 (d) X的晶格宽度 高于某个阈值。此外，当X至多是 3 维或X满足条件 (b)、(c) 之一时，可以通过算法计算 \({{\,\mathrm{rc}\,}}(X)\ )，或 (d) 上述。此外，我们根据X的维数 获得了\({{\,\mathrm{rc}\,}}(X)\)的改进下界.