In this paper we study the rank of polytopes contained in the 0-1 cube with respect to $t$-branch split cuts and $t$-dimensional lattice cuts for a fixed positive integer $t$. These inequalities are the same as split cuts when $t=1$ and generalize split cuts when $t>1$. For polytopes contained in the $n$-dimensional 0-1 cube, the work of Balas implies that the split rank can be at most $n$, and this bound is tight as Cornuéjols and Li gave an example with split rank $n$. All known examples with high split rank – i.e., at least $cn$ for some positive constant $c<1$ – are defined by exponentially many (as a function of $n$) linear inequalities. For any fixed integer $t>0$, we give a family of polytopes contained in ${[0,1]}^n$ for sufficiently large $n$ such that each polytope has empty integer hull, is defined by $O\left(n\right)$ inequalities, and has rank $\Omega \left(n\right)$ with respect to $t$-dimensional lattice cuts. Therefore the split rank of these polytopes is $\Omega \left(n\right)$. It was shown earlier that there exist generalized branch-and-bound proofs, with logarithmic depth, of the nonexistence of integer points in these polytopes. Therefore, our lower bound results on split rank show an exponential separation between the depth of branch-and-bound proofs and split rank.

在本文中，我们研究 0-1 立方体中包含的多面体的等级$t$-分支分裂切割和$t$固定正整数的维晶格切割$t$。这些不等式与分割切割相同$t=1$并概括分裂切割时$t>1$。对于包含在$n$-维0-1立方体，Balas的工作意味着分裂等级最多可以是$n$，并且这个界限很紧，因为 Cornuéjols 和 Li 给出了一个分裂等级的例子$n$。所有已知的具有高分裂等级的例子 - 即，至少$Cn$对于一些正常数$C<1$– 由指数多定义（作为$n$) 线性不等式。对于任意固定整数$t>0$，我们给出了包含在${[0,1]}^n$对于足够大的$n$这样每个多面体都有空的整数外壳，定义为$氧（n）$不平等，并且有等级$\Omega （n）$关于$t$维晶格切割。因此这些多胞体的分裂等级是$\Omega （n）$。前面已经表明，存在具有对数深度的广义分支定界证明，证明这些多胞形中不存在整数点。因此，我们对分割等级的下界结果显示分支定界证明和分割等级的深度之间呈指数分离。