发表机构
Bowdoin College（鲍登学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明格多胞体的单调直径在 $k=3$ 时呈指数增长，否定了多项式上界，并表明 Naddef 结果无法推广至无界情形。
AI 中文摘要
Naddef 在 1989 年的一项有影响力的结果表明，$0/1$-多胞体的直径至多等于其维数。不久之后，Kleinschmidt 和 Onn 将此结果推广到 $[0,k]^{d}$ 中的任意格多胞体，并证明了其直径的上界至多为 $dk$。Naddef 的论证可以轻松扩展到由单纯形法所启发的单调情形，在该情形中要求路径相对于线性目标函数递增。然而，Kleinschmidt-Onn 的论证却无法做到这一点。事实上，在随后的 30 年中，没有任何论证能够填补这一空白。在本工作之前，单调直径是否受 $d$ 和 $k$ 的多项式上界约束这一问题仍然悬而未决，且没有任何下界表明最坏情况直径与最坏情况单调直径之间存在任何分离。对于 $k=1$ 和 $k=2$，线性上界成立。然而，我们通过为每个 $d \geq 1$ 构造一个位于 $[0,3]^{6d}$ 中的格多胞体，其单调直径至少为 $2^{d}-1$，从而在 $k=3$ 处展示了该问题的尖锐阈值。特别地，多项式上界不成立。此外，我们通过构造一族具有 $0/1$-顶点且直径随其维数呈指数增长的无界多面体，表明 Naddef 的结果不能推广到无界情形。
英文摘要
An influential 1989 result of Naddef shows that the diameters of $0/1$-polytopes are at most their dimension. This was extended shortly after by Kleinschmidt and Onn to any lattice polytope in $[0,k]^{d}$, where they showed a bound of at most $dk$. Naddef's argument easily extends to the monotone setting motivated by the simplex method, where one requires paths to increase with respect to a linear objective function. However, the Kleinschmidt-Onn argument does not. In fact, no argument in the 30 years since has managed to fill that gap. Prior to this work, it remained open whether the monotone diameter is bounded by a polynomial in $d$ and $k$ with no lower bounds suggesting any separation between the worst-case diameter and worst-case monotone diameter. Linear upper bounds hold for $k=1$ and $k=2$. However, we exhibit a sharp threshold for this question at $k = 3$ by constructing for each $d \geq 1$ a lattice polytope in $[0,3]^{6d}$ with monotone diameter at least $2^{d}-1$. In particular, the polynomial bound does not hold. Furthermore, we show that Naddef's result does not extend to the unbounded setting by exhibiting a family of unbounded polyhedra with $0/1$-vertices and diameter exponential in their dimension.
Comments7 pages
