0/1立方体中多面体的Chvátal秩的二次上界

中文摘要 AI 辅助

我们证明每个多面体$P\subseteq[0,1]^n$，更一般地，每个紧凸集，其Chvátal秩至多为$12.22n^2+n\log_2 n+2n+4$。这改进了Eisenbrand和Schulz的$O(n^2\log n)$界，并与Rothvoß和Sanità的$\Omega(n^2)$下界一起，表明$[0,1]^n$中多面体的最大Chvátal秩为$\Theta(n^2)$。更精确地，若$P$包含一个整数点，则对每个$c\in\mathbb{Z}^n\setminus\{0\}$，不等式$cx\le\max\{cy: y\in P\cap\mathbb{Z}^n\}$对$P$的$k$次Chvátal闭包有效，其中$k\le 12.22n^2+2n+2\log_2\\|c\\|_\infty+4$。沿用Eisenbrand和Schulz的方法，我们沿着一列越来越粗的整数向量推导此不等式，但不是在每一步将向量减半，而是在每个二进窗口$[2^{-t-1},2^{-t}]$中自由选择尺度$\tau$，对$\tau c$进行舍入。主要的新成分是Dirichlet逼近定理的一个多尺度版本，通过一个初等体积论证证明：对每个$c\in\mathbb{R}^n$，$\tau c$到$\mathbb{Z}^n$的$\ell_1$距离在每个二进窗口内最小化并跨所有窗口求和，总和小于$1.222n^2$，与$\\|c\\|_\infty$无关。

英文摘要

We show that every polytope $P\subseteq[0,1]^n$, and more generally every compact convex set, has Chvátal rank at most $12.22n^2+n\log_2 n+2n+4$. This improves the $O(n^2\log n)$ bound of Eisenbrand and Schulz and, together with the $Ω(n^2)$ lower bound of Rothvoß and Sanità, shows that the maximum Chvátal rank of a polytope in $[0,1]^n$ is $Θ(n^2)$. More precisely, if $P$ contains an integer point, then for every $c\in\mathbb{Z}^n\setminus\{0\}$ the inequality $cx\le\max\{cy: y\in P\cap\mathbb{Z}^n\}$ is valid for the $k$-th Chvátal closure of $P$ for some $k\le 12.22n^2+2n+2\log_2\|c\|_\infty+4$. Following Eisenbrand and Schulz, we derive this inequality along a chain of coarser and coarser integer vectors, but instead of halving the vector in each step, we round $τc$ for a scale $τ$ chosen freely in each dyadic window $[2^{-t-1},2^{-t}]$. The main new ingredient is a multiscale version of Dirichlet's approximation theorem, proved by an elementary volume argument: for every $c\in\mathbb{R}^n$, the $\ell_1$-distances of $τc$ to $\mathbb{Z}^n$, minimized within each dyadic window and summed over all windows, total less than $1.222n^2$, independently of $\|c\|_\infty$.

发表机构

• University of Wisconsin–Madison（威斯康星大学麦迪逊分校）

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