A quadratic upper bound on the Chv\'atal rank of polytopes in the 0/1-cube

2026-09-26

AI Summary

This study addresses the problem of determining a tight upper bound on the maximum Chvátal rank of polytopes within the 0/1 cube, thereby characterizing the convergence rate of cutting-plane methods in integer programming. By integrating combinatorial optimization, convex geometric analysis, and Diophantine approximation theory, this work proposes a multi-scale Dirichlet approximation theorem. The derivation chain is further optimized through refined vector rounding strategies, while volume arguments are employed to achieve error accumulation control independent of vector norms. Consequently, this paper establishes that the tight bound on the Chvátal rank for arbitrary 0/1 polytopes is Θ(n²), significantly improving upon the previously known upper bound of O(n² log n).

Abstract

We show that every polytope $P\subseteq[0,1]^n$, and more generally every compact convex set, has Chv\'atal 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 $\Omega(n^2)$ lower bound of Rothvo{\ss} and Sanit\`a, shows that the maximum Chv\'atal rank of a polytope in $[0,1]^n$ is $\Theta(n^2)$. 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 Chv\'atal 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 $\tau c$ for a scale $\tau$ 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 $\tau c$ to $\mathbb{Z}^n$, minimized within each dyadic window and summed over all windows, total less than $1.222n^2$, independently of $\|c\|_\infty$.

Problem

• Chvátal rank
• 0/1-cube
• polytopes
• upper bound

Innovation

• Chvátal rank
• 0/1-cube
• multiscale Dirichlet approximation
• dyadic window rounding
• integer programming