论文标题
几何缩放的复杂性
The complexity of geometric scaling
论文作者
论文摘要
由Schulz和Weismantel在2002年推出的几何缩放率在P \ Cap \ Cap \ Cap \ Mathbb Z^n \}中解决了整数优化问题$ \ Max \ {C \ Mathord {\ Cdot} X：X \ in P \ Cap \ MathbB Z^n \} $，通过原始增强，其中$ p \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ subset \ mathbb r r^n $ is p poltot is p polote as a p poltot。当$ p $是$ 0/1 $ - polytope时，我们将自己限制在重要情况下。 Schulz和Weismantel表明，需要不超过$ O（N \ log n \ | c \ | _ \ infty）$调用额外的oracle。可以使用Le Bodic，Pavelka，Pfetsch和Pokutta提出的早期停滞策略，将该上限提高到$ O（N \ log \ | c \ | _ \ infty）$。考虑到该方法的最大比率增强变体及其大约版本，我们表明这些上限基本上是通过用$ c $ $ c $在$ n $ dimensional in of of $ n $ dimensional的基本上紧密的，从而使$ \ | c \ | _ \ | _ \ iffty $要么是$ n $或$ 2^n $。
Geometric scaling, introduced by Schulz and Weismantel in 2002, solves the integer optimization problem $\max \{c\mathord{\cdot}x: x \in P \cap \mathbb Z^n\}$ by means of primal augmentations, where $P \subset \mathbb R^n$ is a polytope. We restrict ourselves to the important case when $P$ is a $0/1$-polytope. Schulz and Weismantel showed that no more than $O(n \log n \|c\|_\infty)$ calls to an augmentation oracle are required. This upper bound can be improved to $O(n \log \|c\|_\infty)$ using the early-stopping policy proposed in 2018 by Le Bodic, Pavelka, Pfetsch, and Pokutta. Considering both the maximum ratio augmentation variant of the method as well as its approximate version, we show that these upper bounds are essentially tight by maximizing over a $n$-dimensional simplex with vectors $c$ such that $\|c\|_\infty$ is either $n$ or $2^n$.
