一个彩色Steinitz引理及其在块规划中的应用
A Colorful Steinitz Lemma with Applications to Block Integer Programs
摘要：
维度$d$中的斯坦尼茨常数是最小值$c(d)$，使得对于$\mathbb{R}^{ d}$上的任何范数以及对于单位球中的任何有限零和序列，序列可以被置换，使得每个部分和的范数由$c(d)$界定。Grinberg和Sevastyanov证明了$c(d) \le d$和$d$的界对于任意范数是最好的可能;我们把他们的结果称为斯坦尼茨引理。我们提出了一个变化的斯坦尼茨引理，置换多个序列在同一时间。我们的结果，我们称之为一个丰富多彩的斯坦尼茨引理，证明了上界是独立的序列的数量。整数规划理论中的许多结果都是通过置换有界范数的向量来证明的;这包括邻近结果、Graver基算法和动态规划。由于Eisenbrand和Weismantel最近的一篇论文，关于如何使用Steinitz引理来改进整数规划结果的研究激增。作为应用，我们证明了块结构整数规划的一个邻近结果。
英文摘要：
The Steinitz constant in dimension $d$ is the smallest value $c(d)$ such that for any norm on $\mathbb{R}^{ d}$ and for any finite zero-sum sequence in the unit ball, the sequence can be permuted such that the norm of each partial sum is bounded by $c(d)$. Grinberg and Sevastyanov prove that $c(d) \le d$ and that the bound of $d$ is best possible for arbitrary norms; we refer to their result as the Steinitz Lemma. We present a variation of the Steinitz Lemma that permutes multiple sequences at one time. Our result, which we term a colorful Steinitz Lemma, demonstrates upper bounds that are independent of the number of sequences. Many results in the theory of integer programming are proved by permuting vectors of bounded norm; this includes proximity results, Graver basis algorithms, and dynamic programs. Due to a recent paper of Eisenbrand and Weismantel, there has been a surge of research on how the Steinitz Lemma can be used to improve integer programming results. As an application we prove a proximity result for block-structured integer programs.
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