A colorful Steinitz Lemma with application to block-structured integer programs

• 成果类型: Article
• 署名作者: Oertel, Timm; Paat, Joseph; Weismantel, Robert
• 署名单位: University of Erlangen Nuremberg; University of British Columbia; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-01971-3
• 发表日期: 2024
• 页码: 677-702

摘要

The Steinitz constant in dimension d is the smallest value c(d) such that for any norm on 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) = 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.