Extreme functions with an arbitrary number of slopes

• 成果类型: Article
• 署名作者: Basu, Amitabh; Conforti, Michele; Di Summa, Marco; Paat, Joseph
• 署名单位: Johns Hopkins University; University of Padua; University of Padua; Swiss Federal Institutes of Technology Domain; ETH Zurich
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1184-x
• 发表日期: 2018
• 页码: 303-327
• 关键词: infinite group relaxation

摘要

For the one dimensional infinite group relaxation, we construct a sequence of extreme valid functions that are piecewise linear and such that for every natural number k >= 2, there is a function in the sequence with k slopes. This settles an open question in this area regarding a universal bound on the number of slopes for extreme functions. The function which is the pointwise limit of this sequence is an extreme valid function that is continuous and has an infinite number of slopes. This provides a new and more refined counterexample to an old conjecture of Gomory and Johnson stating that all extreme functions are piecewise linear. These constructions are extended to obtain functions for the higher dimensional group problems via the sequential-merge operation of Dey and Richard.