A \((k+1)\)-slope theorem for the \(k\)-dimensional infinite group relaxation (Q2848184)

The paper focuses on the infinite group relaxation \(f+\sum_{r \in \mathbb{R}^k} rs_r \in \mathbb{Z}^k\), where \(s_r \in \mathbb{Z}_+\) for all \(r\) in \(\mathbb{R}^k\) and \(s\) has finite support. The main result concerns valid functions for this relaxation and states that any minimal valid function \(\pi\) that is piecewise linear with a locally finite cell complex and genuinely \(k\)-dimensional with at most \(k+1\) slopes is a facet and has exactly \(k+1\) slopes. This result generalises results for \(k=1\) by \textit{R. E. Gomory} and \textit{E. L. Johnson} [Math. Program. 96, No. 2 (B), 341--375 (2003; Zbl 1082.90144)] and for \(k=2\) by \textit{G. Cornuéjols} and \textit{M. Molinaro} [Math. Program. 142, No. 1--2 (A), 83--105 (2013; Zbl 1282.90106)].