This paper addresses the triangle width problem, which generalizes the classic two-machine flexible job-shop problem (FJSP) with tooling constraints. This new problem can be studied from three different angles: scheduling, matrix visualization, and vertex ordering in hypergraphs. We prove the equivalence of the different formulations of the problem and use them to establish the \(\mathcal{N}\mathcal{P}\)-Hardness and polynomiality of several of its subcases. This problem allows us to find more elegant (and probably shorter) proofs for several combinatorial problems in our analysis setting. Our study provides an elegant generalization of Johnson’s argument for the two-machine flow shop. It also shows the relation between the question: “Is a matrix triangular?” and the “k-visit of a graph”.

本文解决了三角形宽度问题，它概括了具有工具约束的经典两机柔性作业车间问题（FJSP）。这个新问题可以从三个不同的角度来研究：调度、矩阵可视化和超图中的顶点排序。我们证明了问题的不同表述的等价性，并使用它们来建立几个子情况的\(\mathcal{N}\mathcal{P}\) -硬度和多项式。这个问题使我们能够在我们的分析设置中为几个组合问题找到更优雅（并且可能更短）的证明。我们的研究为约翰逊关于两机流水车间的论点提供了优雅的概括。它还显示了问题之间的关系：“矩阵是三角形的吗？” 和“图的k访问”。