Triangle width problem: at the intersection of graph theory, scheduling, and matrix visualization
Résumé
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 N 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”