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

Khadija Hadj Salem (), Luc Libralesso (), Vincent Jost (), Florian Fontan () and Frédéric Maffray ()
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Khadija Hadj Salem: Université de Tours, LIFAT EA 6300, CNRS, ROOT ERL CNRS 7002
Luc Libralesso: Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP
Vincent Jost: Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP
Florian Fontan: Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP
Frédéric Maffray: Univ. Grenoble Alpes, CNRS, Grenoble INP, G-SCOP

Annals of Operations Research, 2024, vol. 337, issue 2, No 9, 715-730

Abstract: Abstract 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 $$\mathcal{N}\mathcal{P}$$ 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”.

Keywords: Combinatorial optimization; Scheduling; Graph theory; Binary matrix visualization (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s10479-024-05890-0

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