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

Author

Listed:
  • 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)

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”.

Suggested Citation

  • Khadija Hadj Salem & Luc Libralesso & Vincent Jost & Florian Fontan & Frédéric Maffray, 2024. "Triangle width problem: at the intersection of graph theory, scheduling, and matrix visualization," Annals of Operations Research, Springer, vol. 337(2), pages 715-730, June.
    • Handle: RePEc:spr:annopr:v:337:y:2024:i:2:d:10.1007_s10479-024-05890-0
    DOI: 10.1007/s10479-024-05890-0
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    References listed on IDEAS

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    1. Hanan Luss, 1982. "Operations Research and Capacity Expansion Problems: A Survey," Operations Research, INFORMS, vol. 30(5), pages 907-947, October.
    2. M. R. Garey & D. S. Johnson & Ravi Sethi, 1976. "The Complexity of Flowshop and Jobshop Scheduling," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 117-129, May.
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