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Inductive $$k$$ k -independent graphs and c-colorable subgraphs in scheduling: a review

Author

Listed:
  • Matthias Bentert

    (Algorithmics and Computational Complexity, Faculty IV)

  • René Bevern

    (Novosibirsk State University
    Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences)

  • Rolf Niedermeier

    (Algorithmics and Computational Complexity, Faculty IV)

Abstract

Inductive $$k$$ k -independent graphs generalize chordal graphs and have recently been advocated in the context of interference-avoiding wireless communication scheduling. The NP-hard problem of finding maximum-weight induced c-colorable subgraphs, which is a generalization of finding maximum independent sets, naturally occurs when selecting $$c$$ c sets of pairwise non-conflicting jobs (modeled as graph vertices). We investigate the parameterized complexity of this problem on inductive $$k$$ k -independent graphs. We show that the Maximum Independent Set problem is W[1]-hard even on 2-simplicial 3-minoes—a subclass of inductive 2-independent graphs. In contrast, we prove that the more general Max-Weightc-Colorable Subgraph problem is fixed-parameter tractable on edge-wise unions of cluster and chordal graphs, which are 2-simplicial. In both cases, the parameter is the solution size. Aside from this, we survey other graph classes between inductive $$1$$ 1 -independent and inductive $$2$$ 2 -independent graphs with applications in scheduling.

Suggested Citation

  • Matthias Bentert & René Bevern & Rolf Niedermeier, 2019. "Inductive $$k$$ k -independent graphs and c-colorable subgraphs in scheduling: a review," Journal of Scheduling, Springer, vol. 22(1), pages 3-20, February.
    • Handle: RePEc:spr:jsched:v:22:y:2019:i:1:d:10.1007_s10951-018-0595-8
    DOI: 10.1007/s10951-018-0595-8
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    References listed on IDEAS

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    1. Antoon W.J. Kolen & Jan Karel Lenstra & Christos H. Papadimitriou & Frits C.R. Spieksma, 2007. "Interval scheduling: A survey," Naval Research Logistics (NRL), John Wiley & Sons, vol. 54(5), pages 530-543, August.
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    Cited by:

    1. Matthias Bentert & Robert Bredereck & Péter Györgyi & Andrzej Kaczmarczyk & Rolf Niedermeier, 2023. "A multivariate complexity analysis of the material consumption scheduling problem," Journal of Scheduling, Springer, vol. 26(4), pages 369-382, August.
    2. Olga Kuryatnikova & Renata Sotirov & Juan C. Vera, 2022. "The Maximum k -Colorable Subgraph Problem and Related Problems," INFORMS Journal on Computing, INFORMS, vol. 34(1), pages 656-669, January.
    3. Klaus Heeger & Danny Hermelin & George B. Mertzios & Hendrik Molter & Rolf Niedermeier & Dvir Shabtay, 2023. "Equitable scheduling on a single machine," Journal of Scheduling, Springer, vol. 26(2), pages 209-225, April.
    4. Kuryatnikova, Olga & Sotirov, Renata & Vera, J.C., 2022. "The maximum $k$-colorable subgraph problem and related problems," Other publications TiSEM 40e477c0-a78e-4ee1-92de-8, Tilburg University, School of Economics and Management.

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