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Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines

Author

Listed:
  • Claire Hanen

    (Sorbonne Université, CNRS, LIP6
    UPL, Université Paris Nanterre)

  • Alix Munier Kordon

    (Sorbonne Université, CNRS, LIP6)

Abstract

Scheduling problems involving a set of dependent tasks with release dates and deadlines on a limited number of resources have been intensively studied. However, few parameterized complexity results exist for these problems. This paper studies the existence of a feasible schedule for a typed task system with precedence constraints and time intervals $$(r_i,d_i)$$ ( r i , d i ) for each job i. The problem is denoted by $$P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert \star $$ P | M j ( t y p e ) , p r e c , r i , d i | ⋆ . Several parameters are considered: the pathwidth pw(I) of the interval graph I associated with the time intervals $$(r_i, d_i)$$ ( r i , d i ) , the maximum processing time of a task $$p_{\max }$$ p max and the maximum slack of a task $$s\ell _{\max }$$ s ℓ max . This paper establishes that the problem is para- $$\textsf{NP}$$ NP -complete with respect to any of these parameters. It then provides a fixed-parameter algorithm for the problem parameterized by both parameters pw(I) and $$\min (p_{\max },s\ell _{\max })$$ min ( p max , s ℓ max ) . It is based on a dynamic programming approach that builds a levelled graph which longest paths represent all the feasible solutions. Fixed-parameter algorithms for the problems $$P\vert \mathcal{M}_j(type),prec,r_i,d_i\vert C_{\max }$$ P | M j ( t y p e ) , p r e c , r i , d i | C max and $$P\vert \mathcal{M}_j(type),prec,r_i\vert L_{\max }$$ P | M j ( t y p e ) , p r e c , r i | L max are then derived using a binary search.

Suggested Citation

  • Claire Hanen & Alix Munier Kordon, 2024. "Fixed-parameter tractability of scheduling dependent typed tasks subject to release times and deadlines," Journal of Scheduling, Springer, vol. 27(2), pages 119-133, April.
  • Handle: RePEc:spr:jsched:v:27:y:2024:i:2:d:10.1007_s10951-023-00788-4
    DOI: 10.1007/s10951-023-00788-4
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    References listed on IDEAS

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    1. Aurélien Carlier & Claire Hanen & Alix Munier Kordon, 2017. "The equivalence of two classical list scheduling algorithms for dependent typed tasks with release dates, due dates and precedence delays," Journal of Scheduling, Springer, vol. 20(3), pages 303-311, June.
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