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Tight approximation bounds for the LPT rule applied to identical parallel machines with small jobs

Author

Listed:
  • Myungho Lee

    (Postech)

  • Kangbok Lee

    (Postech)

  • Michael Pinedo

    (New York University)

Abstract

We consider a scheduling problem with m identical machines in parallel and the minimum makespan objective. The Longest Processing Time first (LPT) rule is a well-known approximation algorithm for this problem. Although its worst-case approximation ratio has been determined theoretically, it is known that the worst-case approximation ratio of LPT can be smaller with instances of smaller processing times. We assume that each job’s processing time is not longer than 1/k times the optimal makespan for a given integer k. We derive the worst-case approximation ratio of the LPT algorithm in terms of parameters k and m. For that purpose, we divide the whole set of instances of the original problem into classes defined by different values of parameters k and m. On each of those classes, we derive an exact upper bound on the worst-case performance ratio as a function of parameters k and m. We also show that there exist classes of instances for which our worst-case approximation ratio is better than previous bounds. Our bound can complement previous research in terms of the performance analysis of LPT.

Suggested Citation

  • Myungho Lee & Kangbok Lee & Michael Pinedo, 2022. "Tight approximation bounds for the LPT rule applied to identical parallel machines with small jobs," Journal of Scheduling, Springer, vol. 25(6), pages 721-740, December.
    • Handle: RePEc:spr:jsched:v:25:y:2022:i:6:d:10.1007_s10951-022-00742-w
    DOI: 10.1007/s10951-022-00742-w
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    References listed on IDEAS

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    1. Federico Della Croce & Rosario Scatamacchia & Vincent T’kindt, 2019. "A tight linear time $$\frac{13}{12}$$ 13 12 -approximation algorithm for the $$P2 || C_{\max }$$ P 2 | | C max problem," Journal of Combinatorial Optimization, Springer, vol. 38(2), pages 608-617, August.
    2. Brian Thomas Eck & Michael Pinedo, 1993. "On the Minimization of the Makespan Subject to Flowtime Optimality," Operations Research, INFORMS, vol. 41(4), pages 797-801, August.
    3. Federico Della Croce & Rosario Scatamacchia, 2020. "The Longest Processing Time rule for identical parallel machines revisited," Journal of Scheduling, Springer, vol. 23(2), pages 163-176, April.
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