The Longest Processing Time rule for identical parallel machines revisited

  • Federico Della CroceEmail author
  • Rosario Scatamacchia


We consider the \(P_m || C_{\max }\) scheduling problem where the goal is to schedule n jobs on m identical parallel machines \((m < n)\) to minimize makespan. We revisit the famous Longest Processing Time (LPT) rule proposed by Graham in 1969. LPT requires to sort jobs in non-ascending order of processing times and then to assign one job at a time to the machine whose load is smallest so far. We provide new insights into LPT and discuss the approximation ratio of a modification of LPT that improves Graham’s bound from \(\left( \frac{4}{3} - \frac{1}{3m} \right) \) to \(\left( \frac{4}{3} - \frac{1}{3(m-1)} \right) \) for \(m \ge 3\) and from \(\frac{7}{6}\) to \(\frac{9}{8}\) for \(m=2\). We use linear programming to analyze the approximation ratio of our approach. This performance analysis can be seen as a valid alternative to formal proofs based on analytical derivation. Also, we derive from the proposed approach an \(O(n \log n)\) time complexity heuristic. The heuristic splits the sorted job set in tuples of m consecutive jobs (\(1,\dots ,m; m+1,\dots ,2m;\) etc.) and sorts the tuples in non-increasing order of the difference (slack) between largest job and smallest job in the tuple. Then, given this new ordering of the job set, list scheduling is applied. This approach strongly outperforms LPT on benchmark literature instances and is competitive with more involved approaches such as COMBINE and LDM.


Identical parallel machine scheduling LPT rule Linear programming Approximation algorithms 



The authors wish to thank an anonymous referee for pointing out the works on the LPT rule by Dosa (2004) and Dosa and Vizvari (2006). This work has been partially supported by “Ministero dell’Istruzione, dell’Università e della Ricerca” Award “TESUN-83486178370409 finanziamento dipartimenti di eccellenza CAP. 1694 TIT. 232 ART. 6.”


  1. Abolhassani, M., Chan, T. H., Chen, F., Esfandiari, H., Hajiaghayi, M., Hamid, M., et al. (2016). Beating ratio 0.5 for weighted oblivious matching problems. In P. Sankowski & C. Zaroliagis (Eds.), 24th Annual European symposium on algorithms (ESA 2016) (Vol. 57, pp. 3:1–3:18).Google Scholar
  2. Alon, N., Azar, Y., Woeginger, G. J., & Yadid, Y. (1998). Approximation schemes for scheduling on parallel machines. Journal of Scheduling, 1, 55–66.CrossRefGoogle Scholar
  3. Blocher, J. D., & Sevastyanov, D. (2015). A note on the coffman-sethi bound for LPT scheduling. Journal of Scheduling, 18, 325–327.CrossRefGoogle Scholar
  4. Chen, B. (1993). A note on LPT scheduling. Operation Research Letters, 14, 139–142.CrossRefGoogle Scholar
  5. Chen, B., Potts, C. N., & Woeginger, G. J. (1999). A review of machine scheduling: Complexity, algorithms and approximability. In D. Z. Du & P. M. Pardalos (Eds.), Handbook of combinatorial optimization: Volume 1–3. New York: Springer.Google Scholar
  6. Chimani, M., & Wiedera, T. (2016). An ILP-based proof system for the crossing number problem. In P. Sankowski & C. Zaroliagis (Eds.), 24th annual European symposium on algorithms (ESA 2016) (Vol. 57, pp. 29:1–29:13).Google Scholar
  7. Coffman, E. G, Jr., Garey, M. R., & Johnson, D. S. (1978). An application of bin-packing to multiprocessor scheduling. SIAM Journal on Computing, 7, 1–17.CrossRefGoogle Scholar
  8. Coffman, E. G, Jr., & Sethi, R. (1976). A generalized bound on LPT sequencing. Revue Francaise d’Automatique Informatique, Recherche Operationelle Supplement, 10, 17–25.Google Scholar
  9. Della Croce, F., Pferschy, U., & Scatamacchia, R. (2018). Approximation results for the incremental knapsack problem. In Combinatorial algorithms: 28th international workshop, IWOCA 2017, Springer lecture notes in computer science (Vol. 10765, pp. 75–87).Google Scholar
  10. Dosa, G. (2004). Graham example is the only tight one for P \(||\) Cmax (in Hungarian). Annales Univ Sci Budapest, 47, 207–210.Google Scholar
  11. Dosa, G., & Vizvari, A. (2006). The general algorithm lpt(k) for scheduling identical parallel machines. Alkamazott Matematikai Lapok, 23(1), 17–37. (in Hungarian).Google Scholar
  12. Fischetti, M., & Martello, S. (1987). Worst-case analysis of the differencing method for the partition problem. Mathematical Programming, 37, 117–120.CrossRefGoogle Scholar
  13. França, P. M., Gendreau, M., Laporte, G., & Müller, F. (1994). A composite heuristic for the identical parallel machine scheduling problem with minimum makespan objective. Computers & Operations Research, 21, 205–210.CrossRefGoogle Scholar
  14. Frangioni, A., Necciari, E., & Scutellà, M. G. (2004). A multi-exchange neighborhood for minimum makespan parallel machine scheduling problems. Journal of Combinatorial Optimization, 8, 195–220.CrossRefGoogle Scholar
  15. Frenk, J. B. G., & Rinnooy Kan, A. H. G. (1987). The asymptotic optimality of the LPT rule. Mathematics of Operations Research, 12, 241–254.CrossRefGoogle Scholar
  16. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability. New York: W. H. Freeman.Google Scholar
  17. Graham, R. L. (1969). Bounds on multiprocessors timing anomalies. SIAM Journal on Applied Mathematics, 17, 416–429.CrossRefGoogle Scholar
  18. Graham, R. L., Lawler, E. L., Lenstra, J. K., & Rinnooy Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. In P. L. Hammer, E. L. Johnson, & B. H. Korte (Eds.), Discrete optimization II, annals of discrete mathematics (Vol. 5, pp. 287–326).Google Scholar
  19. Gupta, J. N. D., & Ruiz-Torres, A. J. (2001). A listfit heuristic for minimizing makespan on identical parallel machines. Production Planning & Control, 12(1), 28–36.CrossRefGoogle Scholar
  20. He, Y., Kellerer, H., & Kotov, V. (2000). Linear compound algorithms for the partitioning problem. Naval Research Logistics (NRL), 47(7), 593–601.CrossRefGoogle Scholar
  21. Hochbaum, D. S. (Ed.). (1997). Approximation Algorithms for NP-hard Problems. Boston: PWS Publishing Co.Google Scholar
  22. Hochbaum, D. S., & Shmoys, D. B. (1987). Using dual approximation algorithms for scheduling problems theoretical and practical results. Journal of the ACM, 34, 144–162.CrossRefGoogle Scholar
  23. Iori, M., & Martello, S. (2008). Scatter search algorithms for identical parallel machine scheduling problems. In Metaheuristics for scheduling in industrial and manufacturing applications (pp. 41–59).Google Scholar
  24. Jansen, K. (2010). An eptas for scheduling jobs on uniform processors: Using an milp relaxation with a constant number of integral variables. SIAM Journal on Discrete Mathematics, 24, 457–485.CrossRefGoogle Scholar
  25. Jansen, K., Klein, K. M., & Verschae, J. (2017). Improved efficient approximation schemes for scheduling jobs on identical and uniform machines. In Proceedings of the 13th workshop on models and algorithms for planning and scheduling problems (MAPSP 2017) (pp. 77–79).Google Scholar
  26. Karmarkar, N., & Karp, R. M. (1982). The differencing method of set partitioning. Technical Report UCB/CSD 82/113, University of California, Berkeley.Google Scholar
  27. Lee, C. Y., & Massey, J. D. (1988). Multiprocessor scheduling: Combining LPT and MULTIFIT. Discrete Applied Mathematics, 20(3), 233–242.CrossRefGoogle Scholar
  28. Leung, J., Kelly, L., & Anderson, J. H. (2004). Handbook of scheduling: Algorithms, models, and performance analysis. Cambridge: CRC Press, Inc.Google Scholar
  29. Michiels, W., Korst, J., Aarts, E., & van Leeuwen, J. (2007). Performance ratios of the Karmarkar–Karp differencing method. Journal of Combinatorial Optimization, 13(1), 19–32.CrossRefGoogle Scholar
  30. Mireault, P., Orlin, J. B., & Vohra, R. V. (1997). A parametric worst-case analysis of the LPT heuristic for two uniform machines. Operations Research, 45(1), 116–125.CrossRefGoogle Scholar
  31. Paletta, G., & Ruiz-Torres, A. J. (2015). Partial solutions and multifit algorithm for multiprocessor scheduling. Journal of Mathematical Modelling and Algorithms in Operations Research, 14(2), 125–143.CrossRefGoogle Scholar
  32. Pinedo, M. L. (2016). Scheduling: Theory, algorithms, and systems (5th ed.). Berlin: Springer.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria Gestionale e della ProduzionePolitecnico di TorinoTurinItaly
  2. 2.CNR, IEIITTurinItaly

Personalised recommendations