, Volume 57, Issue 2, pp 413–433 | Cite as

New Approximation Bounds for Lpt Scheduling



We provide new bounds for the worst case approximation ratio of the classic Longest Processing Time (Lpt) heuristic for related machine scheduling (Q||C max ). For different machine speeds, Lpt was first considered by Gonzalez et al. (SIAM J. Comput. 6(1):155–166, 1977). The best previously known bounds originate from more than 20 years back: Dobson (SIAM J. Comput. 13(4):705–716, 1984), and independently Friesen (SIAM J. Comput. 16(3):554–560, 1987) showed that the worst case ratio of Lpt is in the interval (1.512,1.583), and in (1.52,1.67), respectively. We tighten the upper bound to \(1+\sqrt{3}/3\approx1.5773\) , and the lower bound to 1.54. Although this improvement might seem minor, we consider the structure of potential lower bound instances more systematically than former works. We present a scheme for a job-exchanging process, which, repeated any number of times, gradually increases the lower bound. For the new upper bound, this systematic method together with a new idea of introducing fractional jobs, facilitated a proof that is surprisingly simple, relative to the result. We present the upper-bound proof in parameterized terms, which leaves room for further improvements.


Approximation algorithms Related machine scheduling LPT 


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  1. 1.
    Chen, B.: Parametric bounds for LPT scheduling on uniform processors. Acta Math. Appl. Sinica 7, 67–73 (1991) MATHCrossRefGoogle Scholar
  2. 2.
    Coffman, E.G. Jr., Garey, M.R., Johnson, D.S.: An application of bin-packing to multiprocessor scheduling. SIAM J. Comput. 7(1), 1–17 (1978) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Dobson, G.: Scheduling independent tasks on uniform processors. SIAM J. Comput. 13(4), 705–716 (1984) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Epstein, L., Sgall, J.: Approximation schemes for scheduling on uniformly related and identical parallel machines. Algorithmica 39(1), 43–57 (2004) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Friesen, D.K.: Tighter bounds for LPT scheduling on uniform processors. SIAM J. Comput. 16(3), 554–560 (1987) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. Freeman, San Francisco (1979) MATHGoogle Scholar
  7. 7.
    Gonzalez, T., Ibarra, O.H., Sahni, S.: Bounds for LPT schedules on uniform processors. SIAM J. Comput. 6(1), 155–166 (1977) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45, 1563–1581 (1966) Google Scholar
  9. 9.
    Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17, 416–429 (1969) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Hochbaum, D.S., Shmoys, D.B.: A polynomial approximation scheme for scheduling on uniform processors: Using the dual approximation approach. SIAM J. Comput. 17(3), 539–551 (1988) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Kovács, A.: Tighter approximation bounds for LPT scheduling in two special cases. CIAC06 Special Issue of the J. Discrete Algorithms. To appear Google Scholar
  12. 12.
    Mireault, P., Orlin, J.B., Vohra, R.V.: A parametric worst case analysis of the LPT heuristic for two uniform machines. Oper. Res. 45(1), 116–125 (1997) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute for Computer ScienceJ.W. Goethe UniversityFrankfurt/MainGermany

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