Doklady Mathematics

, Volume 97, Issue 3, pp 262–265 | Cite as

Estimation of the Absolute Error and Polynomial Solvability for a Classical NP-Hard Scheduling Problem

  • A. A. LazarevEmail author
  • D. I. Arkhipov


A method for finding an approximate solution for NP-hard scheduling problems is proposed. The example of the classical NP-hard in the strong sense problem of minimizing the maximum lateness of job processing with a single machine shows how a metric introduced on the instance space of the problem and polynomially solvable areas can be used to find an approximate solution with a guaranteed absolute error. The method is evaluated theoretically and experimentally and is compared with the ED-heuristic. Additionally, for the problem under consideration, we propose a numerical characteristic of polynomial unsolvability, namely, an upper bound for the guaranteed absolute error for each equivalence class of the instance space.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. A. Lazarev, Scheduling Theory: Estimation of the Absolute Error and Scheme for Approximate Solution of Scheduling Problems (Mosk. Fiz.-Tekh. Inst., Moscow, 2008) [in Russian].Google Scholar
  2. 2.
    J. R. Jackson, “Scheduling a production line to minimize maximum tardiness,” Los Angeles, CA, University of California. Manag. Sci. Res. Project Res. Rep., No. 43 (1955).Google Scholar
  3. 3.
    B. B. Simons, “A fast algorithm for single processor scheduling,” The 19th Annual Symposium on Foundations of Computer Science(Ann Arbor, Mich., 1978), pp. 246–252.Google Scholar
  4. 4.
    N. Vakhania, “A binary search algorithm for a special case of minimizing the lateness on a single machine,” Int. J. Appl. Math. Inf. I.3, 45–50 (2009).Google Scholar
  5. 5.
    J. A. Hoogeven, “Minimizing maximum promptness and maximum lateness on a single machine,” Math. Operat. Res. 21, 100–114 (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, “Complexity of machine scheduling problems,” Ann. Discret. Math. 1, 343–362 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    A. A. Lazarev and D. I. Arkhipov, “Minimization of the maximal lateness for a single machine,” Autom. Remote Control 77 (4), 656–671 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    L. Schrage, “Obtaining optimal solutions to resource constrained network scheduling problems,” unpublished manuscript (1971).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.National Research University Higher School of EconomicsMoscowRussia
  3. 3.Faculty of PhysicsMoscow State UniversityMoscowRussia
  4. 4.Moscow Institute of Physics and Technology (State University)Dolgoprudnyi, Moscow oblastRussia

Personalised recommendations