Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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].
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).
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.
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).
J. A. Hoogeven, “Minimizing maximum promptness and maximum lateness on a single machine,” Math. Operat. Res. 21, 100–114 (1996).
J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, “Complexity of machine scheduling problems,” Ann. Discret. Math. 1, 343–362 (1977).
A. A. Lazarev and D. I. Arkhipov, “Minimization of the maximal lateness for a single machine,” Autom. Remote Control 77 (4), 656–671 (2016).
L. Schrage, “Obtaining optimal solutions to resource constrained network scheduling problems,” unpublished manuscript (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © A.A. Lazarev, D.I. Arkhipov, 2018, published in Doklady Akademii Nauk, 2018, Vol. 480, No. 5, pp. 523–527.
Rights and permissions
About this article
Cite this article
Lazarev, A.A., Arkhipov, D.I. Estimation of the Absolute Error and Polynomial Solvability for a Classical NP-Hard Scheduling Problem. Dokl. Math. 97, 262–265 (2018). https://doi.org/10.1134/S1064562418030201
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1064562418030201