Advertisement

Journal of Heuristics

, Volume 24, Issue 2, pp 173–203 | Cite as

Reduction criteria, upper bounds, and a dynamic programming based heuristic for the max–min \(k_i\)-partitioning problem

  • Alexander Lawrinenko
  • Stefan Schwerdfeger
  • Rico WalterEmail author
Article
  • 131 Downloads

Abstract

This paper addresses the max–min \(k_i\)-partitioning problem that asks for an assignment of n jobs to m parallel machines so that the minimum machine completion time is maximized and the number of jobs on each machine does not exceed a machine-dependent cardinality limit \(k_i\) \((i=1,\ldots ,m)\). We propose different preprocessing as well as lifting procedures and derive several upper bound arguments. Furthermore, we introduce suited construction heuristics as well as an effective dynamic programming based improvement procedure. Results of a comprehensive computational study on a large set of randomly generated instances indicate that our algorithm quickly finds (near-)optimal solutions.

Keywords

Parallel machines Cardinality limits Preprocessing Upper bounds Dynamic programming 

References

  1. Babel, L., Kellerer, H., Kotov, V.: The \(k\)-partitioning problem. Math. Methods Oper. Res. 47, 59–82 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chen, S.P., He, Y., Yao, E.-Y.: Three-partitioning containing kernels: complexity and heuristic. Computing 57, 255–271 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  3. Chen, S.P., He, Y., Lin, G.: 3-Partitioning problems for maximizing the minimum load. J. Comb. Optim. 6, 67–80 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. Csirik, J., Kellerer, H., Woeginger, G.: The exact LPT-bound for maximizing the minimum completion time. Oper. Res. Lett. 11, 281–287 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dell’Amico, M., Martello, S.: Optimal scheduling of tasks on identical parallel processors. ORSA J. Comput. 7, 191–200 (1995)CrossRefzbMATHGoogle Scholar
  6. Dell’Amico, Martello, S.: Bounds for the cardinality constrained \(P, : C_{\rm max}\) problem. J. Sched. 4, 123–138 (2001)Google Scholar
  7. Dell’Amico, M., Iori, M., Martello, S.: Heuristic algorithms and scatter search for the cardinality constrained \(P||C_{{\rm max}}\) problem. J. Heuristics 10, 169–204 (2004)CrossRefGoogle Scholar
  8. Dell’Amico, M., Iori, M., Martello, S., Monaci, M.: Lower bounds and heuristics for the \(k_i\)-partitioning problem. Eur. J. Oper. Res. 171, 725–742 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Deuermeyer, B.L., Friesen, D.K., Langston, M.A.: Scheduling to maximize the minimum processor finish time in a multiprocessor system. SIAM J. Algebr. Discrete Methods 3, 190–196 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Friesen, D.K., Deuermeyer, B.L.: Analysis of greedy solutions for a replacement part sequencing problem. Math. Oper. Res. 6, 74–87 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  12. Haouari, M., Jemmali, M.: Maximizing the minimum completion time on parallel machines. 4OR A Q. J. Oper. Res. 6, 375–392 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. He, Y., Tan, Z., Zhu, J., Yao, E.: \(k\)-Partitioning problems for maximizing the minimum load. Comput. Math. Appl. 46, 1671–1681 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kellerer, H., Kotov, V.: A 7/6-approximation algorithm for 3-partitioning and its application to multiprocessor scheduling. INFOR 37, 48–56 (1999)Google Scholar
  15. Kellerer, H., Kotov, V.: A 3/2-approximation algorithm for \(k_i\)-partitioning. Oper. Res. Lett. 39, 359–362 (2011)MathSciNetzbMATHGoogle Scholar
  16. Kellerer, H., Woeginger, G.J.: A tight bound for 3-partitioning. Discrete Appl. Math. 45, 249–259 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  17. Mertens, S.: A complete anytime algorithm for balanced number partitioning. Preprint arXiv:cs/9903011v1 (1999)
  18. Michiels, W., Aarts, E., Korst, J., van Leeuwen, J., Spieksma, F.C.R.: Computer-assisted proof of performance ratios for the differencing method. Discrete Optim. 9, 1–16 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Tsai, L.-H.: Asymptotic analysis of an algorithm for balanced parallel processor scheduling. SIAM J. Comput. 21, 59–64 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. Walter, R.: Comparing the minimum completion times of two longest-first scheduling-heuristics. CEJOR 21, 125–139 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. Walter, R., Wirth, M., Lawrinenko, A.: Improved approaches to the exact solution of the machine covering problem. J. Sched. 20, 147–164 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. Woeginger, G.J.: A polynomial-time approximation scheme for maximizing the minimum machine completion time. Oper. Res. Lett. 20, 149–154 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Woeginger, G.J.: A comment on scheduling two parallel machines with capacity constraints. Discrete Optim. 2, 269–272 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Zhang, C., Wang, G., Liu, X., Liu, J.: Approximating scheduling machines with capacity constraints. In: Deng, X., Hopcroft, J.E., Xue, J. (eds.) Frontiers in Algorithmics. Lecture Notes in Computer Science 5598, pp. 283–292. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  • Alexander Lawrinenko
    • 1
  • Stefan Schwerdfeger
    • 1
  • Rico Walter
    • 1
    Email author
  1. 1.Chair for Management ScienceFriedrich-Schiller-University JenaJenaGermany

Personalised recommendations