Document Open Access Logo

Cardinality Constrained Scheduling in Online Models

Authors Leah Epstein, Alexandra Lassota , Asaf Levin, Marten Maack , Lars Rohwedder

PDF
Thumbnail PDF

File

LIPIcs.STACS.2022.28.pdf
  • Filesize: 0.72 MB
  • 15 pages

Document Identifiers

Author Details

Leah Epstein
  • Department of Mathematics, University of Haifa, Israel
Alexandra Lassota
  • Chair of Discrete Optimization, EPFL, Lausanne, Switzerland
Asaf Levin
  • Faculty of Industrial Engineering and Management, Technion, Haifa, Israel
Marten Maack
  • Heinz Nixdorf Institute & Department of Computer Science, Paderborn University, Germany
Lars Rohwedder
  • School of Business and Economics, Maastricht University, The Netherlands

Funding

L. Epstein and A. Levin were partially supported by GIF - the German-Israeli Foundation for Scientific Research and Development (grant number I-1366-407. 6/2016 on "Polynomial Migration for Online Scheduling").
  • Lassota, Alexandra: A. Lassota was supported by the Swiss National Science Foundation within the project Lattice algorithms and Integer Programming (200021_185030/1).
  • Levin, Asaf: A. Levin was also partially supported by a grant from ISF - Israeli Science Foundation (grant number 308/18).
  • Maack, Marten: Partially supported by the German Research Foundation (DFG) within the Collaborative Research Centre "On-The-Fly Computing" under the project number 160364472 - SFB 901/3.

Cite AsGet BibTex

Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder. Cardinality Constrained Scheduling in Online Models. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 28:1-28:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.STACS.2022.28

Abstract

Makespan minimization on parallel identical machines is a classical and intensively studied problem in scheduling, and a classic example for online algorithm analysis with Graham’s famous list scheduling algorithm dating back to the 1960s. In this problem, jobs arrive over a list and upon an arrival, the algorithm needs to assign the job to a machine. The goal is to minimize the makespan, that is, the maximum machine load. In this paper, we consider the variant with an additional cardinality constraint: The algorithm may assign at most k jobs to each machine where k is part of the input. While the offline (strongly NP-hard) variant of cardinality constrained scheduling is well understood and an EPTAS exists here, no non-trivial results are known for the online variant. We fill this gap by making a comprehensive study of various different online models. First, we show that there is a constant competitive algorithm for the problem and further, present a lower bound of 2 on the competitive ratio of any online algorithm. Motivated by the lower bound, we consider a semi-online variant where upon arrival of a job of size p, we are allowed to migrate jobs of total size at most a constant times p. This constant is called the migration factor of the algorithm. Algorithms with small migration factors are a common approach to bridge the performance of online algorithms and offline algorithms. One can obtain algorithms with a constant migration factor by rounding the size of each incoming job and then applying an ordinal algorithm to the resulting rounded instance. With this in mind, we also consider the framework of ordinal algorithms and characterize the competitive ratio that can be achieved using the aforementioned approaches. More specifically, we show that in both cases, one can get a competitive ratio that is strictly lower than 2, which is the bound from the standard online setting. On the other hand, we prove that no PTAS is possible.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Cardinality Constrained Scheduling
  • Makespan Minimization
  • Online Algorithms
  • Lower Bounds
  • Pure Online
  • Migration
  • Ordinal Algorithms

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Susanne Albers. Better bounds for online scheduling. SIAM Journal on Computing, 29(2):459-473, 1999. Google Scholar
  2. Luitpold Babel, Bo Chen, Hans Kellerer, and Vladimir Kotov. Algorithms for on-line bin-packing problems with cardinality constraints. Discrete Applied Mathematics, 143(1-3):238-251, 2004. Google Scholar
  3. János Balogh, József Békési, György Dósa, Leah Epstein, and Asaf Levin. Online bin packing with cardinality constraints resolved. Journal of Computer and System Sciences, 112:34-49, 2020. Google Scholar
  4. Nikhil Bansal, Tim Oosterwijk, Tjark Vredeveld, and Ruben van der Zwaan. Approximating vector scheduling: Almost matching upper and lower bounds. Algorithmica, 76(4):1077-1096, 2016. Google Scholar
  5. József Békési, György Dósa, and Leah Epstein. Bounds for online bin packing with cardinality constraints. Information and Computation, 249:190-204, 2016. Google Scholar
  6. Sebastian Berndt, Leah Epstein, Klaus Jansen, Asaf Levin, Marten Maack, and Lars Rohwedder. Online bin covering with limited migration. In Proc. of the 27th Annual European Symposium on Algorithms, ESA 2019, volume 144, pages 18:1-18:14, 2019. Google Scholar
  7. Sebastian Berndt, Klaus Jansen, and Kim-Manuel Klein. Fully dynamic bin packing revisited. Mathematical Programming, 179(1):109-155, 2020. Google Scholar
  8. Lin Chen, Klaus Jansen, Wenchang Luo, and Guochuan Zhang. An efficient PTAS for parallel machine scheduling with capacity constraints. In International Conference on Combinatorial Optimization and Applications, pages 608-623. Springer, 2016. Google Scholar
  9. Mauro Dell'Amico and Silvano Martello. Bounds for the cardinality constrained P||C_max problem. Journal of Scheduling, 4(3):123-138, 2001. Google Scholar
  10. Mauro Dell’Amico, Manuel Iori, Silvano Martello, and Michele Monaci. Lower bounds and heuristic algorithms for the k_i-partitioning problem. European Journal of Operational Research, 171(3):725-742, 2006. Google Scholar
  11. L. Epstein. Online bin packing with cardinality constraints. SIAM Journal on Discrete Mathematics, 20(4):1015-1030, 2006. Google Scholar
  12. Leah Epstein. A survey on makespan minimization in semi-online environments. Journal of Scheduling, 21(3):269-284, 2018. Google Scholar
  13. Leah Epstein, Alexandra Lassota, Asaf Levin, Marten Maack, and Lars Rohwedder. Cardinality constrained scheduling in online models, 2022. URL: http://arxiv.org/abs/2201.05113.
  14. Leah Epstein and Asaf Levin. A robust APTAS for the classical bin packing problem. Mathematical Programming, 119(1):33-49, 2009. Google Scholar
  15. Leah Epstein and Asaf Levin. Robust approximation schemes for cube packing. SIAM Journal on Optimization, 23(2):1310-1343, 2013. Google Scholar
  16. Leah Epstein and Asaf Levin. Robust algorithms for preemptive scheduling. Algorithmica, 69(1):26-57, 2014. Google Scholar
  17. Leah Epstein and Asaf Levin. Robust algorithms for total completion time. Discrete Optimization, 33:70-86, 2019. Google Scholar
  18. Björn Feldkord, Matthias Feldotto, Anupam Gupta, Guru Guruganesh, Amit Kumar, Sören Riechers, and David Wajc. Fully-dynamic bin packing with little repacking. In Proc. of the 45th International Colloquium on Automata, Languages, and Programming, ICALP 2018, pages 51:1-51:24, 2018. Google Scholar
  19. Rudolf Fleischer and Michaela Wahl. Online scheduling revisited. Journal of Scheduling, 3(6):343-353, 2000. Google Scholar
  20. Waldo Gálvez, José A. Soto, and José Verschae. Symmetry exploitation for online machine covering with bounded migration. In Proc. of the 26th European Symposium on Algorithms, ESA 2018, pages 32:1-32:14, 2018. Google Scholar
  21. Ronald L. Graham. Bounds for certain multiprocessing anomalies. Bell System Technical Journal, 45(9):1563-1581, 1966. Google Scholar
  22. Yong He and Zhiyi Tan. Ordinal on-line scheduling for maximizing the minimum machine completion time. Journal of Combinatorial Optimization, 6(2):199-206, 2002. Google Scholar
  23. Yong He, Zhiyi Tan, Jing Zhu, and Enyu Yao. k-partitioning problems for maximizing the minimum load. Computers & Mathematics with Applications, 46(10-11):1671-1681, 2003. Google Scholar
  24. Klaus Jansen and Kim-Manuel Klein. A robust AFPTAS for online bin packing with polynomial migration. SIAM Journal on Discrete Mathematics, 33(4):2062-2091, 2019. Google Scholar
  25. Klaus Jansen, Kim-Manuel Klein, Maria Kosche, and Leon Ladewig. Online strip packing with polynomial migration. In Proc. of the 20th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2017, pages 13:1-13:18, 2017. Google Scholar
  26. Yasushi Kawase, Kei Kimura, Kazuhisa Makino, and Hanna Sumita. Optimal matroid partitioning problems. Algorithmica, 2021. To appear. Google Scholar
  27. Hans Kellerer and Vladimir Kotov. A 3/2-approximation algorithm for k_i-partitioning. Operations Research Letters, 39(5):359-362, 2011. Google Scholar
  28. K. L. Krause, V. Y. Shen, and H. D. Schwetman. Analysis of several task-scheduling algorithms for a model of multiprogramming computer systems. Journal of the ACM, 22(4):522-550, 1975. Google Scholar
  29. Wei-Ping Liu and Jeffrey B Sidney. Bin packing using semi-ordinal data. Operations research letters, 19(3):101-104, 1996. Google Scholar
  30. Wei-Ping Liu, Jeffrey B Sidney, and André Van Vliet. Ordinal algorithms for parallel machine scheduling. Operations Research Letters, 18(5):223-232, 1996. Google Scholar
  31. John F. Rudin III. Improved bounds for the on-line scheduling problem. PhD thesis, The University of Texas at Dallas, 2001. Google Scholar
  32. Peter Sanders, Naveen Sivadasan, and Martin Skutella. Online scheduling with bounded migration. Mathematics of Operations Research, 34(2):481-498, 2009. Google Scholar
  33. Martin Skutella and José Verschae. Robust polynomial-time approximation schemes for parallel machine scheduling with job arrivals and departures. Mathematics of Operations Research, 41(3):991-1021, 2016. Google Scholar
  34. Zhiyi Tan and Yong He. Semi-on-line scheduling with ordinal data on two uniform machines. Operations Research Letters, 28(5):221-231, 2001. Google Scholar
  35. Zhiyi Tan, Yong He, and Leah Epstein. Optimal on-line algorithms for the uniform machine scheduling problem with ordinal data. Information and Computation, 196(1):57-70, 2005. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact