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New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling

Authors Spencer Compton, Slobodan Mitrović, Ronitt Rubinfeld

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Author Details

Spencer Compton
  • Stanford University, CA, USA
Slobodan Mitrović
  • University of California Davis, CA, USA
Ronitt Rubinfeld
  • MIT, Cambridge, MA, USA

Funding

S. Compton was supported in part by the National Defense Science & Engineering Graduate (NDSEG) Fellowship Program. S. Mitrović was supported by the Swiss NSF grant No. P400P2_191122/1, NSF award CCF-1733808, and FinTech@CSAIL. R. Rubinfeld was supported by the NSF TRIPODS program (awards CCF-1740751 and DMS-2022448), NSF award CCF-2006664, and FinTech@CSAIL.

Acknowledgements

We thank Benjamin Qi (MIT) for helpful discussions.

Cite AsGet BibTex

Spencer Compton, Slobodan Mitrović, and Ronitt Rubinfeld. New Partitioning Techniques and Faster Algorithms for Approximate Interval Scheduling. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 45:1-45:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.45

Abstract

Interval scheduling is a basic problem in the theory of algorithms and a classical task in combinatorial optimization. We develop a set of techniques for partitioning and grouping jobs based on their starting and ending times, that enable us to view an instance of interval scheduling on many jobs as a union of multiple interval scheduling instances, each containing only a few jobs. Instantiating these techniques in dynamic and local settings of computation leads to several new results. For (1+ε)-approximation of job scheduling of n jobs on a single machine, we develop a fully dynamic algorithm with O((log n)/ε) update and O(log n) query worst-case time. Further, we design a local computation algorithm that uses only O((log N)/ε) queries when all jobs are length at least 1 and have starting/ending times within [0,N]. Our techniques are also applicable in a setting where jobs have rewards/weights. For this case we design a fully dynamic deterministic algorithm whose worst-case update and query time are poly(log n,1/ε). Equivalently, this is the first algorithm that maintains a (1+ε)-approximation of the maximum independent set of a collection of weighted intervals in poly(log n,1/ε) time updates/queries. This is an exponential improvement in 1/ε over the running time of a randomized algorithm of Henzinger, Neumann, and Wiese [SoCG, 2020], while also removing all dependence on the values of the jobs' starting/ending times and rewards, as well as removing the need for any randomness. We also extend our approaches for interval scheduling on a single machine to examine the setting with M machines.

Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Approximation algorithms analysis
Keywords
  • interval scheduling
  • dynamic algorithms
  • local computation algorithms

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References

  1. Pankaj K Agarwal and Marc J Van Kreveld. Label placement by maximum independent set in rectangles, volume 1998. Utrecht University: Information and Computing Sciences, 1998. Google Scholar
  2. Noga Alon, Ronitt Rubinfeld, Shai Vardi, and Ning Xie. Space-efficient local computation algorithms. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 1132-1139. Society for Industrial and Applied Mathematics, 2012. Google Scholar
  3. Esther M Arkin and Ellen B Silverberg. Scheduling jobs with fixed start and end times. Discrete Applied Mathematics, 18(1):1-8, 1987. Google Scholar
  4. Sujoy Bhore, Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Dynamic geometric independent set. arXiv preprint, 2020. URL: https://arxiv.org/abs/2007.08643.
  5. Giorgio C Buttazzo, Marko Bertogna, and Gang Yao. Limited preemptive scheduling for real-time systems. a survey. IEEE Transactions on Industrial Informatics, 9(1):3-15, 2012. Google Scholar
  6. Jean Cardinal, John Iacono, and Grigorios Koumoutsos. Worst-case efficient dynamic geometric independent set. In 29th Annual European Symposium on Algorithms (ESA 2021), volume 204, page 25. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. Google Scholar
  7. José R Correa and Andreas S Schulz. Single-machine scheduling with precedence constraints. Mathematics of Operations Research, 30(4):1005-1021, 2005. Google Scholar
  8. A FRANK. Some polynomial algorithms for certain graphs and hypergraphs. In Proceedings of the 5th British Combinatorial Conference, 1975. Utilitas Mathematica, 1975. Google Scholar
  9. Alexander Gavruskin, Bakhadyr Khoussainov, Mikhail Kokho, and Jiamou Liu. Dynamic interval scheduling for multiple machines. In International Symposium on Algorithms and Computation, pages 235-246. Springer, 2014. Google Scholar
  10. Alexander Gavruskin, Bakhadyr Khoussainov, Mikhail Kokho, and Jiamou Liu. Dynamic algorithms for monotonic interval scheduling problem. Theoretical Computer Science, 562:227-242, 2015. Google Scholar
  11. Paweł Gawrychowski and Karol Pokorski. Sublinear dynamic interval scheduling (on one or multiple machines). arXiv preprint, 2022. URL: https://arxiv.org/abs/2203.14310.
  12. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In 36th International Symposium on Computational Geometry (SoCG 2020). Schloss Dagstuhl-Leibniz-Zentrum für Informatik, 2020. Google Scholar
  13. Dorit S Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and vlsi. Journal of the ACM (JACM), 32(1):130-136, 1985. Google Scholar
  14. Antoon WJ Kolen, Jan Karel Lenstra, Christos H Papadimitriou, and Frits CR Spieksma. Interval scheduling: A survey. Naval Research Logistics (NRL), 54(5):530-543, 2007. Google Scholar
  15. Jan Karel Lenstra and AHG Rinnooy Kan. Complexity of scheduling under precedence constraints. Operations Research, 26(1):22-35, 1978. Google Scholar
  16. Elaine Levey and Thomas Rothvoss. A (1+ epsilon)-approximation for makespan scheduling with precedence constraints using lp hierarchies. SIAM Journal on Computing, pages STOC16-201, 2019. Google Scholar
  17. Aristide Mingozzi, Marco A Boschetti, Salvatore Ricciardelli, and Lucio Bianco. A set partitioning approach to the crew scheduling problem. Operations Research, 47(6):873-888, 1999. Google Scholar
  18. Michael Pinedo. Scheduling, volume 29. Springer, 2012. Google Scholar
  19. Julien Robert and Nicolas Schabanel. Non-clairvoyant scheduling with precedence constraints. In Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms, pages 491-500, 2008. Google Scholar
  20. Ronitt Rubinfeld, Gil Tamir, Shai Vardi, and Ning Xie. Fast local computation algorithms. arXiv preprint, 2011. URL: https://arxiv.org/abs/1104.1377.
  21. Pinal Salot. A survey of various scheduling algorithm in cloud computing environment. International Journal of Research in Engineering and Technology, 2(2):131-135, 2013. Google Scholar
  22. Raksha Sharma, Vishnu Kant Soni, Manoj Kumar Mishra, and Prachet Bhuyan. A survey of job scheduling and resource management in grid computing. world academy of science, engineering and technology, 64:461-466, 2010. Google Scholar
  23. Martin Skutella and Marc Uetz. Stochastic machine scheduling with precedence constraints. SIAM Journal on Computing, 34(4):788-802, 2005. Google Scholar
  24. Eva Tardos and Jon Kleinberg. Algorithm design, 2005. Google Scholar
  25. Bram Verweij and Karen Aardal. An optimisation algorithm for maximum independent set with applications in map labelling. In European Symposium on Algorithms, pages 426-437. Springer, 1999. Google Scholar
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