Document Open Access Logo

Scheduling Under Non-Uniform Job and Machine Delays

Authors Rajmohan Rajaraman, David Stalfa, Sheng Yang



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2023.98.pdf
  • Filesize: 0.96 MB
  • 20 pages

Document Identifiers

Author Details

Rajmohan Rajaraman
  • Northeastern University, Boston, MA, USA
David Stalfa
  • Northeastern University, Boston, MA, USA
Sheng Yang
  • Shanghai, CN

Cite AsGet BibTex

Rajmohan Rajaraman, David Stalfa, and Sheng Yang. Scheduling Under Non-Uniform Job and Machine Delays. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 98:1-98:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ICALP.2023.98

Abstract

We study the problem of scheduling precedence-constrained jobs on heterogenous machines in the presence of non-uniform job and machine communication delays. We are given a set of n unit size precedence-ordered jobs, and a set of m related machines each with size m_i (machine i can execute at most m_i jobs at any time). Each machine i has an associated in-delay ρ^{in}_i and out-delay ρ^{out}_i. Each job v also has an associated in-delay ρ^{in}_v and out-delay ρ^{out}_v. In a schedule, job v may be executed on machine i at time t if each predecessor u of v is completed on i before time t or on any machine j before time t - (ρ^{in}_i + ρ^{out}_j + ρ^{out}_u + ρ^{in}_v). The objective is to construct a schedule that minimizes makespan, which is the maximum completion time over all jobs. We consider schedules which allow duplication of jobs as well as schedules which do not. When duplication is allowed, we provide an asymptotic polylog(n)-approximation algorithm. This approximation is further improved in the setting with uniform machine speeds and sizes. Our best approximation for non-uniform delays is provided for the setting with uniform speeds, uniform sizes, and no job delays. For schedules with no duplication, we obtain an asymptotic polylog(n)-approximation for the above model, and a true polylog(n)-approximation for symmetric machine and job delays. These results represent the first polylogarithmic approximation algorithms for scheduling with non-uniform communication delays. Finally, we consider a more general model, where the delay can be an arbitrary function of the job and the machine executing it: job v can be executed on machine i at time t if all of v’s predecessors are executed on i by time t-1 or on any machine by time t - ρ_{v,i}. We present an approximation-preserving reduction from the Unique Machines Precedence-constrained Scheduling (umps) problem, first defined in [Sami Davies et al., 2022], to this job-machine delay model. The reduction entails logarithmic hardness for this delay setting, as well as polynomial hardness if the conjectured hardness of umps holds. This set of results is among the first steps toward cataloging the rich landscape of problems in non-uniform delay scheduling.

Subject Classification

ACM Subject Classification
  • Theory of computation → Scheduling algorithms
Keywords
  • Scheduling
  • Approximation Algorithms
  • Precedence Constraints
  • Communication Delay
  • Non-Uniform Delays

Metrics

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

References

  1. Ishfaq Ahmad and Yu-Kwong Kwok. On exploiting task duplication in parallel program scheduling. IEEE Transactions on Parallel and Distributed Systems, 9(9):872-892, September 1998. URL: https://doi.org/10.1109/71.722221.
  2. Adil Amirjanov and Konstantin Sobolev. Scheduling of directed acyclic graphs by a genetic algorithm with a repairing mechanism. Concurrency and Computation: Practice and Experience, 29(5):e3954, 2017. e3954 CPE-16-0237.R1. URL: https://doi.org/10.1002/cpe.3954.
  3. Pau Andrio, Adam Hospital, Javier Conejero, Luis Jordá, Marc Del Pino, Laia Codo, Stian Soiland-Reyes, Carole Goble, Daniele Lezzi, Rosa M Badia, Modesto Orozco, and Josep Gelpi. Bioexcel building blocks, a software library for interoperable biomolecular simulation workflows. Scientific data, 6(1):1-8, 2019. Google Scholar
  4. Evripidis Bampis, Aristotelis Giannakos, and Jean-Claude König. On the complexity of scheduling with large communication delays. European Journal of Operational Research, 94:252-260, 1996. Google Scholar
  5. Nikhil Bansal. Scheduling open problems: Old and new. The 13th Workshop on Models and Algorithms for Planning and Scheduling Problems (MAPSP 2017), 2017. URL: http://www.mapsp2017.ma.tum.de/MAPSP2017-Bansal.pdf.
  6. Nikhil Bansal and Subhash Khot. Optimal long code test with one free bit. 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 453-462, October 2009. URL: https://doi.org/10.1109/focs.2009.23.
  7. Abbas Bazzi and Ashkan Norouzi-Fard. Towards tight lower bounds for scheduling problems. Lecture Notes in Computer Science, pages 118-129, 2015. URL: https://doi.org/10.1007/978-3-662-48350-3_11.
  8. Gregor Behnke, Daniel Höller, and Susanne Biundo. Bringing order to chaos – a compact representation of partial order in sat-based htn planning. Proceedings of the AAAI Conference on Artificial Intelligence, 33(01):7520-7529, July 2019. URL: https://doi.org/10.1609/aaai.v33i01.33017520.
  9. Karthekeyan Chandrasekaran and Chandra Chekuri. Min-max partitioning of hypergraphs and symmetric submodular functions. In Proceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '21, pages 1026-1038, USA, 2021. Society for Industrial and Applied Mathematics. Google Scholar
  10. Chandra Chekuri and Michael Bender. An efficient approximation algorithm for minimizing makespan on uniformly related machines. Journal of Algorithms, 41(2):212-224, November 2001. URL: https://doi.org/10.1006/jagm.2001.1184.
  11. Eden Chlamtáč, Michael Dinitz, and Yury Makarychev. Minimizing the union: Tight approximations for small set bipartite vertex expansion. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 881-899. SIAM, 2017. Google Scholar
  12. Eden Chlamtáč, Michael Dinitz, and Yury Makarychev. Minimizing the union: Tight approximations for small set bipartite vertex expansion. In Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 881-899, 2017. Google Scholar
  13. Fabián A. Chudak and David B. Shmoys. Approximation algorithms for precedence-constrained scheduling problems on parallel machines that run at different speeds. Journal of Algorithms, 30(2):323-343, 1999. Google Scholar
  14. Sekhar Darbha and Dharma P. Agrawal. Optimal scheduling algorithm for distributed-memory machines. IEEE Transactions on Parallel and Distributed Systems, 9:87-95, 1998. Google Scholar
  15. Sami Davies, Janardhan Kulkarni, Thomas Rothvoss, Sai Sandeep, Jakub Tarnawski, and Yihao Zhang. On the hardness of scheduling with non-uniform communication delays. In Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2958-2977, 2022. URL: https://epubs.siam.org/doi/abs/10.1137/1.9781611976465.176.
  16. Sami Davies, Janardhan Kulkarni, Thomas Rothvoss, Jakub Tarnawski, and Yihao Zhang. Scheduling with communication delays via lp hierarchies and clustering. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 822-833, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00081.
  17. Sami Davies, Janardhan Kulkarni, Thomas Rothvoss, Jakub Tarnawski, and Yihao Zhang. Scheduling with communication delays via lp hierarchies and clustering ii: Weighted completion times on related machines. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2958-2977, 2021. URL: https://doi.org/10.1137/1.9781611976465.176.
  18. Yuanxiang Gao, Li Chen, and Baochun Li. Spotlight: Optimizing device placement for training deep neural networks. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine Learning, volume 80 of Proceedings of Machine Learning Research, pages 1676-1684. PMLR, 10-15 July 2018. URL: https://proceedings.mlr.press/v80/gao18a.html.
  19. M. R. Garey and David S. Johnson. Scheduling tasks with nonuniform deadlines on two processors. J. ACM, 23(3):461-467, 1976. URL: https://doi.org/10.1145/321958.321967.
  20. Ronald L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17:416-429, 1969. Google Scholar
  21. Ubaid Ullah Hafeez, Xiao Sun, Anshul Gandhi, and Zhenhua Liu. Towards optimal placement and scheduling of dnn operations with pesto. In Proceedings of the 22nd International Middleware Conference, pages 39-51, 2021. Google Scholar
  22. Leslie A. Hall, Andreas S. Schulz, David B. Shmoys, and Joel Wein. Scheduling to minimize average completion time: Off-line and on-line approximation algorithms. Mathematics of Operations Research, 22(3):513-544, August 1997. URL: https://doi.org/10.1287/moor.22.3.513.
  23. J.A. Hoogeveen, Jan Karel Lenstra, and Bart Veltman. Three, four, five, six, or the complexity of scheduling with communication delays. Operations Research Letters, 16(3):129-137, 1994. URL: https://doi.org/10.1016/0167-6377(94)90024-8.
  24. Jeffrey M. Jaffe. Efficient scheduling of tasks without full use of processor resources. Theoretical Computer Science, 12(1):1-17, September 1980. URL: https://doi.org/10.1016/0304-3975(80)90002-x.
  25. Jan Karel Lenstra and A. H. G. Rinnooy Kan. Complexity of scheduling under precedence constraints. Operations Research, 26(1):22-35, 1978. URL: https://doi.org/10.1287/opre.26.1.22.
  26. Renaud Lepere and Christophe Rapine. An asymptotic 𝒪 (lnρ / lnlnρ)-approximation algorithm for the scheduling problem with duplication on large communication delay graphs. In Annual Symposium on Theoretical Aspects of Computer Science, pages 154-165. Springer, 2002. Google Scholar
  27. Shi Li. Scheduling to minimize total weighted completion time via time-indexed linear programming relaxations. In 2017 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, pages 283-294, 2017. URL: https://doi.org/10.1109/FOCS.2017.34.
  28. Guanfeng Liang and Ulaş C. Kozat. Fast cloud: Pushing the envelope on delay performance of cloud storage with coding. IEEE/ACM Transactions on Networking, 22(6):2012-2025, December 2014. URL: https://doi.org/10.1109/TNET.2013.2289382.
  29. Quanquan C. Liu, Manish Purohit, Zoya Svitkina, Erik Vee, and Joshua R. Wang. Scheduling with communication delay in near-linear time. In STACS, 2022. Google Scholar
  30. Ashraf Mahgoub, Edgardo Barsallo Yi, Karthick Shankar, Eshaan Minocha, Sameh Elnikety, Saurabh Bagchi, and Somali Chaterji. Wisefuse: Workload characterization and dag transformation for serverless workflows. Proceedings of the ACM on Measurement and Analysis of Computing Systems, 6(2), June 2022. URL: https://doi.org/10.1145/3530892.
  31. Biswaroop Maiti, Rajmohan Rajaraman, David Stalfa, Zoya Svitkina, and Aravindan Vijayaraghavan. Scheduling precedence-constrained jobs on related machines with communication delay. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 834-845, 2020. URL: https://doi.org/10.1109/FOCS46700.2020.00082.
  32. Azalia Mirhoseini, Anna Goldie, Hieu Pham, Benoit Steiner, Quoc V. Le, and Jeff Dean. Hierarchical planning for device placement. In International Conference on Learning Representations, 2018. URL: https://openreview.net/pdf?id=Hkc-TeZ0W.
  33. Azalia Mirhoseini, Hieu Pham, Quoc V. Le, Benoit Steiner, Rasmus Larsen, Yuefeng Zhou, Naveen Kumar, Mohammad Norouzi, Samy Bengio, and Jeff Dean. Device placement optimization with reinforcement learning. In Proceedings of the 34th International Conference on Machine Learning, ICML 2017, pages 2430-2439, 2017. URL: http://proceedings.mlr.press/v70/mirhoseini17a.html.
  34. Alix Munier. Approximation algorithms for scheduling trees with general communication delays. Parallel Computing, 25(1):41-48, 1999. Google Scholar
  35. Alix Munier and Claire Hanen. Using duplication for scheduling unitary tasks on m processors with unit communication delays. Theoretical Computer Science, 178(1):119-127, 1997. URL: https://doi.org/10.1016/S0304-3975(97)88194-7.
  36. Alix Munier and Jean-Claude König. A heuristic for a scheduling problem with communication delays. Operations Research, 45(1):145-147, 1997. Google Scholar
  37. Michael A. Palis, Jing-Chiou Liou, and David S. L. Wei. Task clustering and scheduling for distributed memory parallel architectures. IEEE Transactions on Parallel and Distributed Systems, 7(1):46-55, 1996. Google Scholar
  38. Christos H. Papadimitriou and Mihalis Yannakakis. Scheduling interval-ordered tasks. SIAM J. Comput., 8(3):405-409, 1979. URL: https://doi.org/10.1137/0208031.
  39. Christos H. Papadimitriou and Mihalis Yannakakis. Towards an architecture-independent analysis of parallel algorithms. SIAM journal on computing, 19(2):322-328, 1990. Google Scholar
  40. Christophe Picouleau. Two new NP-complete scheduling problems with communication delays and unlimited number of processors. Inst. Blaise Pascal, Univ., 1991. Google Scholar
  41. Maurice Queyranne and Maxim Sviridenko. Approximation algorithms for shop scheduling problems with minsum objective. Journal of Scheduling, 5(4):287-305, 2002. URL: https://doi.org/10.1002/jos.96.
  42. Rajmohan Rajaraman, David Stalfa, and Sheng Yang. Scheduling under non-uniform job and machine delays, 2022. URL: https://arxiv.org/abs/2207.13121.
  43. Victor J Rayward-Smith. UET scheduling with unit interprocessor communication delays. Discrete Applied Mathematics, 18(1):55-71, 1987. Google Scholar
  44. Juan A. Rico-Gallego, Juan C. Díaz-Martín, Ravi Reddy Manumachu, and Alexey L. Lastovetsky. A survey of communication performance models for high-performance computing. ACM Computing Surveys, 51(6), January 2019. URL: https://doi.org/10.1145/3284358.
  45. Petra Schuurman and Gerhard J. Woeginger. Polynomial time approximation algorithms for machine scheduling: ten open problems. Journal of Scheduling, 2(5):203-213, 1999. Google Scholar
  46. Bastian Seifert, Chris Wendler, and Markus Püschel. Causal fourier analysis on directed acyclic graphs and posets. CoRR, abs/2209.07970, 2022. URL: https://doi.org/10.48550/arXiv.2209.07970.
  47. Ola Svensson. Conditional hardness of precedence constrained scheduling on identical machines. Proceedings of the 42nd ACM symposium on Theory of computing - STOC ’10, pages 745-754, 2010. URL: https://doi.org/10.1145/1806689.1806791.
  48. Bart Veltman, B. J. Lageweg, and Jan Lenstra. Multiprocessor scheduling with communication delays. Parallel Computing, 16:173-182, 1990. Google Scholar
  49. Laurens Versluis, Erwin Van Eyk, and Alexandru Iosup. An analysis of workflow formalisms for workflows with complex non-functional requirements. In Companion of the 2018 ACM/SPEC International Conference on Performance Engineering, pages 107-112, 2018. Google Scholar
  50. Guangyuan Wu, Fangming Liu, Haowen Tang, Keke Huang, Qixia Zhang, Zhenhua Li, Ben Y. Zhao, and Hai Jin. On the performance of cloud storage applications with global measurement. In 2016 IEEE/ACM 24th International Symposium on Quality of Service (IWQoS), pages 1-10, 2016. URL: https://doi.org/10.1109/IWQoS.2016.7590449.
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