Skip to main content
SpringerLink
Log in

Greedy is Optimal for Online Restricted Assignment and Smart Grid Scheduling for Unit Size Jobs

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

Abstract

We study online scheduling of unit-sized jobs in two related problems, namely, restricted assignment problem and smart grid problem. The input to the two problems are in close analogy but the objective functions are different. We show that the greedy algorithm is an optimal online algorithm for both problems. Typically, an online algorithm is proved to be an optimal online algorithm through bounding its competitive ratio and showing a lower bound with matching competitive ratio. However, our analysis does not take this approach. Instead, we prove the optimality without giving the exact bounds on competitive ratio. Roughly speaking, given any online algorithm and a job instance, we show the existence of another job instance for greedy such that (i) the two instances admit the same optimal offline schedule; (ii) the cost of the online algorithm is at least that of the greedy algorithm on the respective job instance. With these properties, we can show that the competitive ratio of the greedy algorithm is the smallest possible.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. We note that we can refer to the simulation of \(\mathcal {A}\) since \(\mathcal {A}\) is an online algorithm.

References

  1. Liu, F.-H., Liu, H.-H., Wong, P.W.H.: Greedy is optimal for online restricted assignment and smart grid scheduling for unit size jobs. In: WAOA, pp. 217–231 (2019)

  2. Hamilton, K., Gulhar, N.: Taking demand response to the next level. IEEE Power Energy Mag. 8(3), 60–65 (2010)

    Article  Google Scholar 

  3. Ipakchi, A., Albuyeh, F.: Grid of the future. IEEE Power Energy Mag. 7(2), 52–62 (2009)

    Article  Google Scholar 

  4. Lui, T.J., Stirling, W., Marcy, H.O.: Get smart. IEEE Power Energy Mag. 8(3), 66–78 (2010)

    Article  Google Scholar 

  5. Zpryme Research & Consulting: Power systems of the future: The case for energy storage, distributed generation, and microgrids. http://smartgrid.ieee.org/images/features/smart_grid_survey.pdf (2012)

  6. Fang, X., Misra, S., Xue, G., Yang, D.: Smart grid – the new and improved power grid: A survey. IEEE Commun. Surv. Tutorials 14(4), 944–980 (2012)

    Article  Google Scholar 

  7. US Department of Energy: The Smart Grid: An Introduction. http://www.oe.energy.gov/SmartGridIntroduction.htm (2009)

  8. European Commission: Europen smartgrids technology platform. ftp://ftp.cordis.europa.eu/pub/fp7/energy/docs/smartgrids_en.pdf (2006)

  9. UK Department of Energy & Climate Change: Smart grid: A more energy-efficient electricity supply for the UK. https://www.gov.uk/smart-grid-a-more-energy-efficient-electricity-supply-for-the-uk (2013)

  10. Farhangi, H.: The path of the smart grid. IEEE Power Energy Mag. 8(1), 18–28 (2010)

    Article  MathSciNet  Google Scholar 

  11. Masters, G.M.: Renewable and efficient electric power systems. Wiley (2013)

  12. Chen, C., Nagananda, K.G., Xiong, G., Kishore, S., Snyder, L.V.: A communication-based appliance scheduling scheme for consumer-premise energy management systems. IEEE Trans. Smart Grid 4(1), 56–65 (2013)

    Article  Google Scholar 

  13. Logenthiran, T., Srinivasan, D., Shun, T.Z.: Demand side management in smart grid using heuristic optimization. IEEE Trans. Smart Grid 3(3), 1244–1252 (2012)

    Article  Google Scholar 

  14. Maharjan, S., Zhu, Q., Zhang, Y., Gjessing, S., Basar, T.: Dependable demand response management in the smart grid: A stackelberg game approach. IEEE Trans. Smart Grid 4(1), 120–132 (2013)

    Article  Google Scholar 

  15. Koutsopoulos, I., Tassiulas, L.: Control and optimization meet the smart power grid: Scheduling of power demands for optimal energy management. In: e-Energy, pp. 41–50. ACM (2011)

  16. Caron, S., Kesidis, G.: Incentive-based energy consumption scheduling algorithms for the smart grid. In: SmartGridComm, pp. 391–396. IEEE (2010)

  17. Salinas, S., Li, M., Li, P.: Multi-objective optimal energy consumption scheduling in smart grids. IEEE Trans. Smart Grid 4(1), 341–348 (2013)

    Article  Google Scholar 

  18. Mohsenian-Rad, A.-H., Wong, V.W.S., Jatskevich, J., Schober, R., Leon-Garcia, A.: Autonomous demand-side management based on game-theoretic energy consumption scheduling for the future smart grid. IEEE Trans. Smart Grid 1(3), 320–331 (2010)

    Article  Google Scholar 

  19. Krishnan, R.: Meters of tomorrow [in my view]. IEEE Power Energy Mag. 6(2), 96–94 (2008)

    Article  Google Scholar 

  20. Kannberg, L.D., Chassin, D.P., DeSteese, J.G., Hauser, S.G., Kintner-Meyer, M., (U.S.), P N N L, of Energy, U.S.D.: GridWise: The benefits of a transformed energy system. Pacific Northwest National Laboratory (2003)

  21. Lockheed Martin: SEELoad™ Solution. http://www.lockheedmartin.co.uk/us/products/energy-solutions/seesuite/seeload.html

  22. REGEN Energy Inc: ENVIROGRID™ SMART GRID BUNDLE. http://www.regenenergy.com/press/announcing-the-envirogrid-smart-grid-bundle/

  23. Toronto Hydro Corporation: Peaksaver Program. http://www.peaksaver.com/peaksaver_THESL.html

  24. Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45(9), 1563–1581 (1966)

    Article  Google Scholar 

  25. Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)

    Article  MathSciNet  Google Scholar 

  26. Azar, Y., Naor, J., Rom, R.: The competitiveness of on-line assignments. J. Algorithm. 18(2), 221–237 (1995)

    Article  MathSciNet  Google Scholar 

  27. Burcea, M., Hon, W.-K., Liu, H.-H., Wong, P.W.H., Yau, D.K.Y.: Scheduling for electricity cost in a smart grid. J. Sched. 19(6), 687–699 (2016)

    Article  MathSciNet  Google Scholar 

  28. Feng, X., Xu, Y., Zheng, F.: Online scheduling for electricity cost in smart grid. In: COCOA, pp. 783–793. Springer (2015)

  29. Liu, F.-H., Liu, H.-H., Wong, P.W.H.: Non-preemptive scheduling in a smart grid model and its implications on machine minimization. Algorithmica 82(12), 3415–3457 (2020)

    Article  MathSciNet  Google Scholar 

  30. Liu, F.-H., Liu, H.-H., Wong, P.W.H.: Optimal nonpreemptive scheduling in a smart grid model. In: ISAAC, pp. 53:1–53:13. LIPIcs (2016)

  31. Chau, V., Feng, S., Thang, N.K.: Competitive algorithms for demand response management in smart grid. In: LATIN, pp. 303–316 (2018)

  32. Bartal, Y., Fiat, A., Karloff, H.J., Vohra, R.: New algorithms for an ancient scheduling problem. J. Comput. Syst. Sci. 51(3), 359–366 (1995)

    Article  MathSciNet  Google Scholar 

  33. Caragiannis, I.: Better bounds for online load balancing on unrelated machines. In: SODA, pp. 972–981 (2008)

  34. Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. Algorithmica 61(3), 606–637 (2011)

    Article  MathSciNet  Google Scholar 

  35. Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259–271 (1990)

    Article  MathSciNet  Google Scholar 

  36. Svensson, O.: Santa claus schedules jobs on unrelated machines. SIAM J. Comput. 41(5), 1318–1341 (2012)

    Article  MathSciNet  Google Scholar 

  37. Ebenlendr, T., Krcál, M., Sgall, J.: Graph balancing: A special case of scheduling unrelated parallel machines. Algorithmica 68(1), 62–80 (2014)

    Article  MathSciNet  Google Scholar 

  38. Wang, C., Sitters, R.: On some special cases of the restricted assignment problem. Inf. Process. Lett. 116(11), 723–728 (2016)

    Article  MathSciNet  Google Scholar 

  39. Lee, K., Leung, J.Y.-T., Pinedo, M.L.: A note on graph balancing problems with restrictions. Inf. Process. Lett. 110(1), 24–29 (2009)

    Article  MathSciNet  Google Scholar 

  40. Verschae, J., Wiese, A.: On the configuration-lp for scheduling on unrelated machines. J. Sched. 17(4), 371–383 (2014)

    Article  MathSciNet  Google Scholar 

  41. Huo, Y., Leung, J.Y.-T.: Fast approximation algorithms for job scheduling with processing set restrictions. Theor. Comput. Sci. 411(44-46), 3947–3955 (2010)

    Article  MathSciNet  Google Scholar 

  42. Muratore, G., Schwarz, U.M., Woeginger, G.J.: Parallel machine scheduling with nested job assignment restrictions. Oper. Res. Lett. 38(1), 47–50 (2010)

    Article  MathSciNet  Google Scholar 

  43. Kolliopoulos, S.G., Moysoglou, Y.: The 2-valued case of makespan minimization with assignment constraints. Inf. Process. Lett. 113(1-2), 39–43 (2013)

    Article  MathSciNet  Google Scholar 

  44. Azar, Y., Broder, A.Z., Karlin, A.R.: On-line load balancing. Theor. Comput. Sci. 130(1), 73–84 (1994)

    Article  MathSciNet  Google Scholar 

  45. Azar, Y., Kalyanasundaram, B., Plotkin, S.A., Pruhs, K., Waarts, O.: On-line load balancing of temporary tasks. J. Algorithm. 22(1), 93–110 (1997)

    Article  MathSciNet  Google Scholar 

  46. Aspnes, J., Azar, Y., Fiat, A., Plotkin, S.A., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44(3), 486–504 (1997)

    Article  MathSciNet  Google Scholar 

  47. Azar, Y.: On-line load balancing. In: Fiat, A, Woeginger, J (eds.) Online Algorithms: The State of the Art, pp 178–195. Springer (1998)

  48. Lam, T.W., Ting, H.-F., To, K.-K., Wong, P.W.H.: On-line load balancing of temporary tasks revisited. Theor. Comput. Sci. 270(1-2), 325–340 (2002)

    Article  MathSciNet  Google Scholar 

  49. Borodin, A., El-Yaniv, R.: Online computation and competitive analysis. Cambridge University Press (1998)

  50. Yao, F.F., Demers, A.J., Shenker, S.: A scheduling model for reduced CPU energy. In: 36th Annual Symposium on Foundations of Computer Science, pp. 374–382, Milwaukee (1995)

Download references

Acknowledgements

The authors would like to thank Marcin Bienkowski for helpful discussion.

Author information

Authors and Affiliations

  1. Department of Computer Science, National Tsing Hua University, Tsing Hua, Taiwan

    Fu-Hong Liu

  2. Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands

    Hsiang-Hsuan Liu

  3. Institute of Computer Science, Wroclaw University, Wroclaw, Poland

    Hsiang-Hsuan Liu

  4. Department of Computer Science, University of Liverpool, Liverpool, UK

    Prudence W. H. Wong

Authors
  1. Fu-Hong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hsiang-Hsuan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Prudence W. H. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fu-Hong Liu.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article belongs to the Topical Collection: Special Issue on Approximation and Online Algorithms (2019)

Guest Editors: Evripidis Bampis and Nicole Megow

This work is partially supported by Polish National Science Centre grant 2016/22/E/ST6/00499. This work is supported by Networks Sciences & Technologies(NeST), School of EEECS, University of Liverpool. This work was partially done when Hsiang-Hsuan Liu worked in Wroclaw University, Poland. A preliminary version of this paper was published in WAOA 2019 [1].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, FH., Liu, HH. & Wong, P.W.H. Greedy is Optimal for Online Restricted Assignment and Smart Grid Scheduling for Unit Size Jobs. Theory Comput Syst 65, 1009–1032 (2021). https://doi.org/10.1007/s00224-021-10037-w

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-021-10037-w

Keywords

Search

Navigation