Primal-Dual and Dual-Fitting Analysis of Online Scheduling Algorithms for Generalized Flow Time Problems

  • Spyros Angelopoulos
  • Giorgio Lucarelli
  • Kim Thang Nguyen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9294)


We study online scheduling problems on a single processor that can be viewed as extensions of the well-studied problem of minimizing total weighted flow time. In particular, we provide a framework of analysis that is derived by duality properties, does not rely on potential functions and is applicable to a variety of scheduling problems. A key ingredient in our approach is bypassing the need for “black-box” rounding of fractional solutions, which yields improved competitive ratios.

We begin with an interpretation of Highest-Density-First (HDF) as a primal-dual algorithm, and a corresponding proof that HDF is optimal for total fractional weighted flow time (and thus scalable for the integral objective). Building upon the salient ideas of the proof, we show how to apply and extend this analysis to the more general problem of minimizing ∑ jwjg(Fj), where wj is the job weight, Fj is the flow time and g is a non-decreasing cost function. Among other results, we present improved competitive ratios for the setting in which g is a concave function, and the setting of same-density jobs but general cost functions. We further apply our framework of analysis to online weighted completion time with general cost functions as well as scheduling under polyhedral constraints.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anand, S., Garg, N., Kumar, A.: Resource augmentation for weighted flow-time explained by dual fitting. In: SODA, pp. 1228–1241 (2012)Google Scholar
  2. 2.
    Angelopoulos, S., Lucarelli, G., Thang, N.K.: Primal-dual and dual-fitting analysis of online scheduling algorithms for generalized flow-time problems. CoRR, abs/1502.03946 (2015)Google Scholar
  3. 3.
    Antoniadis, A., Barcelo, N., Consuegra, M., Kling, P., Nugent, M., Pruhs, K., Scquizzato, M.: Efficient computation of optimal energy and fractional weighted flow trade-off schedules. In: STACS. LIPIcs, vol. 25, pp. 63–74 (2014)Google Scholar
  4. 4.
    Bansal, N., Chan, H.-L.: Weighted flow time does not admit o(1)-competitive algorithms. In: SODA, pp. 1238–1244 (2009)Google Scholar
  5. 5.
    Bansal, N., Pruhs, K.R.: Server scheduling in the weighted ℓp norm. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 434–443. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Bansal, N., Pruhs, K.: The geometry of scheduling. In: FOCS, pp. 407–414 (2010)Google Scholar
  7. 7.
    Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Pruhs, K.: Online weighted flow time and deadline scheduling. J. Discrete Algorithms 4(3), 339–352 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Devanur, N.R., Huang, Z.: Primal dual gives almost optimal energy efficient online algorithms. In: SODA, pp. 1123–1140 (2014)Google Scholar
  9. 9.
    Fox, K., Im, S., Kulkarni, J., Moseley, B.: Online non-clairvoyant scheduling to simultaneously minimize all convex functions. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds.) RANDOM 2013 and APPROX 2013. LNCS, vol. 8096, pp. 142–157. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Gupta, A., Krishnaswamy, R., Pruhs, K.: Online primal-dual for non-linear optimization with applications to speed scaling. In: Erlebach, T., Persiano, G. (eds.) WAOA 2012. LNCS, vol. 7846, pp. 173–186. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Im, S., Kulkarni, J., Munagala, K.: Competitive algorithms from competitive equilibria: Non-clairvoyant scheduling under polyhedral constraints. In: STOC, pp. 313–322 (2014)Google Scholar
  12. 12.
    Im, S., Kulkarni, J., Munagala, K., Pruhs, K.: Selfishmigrate: A scalable algorithm for non-clairvoyantly scheduling heterogeneous processors. In: FOCS, pp. 531–540 (2014)Google Scholar
  13. 13.
    Im, S., Moseley, B., Pruhs, K.: A tutorial on amortized local competitiveness in online scheduling. SIGACT News 42(2), 83–97 (2011)CrossRefGoogle Scholar
  14. 14.
    Im, S., Moseley, B., Pruhs, K.: Online scheduling with general cost functions. SIAM Journal on Computing 43(1), 126–143 (2014)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kalyanasundaram, B., Pruhs, K.: Speed is as powerful as clairvoyance. Journal of the ACM 47(4), 617–643 (2000)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Nguyen, K.T.: Lagrangian duality in online scheduling with resource augmentation and speed scaling. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 755–766. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Spyros Angelopoulos
    • 1
    • 2
  • Giorgio Lucarelli
    • 3
  • Kim Thang Nguyen
    • 4
  1. 1.UPMC Univ Paris 06Sorbonne UniversitésParisFrance
  2. 2.CNRSParisFrance
  3. 3.LIG, University Grenoble-AlpesGrenobleFrance
  4. 4.IBISC, University of Evry Val d’EssonneEssonneFrance

Personalised recommendations