Energy Minimization via a Primal-Dual Algorithm for a Convex Program

  • Evripidis Bampis
  • Vincent Chau
  • Dimitrios Letsios
  • Giorgio Lucarelli
  • Ioannis Milis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7933)


We present an optimal primal-dual algorithm for the energy minimization preemptive open-shop problem in the speed-scaling setting. Our algorithm uses the approach of Devanur et al. [JACM 2008], by applying the primal-dual method in the setting of convex programs and KKT conditions. We prove that our algorithm converges and that it returns an optimal solution, but we were unable to prove that it converges in polynomial time. For this reason, we conducted a series of experiments showing that the number of iterations of our algorithm increases linearly with the number of jobs, n, when n is greater than the number of machines, m. We also compared the speed of our method with respect to the time spent by a commercial solver to directly solve the corresponding convex program. The computational results give evidence that for n > m, our algorithm is clearly faster. However, for the special family of instances where n = m, our method is slower.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Albers, S.: Energy-efficient algorithms. Communications of ACM 53, 86–96 (2010)CrossRefGoogle Scholar
  2. 2.
    Albers, S.: Algorithms for dynamic speed scaling. In: STACS 2011, pp. 1–11 (2011)Google Scholar
  3. 3.
    Bampis, E., Letsios, D., Lucarelli, G.: Green scheduling, flows and matchings. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 106–115. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge Un. Press (2004)Google Scholar
  5. 5.
    Brucker, P.: Scheduling algorithms, 4th edn. Springer (2004)Google Scholar
  6. 6.
    Devanur, N.R., Papadimitriou, C.H., Saberi, A., Vazirani, V.V.: Market equilibrium via a primal-dual algorithm for a convex program. Journal of the ACM 55(5) (2008)Google Scholar
  7. 7.
    Edmonds, J.: Maximum matching and a polyhedron with 0,1-vertices. Journal of Research of the National Bureau of Standards, Section B 69, 125–130 (1965)MathSciNetMATHGoogle Scholar
  8. 8.
    Ford, L.R., Fulkerson, D.R.: Maximal flow through a network. Canadian Journal of Mathematics 8, 399–404 (1956)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems, ch. 4, pp. 144–191. PWS Publishing Company (1997)Google Scholar
  10. 10.
    Gonzalez, T.: A note on open shop preemptive schedules. IEEE Transactions on Computers C-28, 782–786 (1979)CrossRefGoogle Scholar
  11. 11.
    Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling. Annals of Discrete Mathematics 5, 287–326 (1979)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gupta, A., Krishnaswamy, R., Pruhs, K.: Online primal-dual for non-linear optimization with applications to speed scaling. CoRR, abs/1109.5931 (2011)Google Scholar
  13. 13.
    Kuhn, H.W.: The hungarian method for the assignment problem. Naval Research Logistics Quarterly 2, 83–97 (1955)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Vazirani, V.V.: Approximation algorithms, ch. 12. Springer (2001)Google Scholar
  15. 15.
    Yao, F., Demers, A., Shenker, S.: A scheduling model for reduced CPU energy. In: FOCS 1995, pp. 374–382 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Evripidis Bampis
    • 1
  • Vincent Chau
    • 2
  • Dimitrios Letsios
    • 1
    • 2
  • Giorgio Lucarelli
    • 1
    • 2
  • Ioannis Milis
    • 3
  1. 1.LIP6Université Pierre et Marie CurieFrance
  2. 2.IBISCUniversité d’ÉvryFrance
  3. 3.Dept. of InformaticsAUEBAthensGreece

Personalised recommendations