Abstract
We give a principled method to design online algorithms (for potentially non-linear problems) using a mathematical programming formulation of the problem, and also to analyze the competitiveness of the resulting algorithm using the dual program. This method can be viewed as an extension of the online primal-dual method for linear programming problems, to nonlinear programs. We show the application of this method to two online speed-scaling problems: one involving scheduling jobs on a speed scalable processor so as to minimize energy plus an arbitrary sum scheduling objective, and one involving routing virtual circuit connection requests in a network of speed scalable routers so as to minimize the aggregate power or energy used by the routers. This analysis shows that competitive algorithms exist for problems that had resisted analysis using the dominant potential function approach in the speed-scaling literature, and provides alternate cleaner analysis for other known results. This gives us another tool in the design and analysis of primal-dual algorithms for online problems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Im, S., Moseley, B., Pruhs, K.: A tutorial on amortized local competitiveness in online scheduling. SIGACT News 42(2), 83–97 (2011)
Becchetti, L., Leonardi, S., Marchetti-Spaccamela, A., Pruhs, K.: Online weighted flow time and deadline scheduling. Journal of Discrete Algorithms 4(3), 339–352 (2006)
Bansal, N., Pruhs, K., Stein, C.: Speed scaling for weighted flow time. SIAM J. Comput. 39(4), 1294–1308 (2009)
Andrews, M., Antonakopoulos, S., Zhang, L.: Energy-aware scheduling algorithms for network stability. In: INFOCOM, pp. 1359–1367 (2011)
Buchbinder, N., Naor, J.S.: The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science 3(2-3), 93–263 (2009)
Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, New York (2004)
Yao, F., Demers, A., Shenker, S.: A scheduling model for reduced CPU energy. In: Proc. IEEE Symposium on Foundations of Computer Science, pp. 374–382 (1995)
Bansal, N., Kimbrel, T., Pruhs, K.: Speed scaling to manage energy and temperature. Journal of the ACMÂ 54(1) (2007)
Bansal, N., Bunde, D.P., Chan, H.L., Pruhs, K.: Average rate speed scaling. Algorithmica 60(4), 877–889 (2011)
Bansal, N., Chan, H.-L., Pruhs, K., Katz, D.: Improved bounds for speed scaling in devices obeying the cube-root rule. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 144–155. Springer, Heidelberg (2009)
Albers, S., Fujiwara, H.: Energy-efficient algorithms for flow time minimization. ACM Transactions on Algorithms 3(4), 49 (2007)
Bansal, N., Chan, H.L., Pruhs, K.: Speed scaling with an arbitrary power function. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 693–701 (2009)
Lam, T., Lee, L., To, I., Wong, P.: Speed scaling functions based for flow time scheduling based on active job count. In: European Symposium on Algorithms, pp. 647–659 (2008)
Andrew, L.L.H., Lin, M., Wierman, A.: Optimality, fairness, and robustness in speed scaling designs. In: SIGMETRICS, pp. 37–48 (2010)
Chan, H.L., Edmonds, J., Lam, T.W., Lee, L.K., Marchetti-Spaccamela, A., Pruhs, K.: Nonclairvoyant speed scaling for flow and energy. In: Symposium on Theoretical Aspects of Computer Science, pp. 255–264 (2009)
Chan, S.H., Lam, T.W., Lee, L.K.: Non-clairvoyant speed scaling for weighted flow time. In: European Symposium on Algorithms, pp. 23–35 (2010)
Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. Journal of the ACM 44(3), 486–504 (1997)
Awerbuch, B., Azar, Y., Grove, E.F., Kao, M.Y., Krishnan, P., Vitter, J.S.: Load balancing in the Lp norm. In: IEEE Symposium on Foundations of Computer Science, pp. 383–391 (1995)
Albers, S.: Energy-efficient algorithms. Communications of the ACM 53(5), 86–96 (2010)
Caragiannis, I.: Better bounds for online load balancing on unrelated machines. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 972–981. ACM, New York (2008)
Azar, Y., Epstein, A.: Convex programming for scheduling unrelated parallel machines. In: ACM Symposium on Theory of Computing, pp. 331–337. ACM, New York (2005)
Anil Kumar, V.S., Marathe, M.V., Parthasarathy, S., Srinivasan, A.: A unified approach to scheduling on unrelated parallel machines. Journal of the ACMÂ 56(5), Art. 28, 31 (2009)
Andrews, M., Antonakopoulos, S., Zhang, L.: Minimum-cost network design with (dis)economies of scale. In: IEEE Symposium on Foundations of Computer Science, pp. 585–592 (2010)
Gupta, A., Im, S., Krishnaswamy, R., Moseley, B., Pruhs, K.: Scheduling jobs with varying parallelizability to reduce variance. In: ACM Symposium on Parallelism in Algorithms and Architectures, pp. 11–20 (2010)
Im, S., Moseley, B.: Online scalable algorithm for minimizing lk-norms of weighted flow time on unrelated machines. In: ACM-SIAM Symposium on Discrete Algorithms, pp. 95–108 (2011)
Anand, S., Garg, N., Kumar, A.: Resource augmentation for weighted flow-time explained by dual fitting. In: ACM-SIAM Symposium on Discrete Algorithms (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gupta, A., Krishnaswamy, R., Pruhs, K. (2013). Online Primal-Dual for Non-linear Optimization with Applications to Speed Scaling. In: Erlebach, T., Persiano, G. (eds) Approximation and Online Algorithms. WAOA 2012. Lecture Notes in Computer Science, vol 7846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38016-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-38016-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-38015-0
Online ISBN: 978-3-642-38016-7
eBook Packages: Computer ScienceComputer Science (R0)