Abstract
We consider the on-line version of the preemptive scheduling problem that minimizes the machine completion time vector in the ℓ p norm (a direct extension of the l ∞ norm: the makespan) on m parallel identical machines. We present a lower bound on the competitive ratio of any randomized on-line algorithm with respect to the general ℓ p norm. This lower bound amounts to calculating a (non-convex) mathematical program and generalizes the existing result on makespan. While similar technique has been utilized to provide the best possible lower bound for makespan, the proposed lower bound failed to achieve the best possible lower bound for general ℓ p norm (though very close), and hence revealing intricate and essential difference between the general ℓ p norm and the makespan.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alon, N., Azar, Y., Woeginger, G.J., Yadid, T.: Approximation Schemes for Scheduling. In: SODA 1997, pp. 493–500 (1997)
Avidor, A., Azar, Y., Sgall, J.: Ancient and New Algorithms for Load Balancing in the ℓ p Norm. Algorithmica 29, 422–441 (2001)
Azar, Y., Epstein, A.: Convex programming for Scheduling Unrelated Parallel Machines. In: STOC 2005, pp. 331–337 (2005)
Azar, Y., Epstein, A., Epstein, L.: Load Balancing of Temporary Tasks in the ℓ p Norm. Theoretical Computer Science 361(2-3), 314–328 (2006)
Azar, Y., Epstein, L., Richter, Y., Woeginger, G.J.: All-Norm Approximation Algorithms. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, pp. 288–297. Springer, Heidelberg (2002)
Azar, Y., Taub, S.: All-Norm Approximation for Scheduling on Identical Machines. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 298–310. Springer, Heidelberg (2004)
Chandra, A.K., Wong, C.K.: Worst-Case Analysis of a Placement Algorithm Related to Storage Allocation. SIAM Journal on Computing 1, 249–263 (1975)
Chen, B., van Vliet, A., Woeginger, G.J.: An Optimal Algorithm for Preemptive On-Line Scheduling. Operations Research Letters 18(3), 127–131 (1995)
Du, D.-L., Jiang, X., Zhang, G.: Optimal Preemptive Online Scheduling to Minimize ℓ p Norm on Two Processors. Journal of Manufacturing and Management Optimization 1(3), 345–351 (2005)
Ebenlendr, T., Jawor, W., Sgall, J.: Preemptive Online Scheduling: Optimal Algorithms for All Speeds. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 327–339. Springer, Heidelberg (2006)
Epstein, L., Sgall, J.: A Lower Bound for On-Line Scheduling on Uniformly Related Machines. Operations Research Letters 26(1), 17–22 (2000)
Epstein, L., Tassa, T.: Optimal Preemptive Scheduling for General Target Functions. Journal of Computer and System Sciences 72(1), 132–162 (2006)
Kumar, V.S.A., Marathe, M.V., Parthasarathy, S., Srinivasan, A.: Approximation Algorithms for Scheduling on Multiple Machines. In: FOCS 2005, pp. 254–263 (2005)
Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B.: Sequencing and Scheduling: Algorithms and Complexity. In: Graves, S.C., Rinnooy Kan, A.H.G., Zipkin, P.H. (eds.) Logistics of Production and Inventory, pp. 445–522. North-Holland, Amsterdam (1993)
Lin, L.: Semi-Online Scheduling Algorithm under the ℓ p Norm on Two Identical Machines. Journal of Zhejiang University (Science Edition) 34(2), 148–151 (2007) (in Chinese)
Lin, L., Tan, Z.Y., He, Y.: Deterministic and Randomized Scheduling Problems under the ℓ p Norm on Two Identical Machines. Journal of Zhejiang University Science 6(1), 20–26 (2005)
McNaughton, R.: Scheduling with Deadlines and Loss Functions. Management Science 6(1), 1–12 (1959)
Tan, Z., He, Y., Epstein, L.: Optimal On-line Algorithms for the Uniform Machine Scheduling Problem with Ordinal Data. Information and Computation 196(1), 57–70 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shuai, T., Du, D. (2008). A Lower Bound for the On-Line Preemptive Machine Scheduling with ℓ p Norm. In: Hu, X., Wang, J. (eds) Computing and Combinatorics. COCOON 2008. Lecture Notes in Computer Science, vol 5092. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69733-6_65
Download citation
DOI: https://doi.org/10.1007/978-3-540-69733-6_65
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69732-9
Online ISBN: 978-3-540-69733-6
eBook Packages: Computer ScienceComputer Science (R0)