Robust Algorithms for Preemptive Scheduling
 Leah Epstein,
 Asaf Levin
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Preemptive scheduling problems on parallel machines are classic problems. Given the goal of minimizing the makespan, they are polynomially solvable even for the most general model of unrelated machines. In these problems, a set of jobs is to be assigned to run on a set of m machines. A job can be split into parts arbitrarily and these parts are to be assigned to time slots on the machines without parallelism, that is, for every job, at most one of its parts can be processed at each time.
Motivated by sensitivity analysis and online algorithms, we investigate the problem of designing robust algorithms for constructing preemptive schedules. Robust algorithms receive one piece of input at a time. They may change a small portion of the solution as an additional part of the input is revealed. The capacity of change is based on the size of the new piece of input. For scheduling problems, the supremum ratio between the total size of the jobs (or parts of jobs) which may be rescheduled upon the arrival of a new job j, and the size of j, is called migration factor.
We design a strongly optimal algorithm with the migration factor \(1\frac{1}{m}\) for identical machines. Strongly optimal algorithms avoid idle time and create solutions where the (nonincreasingly) sorted vector of completion times of the machines is lexicographically minimal. In the case of identical machines this results not only in makespan minimization, but the created solution is also optimal with respect to any ℓ _{ p } norm (for p>1). We show that an algorithm of a smaller migration factor cannot be optimal with respect to makespan or any other ℓ _{ p } norm, thus the result is best possible in this sense as well. We further show that neither uniformly related machines nor identical machines with restricted assignment admit an optimal algorithm with a constant migration factor. This lower bound holds both for makespan minimization and for any ℓ _{ p } norm. Finally, we analyze the case of two machines and show that in this case it is still possible to maintain an optimal schedule with a small migration factor in the cases of two uniformly related machines and two identical machines with restricted assignment.
 Alon, N., Azar, Y., Woeginger, G.J., Yadid, T. (1997) Approximation schemes for scheduling. Proc. 8th Symp. on Discrete Algorithms (SODA). ACM/SIAM, New York/Philadelphia, pp. 493500
 Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O. (1997) Online load balancing with applications to machine scheduling and virtual circuit routing. J. ACM 44: pp. 486504
 Azar, Y., Naor, J., Rom, R. (1995) The competitiveness of online assignments. J. Algorithms 18: pp. 221237 CrossRef
 Berman, P., Charikar, M., Karpinski, M. (2000) Online load balancing for related machines. J. Algorithms 35: pp. 108121 CrossRef
 Caprara, A., Kellerer, H., Pferschy, U. (2003) Approximation schemes for ordered vector packing problems. Nav. Res. Logist. 50: pp. 5869 CrossRef
 Chen, B., Vliet, A., Woeginger, G.J. (1995) An optimal algorithm for preemptive online scheduling. Oper. Res. Lett. 18: pp. 127131 CrossRef
 Correa, J.R., Skutella, M., Verschae, J. (2012) The power of preemption on unrelated machines and applications to scheduling orders. Math. Oper. Res. 37: pp. 379398 CrossRef
 Dósa, G., Epstein, L. (2011) Preemptive online scheduling with reordering. SIAM J. Discrete Math. 25: pp. 2149 CrossRef
 Ebenlendr, T., Jawor, W., Sgall, J. (2009) Preemptive online scheduling: optimal algorithms for all speeds. Algorithmica 53: pp. 504522 CrossRef
 Ebenlendr, T., Sgall, J. (2009) Optimal and online preemptive scheduling on uniformly related machines. J. Sched. 12: pp. 517527 CrossRef
 Englert, M., Özmen, D., Westermann, M. (2008) The power of reordering for online minimum makespan scheduling. Proc. 48th Symp. Foundations of Computer Science (FOCS). pp. 603612
 Epstein, L. (2001) Optimal preemptive online scheduling on uniform processors with nondecreasing speed ratios. Oper. Res. Lett. 29: pp. 9398 CrossRef
 Epstein, L., Levin, A. (2009) A robust APTAS for the classical bin packing problem. Math. Program. 119: pp. 3349 CrossRef
 Epstein, L., Levin, A. (2010) AFPTAS results for common variants of bin packing: a new method for handling the small items. SIAM J. Optim. 20: pp. 31213145 CrossRef
 Epstein, L., Levin, A.: Robust approximation schemes for cube packing. Manuscript (2010, in review)
 Epstein, L., Noga, J., Seiden, S.S., Sgall, J., Woeginger, G.J. (2001) Randomized online scheduling on two uniform machines. J. Sched. 4: pp. 7192 CrossRef
 Epstein, L., Sgall, J. (2000) A lower bound for online scheduling on uniformly related machines. Oper. Res. Lett. 26: pp. 1722 CrossRef
 Epstein, L., Tassa, T. (2006) Optimal preemptive scheduling for general target functions. J. Comput. Syst. Sci. 72: pp. 132162 CrossRef
 Fleischer, R., Wahl, M. (2000) Online scheduling revisited. J. Sched. 3: pp. 343353 CrossRef
 Gonzales, T.F., Sahni, S. (1978) Preemptive scheduling of uniform processor systems. J. ACM 25: pp. 92101
 Graham, R.L. (1966) Bounds for certain multiprocessing anomalies. Bell Syst. Tech. J. 45: pp. 15631581 CrossRef
 Horvath, E.C., Lam, S., Sethi, R. (1977) A level algorithm for preemptive scheduling. J. ACM 24: pp. 3243
 Huo, Y., Leung, J.Y.T., Wang, X. (2009) Preemptive scheduling algorithms with nested processing set restriction. Int. J. Found. Comput. Sci. 20: pp. 11471160 CrossRef
 Lawler, E.L., Labetoulle, J. (1978) On preemptive scheduling of unrelated parallel processors by linear programming. J. ACM 25: pp. 612619
 Lenstra, J.K., Shmoys, D.B., Tardos, E. (1990) Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46: pp. 259271 CrossRef
 Liu, J.W.S., Liu, C.L. Bounds on scheduling algorithms for heterogeneous computing systems. In: Rosenfeld, J.L. eds. (1974) Proceedings of IFIP Congress. pp. 349353
 Liu, J.W.S., Yang, A.T. (1974) Optimal scheduling of independent tasks on heterogeneous computing systems. Proceedings of the ACM National Conference. ACM, New York, pp. 3845
 McNaughton, R. (1959) Scheduling with deadlines and loss functions. Manag. Sci. 6: pp. 112 CrossRef
 Muntz, R.R., Coffman, E.G. (1969) Optimal preemptive scheduling on twoprocessor systems. IEEE Trans. Comput. 18: pp. 10141020 CrossRef
 Muntz, R.R., Coffman, E.G. (1970) Preemptive scheduling of realtime tasks on multiprocessor systems. J. ACM 17: pp. 324338
 Sanders, P., Sivadasan, N., Skutella, M. (2009) Online scheduling with bounded migration. Math. Oper. Res. 34: pp. 481498 CrossRef
 Sgall, J. (1997) A lower bound for randomized online multiprocessor scheduling. Inf. Process. Lett. 63: pp. 5155 CrossRef
 Shachnai, H., Tamir, T., Woeginger, G.J. (2005) Minimizing makespan and preemption costs on a system of uniform machines. Algorithmica 42: pp. 309334 CrossRef
 Skutella, M., Verschae, J. (2010) A robust PTAS for machine covering and packing. Proc. 18th European Symp. on Algorithms (ESA). pp. 3647
 Wen, J., Du, D. (1998) Preemptive online scheduling for two uniform processors. Oper. Res. Lett. 23: pp. 113116 CrossRef
 Title
 Robust Algorithms for Preemptive Scheduling
 Journal

Algorithmica
Volume 69, Issue 1 , pp 2657
 Cover Date
 20140501
 DOI
 10.1007/s0045301297183
 Print ISSN
 01784617
 Online ISSN
 14320541
 Publisher
 Springer US
 Additional Links
 Topics
 Keywords

 Preemptive scheduling
 Migration factor
 Robust algorithms
 Industry Sectors
 Authors

 Leah Epstein ^{(1)}
 Asaf Levin ^{(2)}
 Author Affiliations

 1. Department of Mathematics, University of Haifa, 31905, Haifa, Israel
 2. Faculty of Industrial Engineering and Management, The Technion, 32000, Haifa, Israel