Abstract
We consider a scheduling problem where a set of jobs is a-priori distributed over parallel machines. The processing time of any job is dependent on the usage of a scarce renewable resource, e.g. personnel. An amount of k units of that resource can be allocated to the jobs at any time, and the more of that resource is allocated to a job, the smaller its processing time. The dependence of processing times on the amount of resources is linear for any job. The objective is to find a resource allocation and a schedule that minimizes the makespan. Utilizing an integer quadratic programming relaxation, we show how to obtain a (3 + ε) -approximation algorithm for that problem, for any ε > 0. This generalizes and improves previous results, respectively. Our approach relies on a fully polynomial time approximation scheme to solve the quadratic programming relaxation. This result is interesting in itself, because the underlying quadratic program is NP-hard to solve. We also derive lower bounds, and discuss further generalizations of the results.
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
Blazewicz, J., Lenstra, J.K., Rinnooy Kan, A.H.G.: Scheduling subject to resource constraints: Classification and complexity. Discr. Appl. Math. 5, 11–24 (1983)
Chen, Z.-L.: Simultaneous Job Scheduling and Resource Allocation on Parallel Machines. Ann. Oper. Res. 129, 135–153 (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completenes. W. H. Freeman, New York (1979)
Graham, R.L.: Bounds for certain multiprocessing anomalies. Bell System Technical Journal 45, 1563–1581 (1966); See also [5]
Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Applied Math. 17, 416–429 (1969)
Graham, R.L., Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G.: Optimization and approximation in deterministic sequencing and scheduling: A survey. Ann. Discr. Math. 5, 287–326 (1979)
Grigoriev, A., Kellerer, H., Strusevich, V.A.: Scheduling parallel dedicated machines with the speeding-up resource (manuscript, 2003); Proceedings of the 6th Workshop on Models and Algorithms for Planning and Scheduling Problems, Aussois, France, pp. 131–132 (2003) (extended abstract)
Grigoriev, A., Sviridenko, M., Uetz, M.: Unrelated Parallel Machine Scheduling with Resource Dependent Processing Times. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 182–195. Springer, Heidelberg (2005)
Horst, R., Pardalos, P.M. (eds.): Handbook of Global Optimization. Nonconvex Optimization and Its Applications, vol. 2. Springer, Heidelberg (1995)
Jansen, K.: Scheduling Malleable Parallel Tasks: An Asymptotic Fully Polynomial Time Approximation Scheme. Algorithmica 39, 59–81 (2004)
Jansen, K., Mastrolilli, M., Solis-Oba, R.: Approximation Schemes for Job Shop Scheduling Problems with Controllable Processing Times. European Journal of Operational Research 167, 297–319 (2005)
Kelley, J.E., Walker, M.R.: Critical path planning and scheduling: An introduction. Mauchly Associates, Ambler (PA) (1959)
Kellerer, H., Strusevich, V.A.: Scheduling parallel dedicated machines under a single non-shared resource. Europ. J. Oper. Res. 147, 345–364 (2003)
Kellerer, H., Strusevich, V.A.: Scheduling problems for parallel dedicated machines under multiple resource constraints. Discr. Appl. Math. 133, 45–68 (2004)
Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Math. Prog. 46, 259–271 (1990)
Mounie, G., Rapine, C., Trystram, D.: Efficient Approximation Algorithms for Scheduling Malleable Tasks. In: Proceedings of the 11th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 23–32 (1999)
Mounie, G., Rapine, C., Trystram, D.: A 3/2-Dual Approximation Algorithm for Scheduling Independent Monotonic Malleable Tasks (manuscript), Retrieved from: http://citeseer.csail.mit.edu/558879.html
Pardalos, P.M., Schnitger, G.: Checking Local Optimality in Constrained Quadratic Programming is NP-hard. Oper. Res. Lett. 7, 33–35 (1988)
Pruhs, K., Woeginger, G.J.: Approximation Schemes for a Class of Subset Selection Problems. In: Farach-Colton, M. (ed.) LATIN 2004. LNCS, vol. 2976, pp. 203–211. Springer, Heidelberg (2004)
Shmoys, D.B., Tardos, E.: An approximation algorithm for the generalized assignment problem. Math. Prog. 62, 461–474 (1993)
Skutella, M.: Approximation algorithms for the discrete time-cost tradeoff problem. Math. Oper. Res. 23, 909–929 (1998)
Turek, J., Wolf, J.L., Yu, P.S.: Approximate Algorithms for Scheduling Parallelizable Tasks. In: Proceedings of the 4th Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 323–332 (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Grigoriev, A., Uetz, M. (2006). Scheduling Parallel Jobs with Linear Speedup. In: Erlebach, T., Persinao, G. (eds) Approximation and Online Algorithms. WAOA 2005. Lecture Notes in Computer Science, vol 3879. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11671411_16
Download citation
DOI: https://doi.org/10.1007/11671411_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-32207-8
Online ISBN: 978-3-540-32208-5
eBook Packages: Computer ScienceComputer Science (R0)