The local–global conjecture for scheduling with non-linear cost
- First Online:
We consider the classical scheduling problem on a single machine, on which we need to schedule sequentially n given jobs. Every job j has a processing time \(p_j\) and a priority weight \(w_j\), and for a given schedule a completion time \(C_j\). In this paper, we consider the problem of minimizing the objective value \(\sum _j w_j C_j^\beta \) for some fixed constant \(\beta >0\). This non-linearity is motivated for example by the learning effect of a machine improving its efficiency over time, or by the speed scaling model. For \(\beta =1\), the well-known Smith’s rule that orders job in the non-increasing order of \(w_j/p_j\) gives the optimum schedule. However, for \(\beta \ne 1\), the complexity status of this problem is open. Among other things, a key issue here is that the ordering between a pair of jobs is not well defined, and might depend on where the jobs lie in the schedule and also on the jobs between them. We investigate this question systematically and substantially generalize the previously known results in this direction. These results lead to interesting new dominance properties among schedules which lead to huge speed up in exact algorithms for the problem. An experimental study evaluates the impact of these properties on the exact algorithm A*.
KeywordsScheduling Single machine Non-linear cost function Pruning rules Algorithm A*
- Bansal, N., & Pruhs, K. (2010). The geometry of scheduling. In Proceedings of the IEEE 51st Annual Symposium on Foundations of Computer Science (FOCS) (pp. 407–414).Google Scholar
- Cheung, M., & Shmoys, D. (2011). A primal-dual approximation algorithm for min-sum single-machine scheduling problems. In Proceedings of the 14th International Workshop APPROX and 15th International Workshop RANDOM (pp. 135–146).Google Scholar
- Croce, F., Tadei, R., Baracco, P., Di Tullio, R. (1993). On minimizing the weighted sum of quadratic completion times on a single machine. In Proceedings of the IEEE International Conference on Robotics and Automation (pp. 816–820).Google Scholar
- Dürr, C., Jeż, Ł., & Vásquez, O. C. (2014). Scheduling under dynamic speed-scaling for minimizing weighted completion time and energy consumption. Discrete Applied Mathematics, 196, 20–27.Google Scholar
- Epstein, L., Levin, A., Marchetti-Spaccamela, A., Megow, N., Mestre, J., Skutella, M., & Stougie, L. (2010). Universal sequencing on a single machine. In Proceedings of the 14th International Conference of Integer Programming and Combinatorial Optimization (IPCO) (pp. 230–243).Google Scholar
- Höhn, W., & Jacobs, T. (2012a). An experimental and analytical study of order constraints for single machine scheduling with quadratic cost. In Proceedings of the 14th Workshop on Algorithm Engineering and Experiments (ALENEX’12) (pp. 103–117).Google Scholar
- Höhn, W., & Jacobs, T. (2012b). Generalized min sum scheduling instance library. http://www.coga.tu-berlin.de/v-menue/projekte/complex_scheduling/generalized_min-sum_scheduling_instance_library/.
- Höhn, W., & Jacobs, T. (2012c). On the performance of Smith’s rule in single-machine scheduling with nonlinear cost. In Proceedings of the 10th Latin American Theoretical Informatics Symposium (LATIN) (pp. 482–493).Google Scholar
- Megow, N., & Verschae, J. (2013). Dual techniques for scheduling on a machine with varying speed. In Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP) (pp. 745–756).Google Scholar
- Szwarc, W. (1998). Decomposition in single-machine scheduling. Annals of Operations Research, 83, 271–287.Google Scholar
- Vásquez, O. C. (2014). For the airplane refueling problem local precedence implies global precedence. Optimization Letters, 9(4), 663–675.Google Scholar