# Tight lower bounds for semi-online scheduling on two uniform machines with known optimum

- 17 Downloads

## Abstract

This problem is about scheduling a number of jobs on two uniform machines with given speeds 1 and \(s\ge 1\), so that the overall finishing time, i.e., the makespan, is earliest possible. We consider a semi-online variant (introduced for equal speeds) by Azar and Regev, where the jobs are arriving one after the other, while the scheduling algorithm knows the optimum value of the corresponding offline problem. One can ask how close any possible algorithm could get to the optimum value, that is, to give a lower bound on the competitive ratio: the supremum over ratios between the value of the solution given by the algorithm and the optimal offline solution. We contribute to this question by constructing tight lower bounds for all values of *s* in the intervals \([\frac{1+\sqrt{21}}{4},\frac{3+\sqrt{73}}{8}]\approx [1.3956,1.443]\) and \([\frac{5}{3},\frac{4+\sqrt{133}}{9}]\approx [\frac{5}{3},1.7258]\), except a very narrow interval.

## Keywords

Semi-online scheduling Makespan minimization Machine scheduling Lower bounds## Notes

### Acknowledgements

Gyorgy Dosa acknowledges the financial support of Szechenyi 2020 under the EFOP-3.6.1-16-2016-00015. György Dósa’s and Zsolt Tuza’s work was jointly funded by the National Research, Development and Innovation Office – NKFIH under the grant SNN 116095. Zhiyi Tan’s work was supported by the National Natural Science Foundation of China (11671356, 11271324, 11471286). Krzysztof Węsek’s work was partially supported by the European Union in the framework of European Social Fund through the Warsaw University of Technology Development Programme, realized by Center for Advanced Studies. Furthermore, Węsek’s work was conducted as a guest researcher at the Helmut Schmidt University. Armin Fügenschuh’s work was mostly carried out while being affiliated with the Helmut Schmidt University. Fügenschuh acknowledges the support by the German Research Association (DFG), grant number FU 860/1-1. Last but not least, we are grateful to the two anonymous referees for their various helpful comments on our manuscript.

## References

- Albers S (1999) Better bounds for online scheduling. SIAM Journal of Computing 29:459–473CrossRefGoogle Scholar
- Angelelli E, Speranza MG, Tuza Z (2007) Semi online scheduling on three processors with known sum of the tasks. J Sched 10:263–269CrossRefGoogle Scholar
- Angelelli E, Speranza MG, Tuza Z (2008) Semi-online scheduling on two uniform processors. Theoret Comput Sci 393:211–219CrossRefGoogle Scholar
- Azar Y, Regev O (2001) Online bin-stretching. Theoret Comput Sci 268(1):17–41CrossRefGoogle Scholar
- Berman P, Charikar M, Karpinski M (2000) On-line load balancing for related machines. Journal of Algorithms 35:108–121CrossRefGoogle Scholar
- Böhm M, Sgall J, van Stee R, Veselý P (2015) Better algorithms for online bin stretching. In: Proceeding of the 12th international workshop on approximation and online algorithms, (Lecture notes in computer science), vol 8952. Springer, New York, pp 23–34Google Scholar
- Böhm M, Sgall J, van Stee R, Veselý P (2017) A two-phase algorithm for bin stretching with stretching factor 1.5. J Comb Optim 34(3):810–828CrossRefGoogle Scholar
- Böhm M, Sgall J, van Stee R, Veselý P (2017) Online bin stretching with three bins. J Sched 20:601–621CrossRefGoogle Scholar
- Cheng TCE, Kellerer H, Kotov V (2005) Semi-on-line multi-processor scheduling with given total processing time. Theor Comput Sci 337:134–146CrossRefGoogle Scholar
- Dósa G, Fügenschuh A, Tan Z, Tuza Z, Węsek K (2018) Tight upper bounds for semi-online scheduling on two uniform machines with known optimum. CEJOR 26(1):161–180CrossRefGoogle Scholar
- Dósa G, Speranza MG, Tuza Z (2011) Two uniform machines with nearly equal speeds: unified approach to known sum and known optimum in semi on-line scheduling. J Comb Optim 21:458–480CrossRefGoogle Scholar
- Ebenlendr T, Sgall J (2015) A lower bound on deterministic online algorithms for scheduling on related machines without preemption. Theory Comput Syst 56(1):73–81CrossRefGoogle Scholar
- Epstein L (2003) Bin stretching revisited. Acta Inform. 39:97–117CrossRefGoogle Scholar
- Epstein L, Noga J, Seiden S, Sgall J, Woeginger GJ (2001) Randomized on-line scheduling on two uniform machines. J Sched 4:71–92CrossRefGoogle Scholar
- Faigle U, Kern W, Turán G (1989) On the performance of on-line algorithm for particular problem. Acta Cybern. 9:107–119Google Scholar
- Fleischer R, Wahl M (2000) Online scheduling revisited. J Sched 3:343–353CrossRefGoogle Scholar
- Gabay M, Kotov V, Brauner N (2015) Online bin stretching with bunch techniques. Theoret Comput Sci 602:103–113CrossRefGoogle Scholar
- Gormley T, Reingold N, Torng E, Westbrook J (2000) Generating adversaries for request-answer games. In: Proceeding of the 11th ACM-SIAM symposium on discrete algorithms. Society for industrial and applied mathematics ACM, New YorkGoogle Scholar
- Graham RL (1966) Bounds for certain multiprocessing anomalies. Bell Syst Tech J 45:1563–1581CrossRefGoogle Scholar
- Kellerer H, Kotov V (2013) An efficient algorithm for bin stretching. Oper Res Lett 41(4):343–346CrossRefGoogle Scholar
- Kellerer H, Kotov V, Gabay M (2015) An efficient algorithm for semi-online multiprocessor scheduling with given total processing time. J Sched 18(6):623–630CrossRefGoogle Scholar
- Kellerer H, Kotov V, Speranza MG, Tuza Z (1997) Semi on-line algorithms for the partition problem. Oper Res Lett 21:235–242CrossRefGoogle Scholar
- Lee K, Lim K (2013) Semi-online scheduling problems on a small number of machines. J Sched 16:461–477CrossRefGoogle Scholar
- Ng CT, Tan Z, He Y, Cheng TCE (2009) Two semi-online scheduling problems on two uniform machines. Theoret Comput Sci 410(8–10):776–792CrossRefGoogle Scholar
- Tan Z, Zhang A (2013) Online and semi-online scheduling. In: Pardalos PM et al (eds) Handbook of combinatorial optimization. Springer, New YorkGoogle Scholar