Some results on Liu's conjecture
SIAM Journal on Discrete Mathematics
Proof of the 4/3 conjecture for preemptive vs. nonpreemptive two-processor scheduling
Journal of the ACM (JACM)
A Level Algorithm for Preemptive Scheduling
Journal of the ACM (JACM)
Preemptive Scheduling of Uniform Processor Systems
Journal of the ACM (JACM)
Parallel Processor Scheduling with Limited Number of Preemptions
SIAM Journal on Computing
Optimal scheduling of independent tasks on heterogeneous computing systems
ACM '74 Proceedings of the 1974 annual conference - Volume 1
Optimal scheduling on multi-processor computing systems
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
Optimal and online preemptive scheduling on uniformly related machines
Journal of Scheduling
The Power of Preemption on Unrelated Machines and Applications to Scheduling Orders
Mathematics of Operations Research
A comment on scheduling on uniform machines under chain-type precedence constraints
Operations Research Letters
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In previous study on comparing the makespan of the schedule allowed to be preempted at most i times and that of the optimal schedule with unlimited number of preemptions, the worst case ratio was usually obtained by analyzing the structures of the optimal schedules. For m identical machines case, the worst case ratio was shown to be 2m/(m+i+1) for any 0≤i≤m驴1 (Braun and Schmidt in SIAM J. Comput. 32(3):671---680, 2003), and they showed that LPT algorithm is an exact algorithm which can guarantee the worst case ratio for i=0. In this paper, we propose a simpler method which is based on the design and analysis of the algorithm and finding an instance in the worst case. It can not only obtain the worst case ratio but also give a linear algorithm which can guarantee this ratio for any 0≤i≤m驴1, and thus we generalize the previous results. We also make a discussion on the trade-off between the objective value and the number of preemptions. In addition, we consider the i-preemptive scheduling on two uniform machines. For both i=0 and i=1, we give two linear algorithms and present the worst-case ratios with respect to s, i.e., the ratio of the speeds of two machines.