Preemptive Online Scheduling: Optimal Algorithms for All Speeds

Abstract

Our main result is an optimal online algorithm for preemptive scheduling on uniformly related machines with the objective to minimize makespan. The algorithm is deterministic, yet it is optimal even among all randomized algorithms. In addition, it is optimal for any fixed combination of speeds of the machines, and thus our results subsume all the previous work on various special cases. Together with a new lower bound it follows that the overall competitive ratio of this optimal algorithm is between 2.054 and e ≈ 2.718. We also give a complete analysis of the competitive ratio for three machines.

Better Algorithms for Online Bin Stretching

“…In fact, assuming that no good situation is reached before Step (4), we observe the following: [4,6], we reach GS2 by assigning j to C. So s(j) < 4 and j is assigned to the least loaded bin, which must have load more than 2 already, else we reach GS2 or GSFF(A| 4 , B| 4 ) does not fail on j. This implies that both A and B received items in Step [4,6] leads to GS2. Any item with size more than 6 is assigned to A if it fits there, reaching GS5, and else to B or C, reaching GS7 since s(A) < 9 + 4 − s(B)) .…”

“…Larger items are fine, as one per bin is sufficient, and the smaller ones are fine as well as we can always fit at least two of them and this guarantees that we have only two bins filled below 4. This motivates our classification of items: Only the regular items of size in (0, 3] ∪ (4,6] are packed in the bins filled up to size 6. The medium items of size in (3,4] are packed in their own bins (four or five per bin).…”

“…Such algorithms were studied and are important in designing constant-competitive algorithms without the additional knowledge, e.g., for scheduling in the more general model of uniformly related machines [2,4,6].…”

“…The medium items of size in (3,4] are packed in their own bins (four or five per bin). Similarly, large items of size in (6,9] are packed in pairs in their own bins. Finally, the huge items of size larger than 9 are handled similarly as in the previous papers: If possible, they are packed with the regular items, otherwise each in their own bin.…”

“…[3,6], and C is empty, there exists an algorithm that packs all remaining items into three bins of capacity 22. [3,4), otherwise we reach GS2.…”