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

Abstract

We consider a semi-online version of the problem of scheduling a sequence of jobs of different lengths on two uniform machines with given speeds 1 and s. Jobs are revealed one by one (the assignment of a job has to be done before the next job is revealed), and the objective is to minimize the makespan. In the considered variant the optimal offline makespan is known in advance. The most studied question for this online-type problem is to determine the optimal competitive ratio, that is, the worst-case ratio of the solution given by an algorithm in comparison to the optimal offline solution. In this paper, we make a further step towards completing the answer to this question by determining the optimal competitive ratio for s between 5+ √ 241 12 ≈ 1.7103 and √ 3 ≈ 1.7321, one of the intervals that were still open. Namely, we present and analyze a compound algorithm achieving the previously known lower bounds.

Semi-online scheduling: A survey

“…By traversing the lower bound binary tree from root to the leaf nodes, one can obtain the instances, for which any semi-online algorithm achieves a CR of at least the defined LB. In [89,94], the authors considered the setup studied in [69] and obtained lower bounds in terms of an algebraic function r(s) for the following unexplored speed ratio intervals.…”

“…In [91], they studied for the interval s ∈ [1.710, 1.732] and achieved tight bounds of 2s+10 9s+7 for s=1.7258 and s+1 2 for 1.725 ≤ s ≤ 1.732 respectively. The obtained results draw an insight that a single algebraic function can not formulate the tightness of the LB.…”

Online scheduling on parallel machines: A survey

“…因此给出问题的参 数下界和算法的参数竞争比, 进而得到参数最好算法异常困难. Dósa 等 [43,44] [59] 研究了 Q2|UB&LB|C max , 该问题涉及 两台机器速度比和工件加工时间上下界比两个参数. LS 算法仍是多数参数组合下的最好算法, 但在 部分参数组合下, 存在优于 LS 的算法.…”

Online Scheduling with Makespan Minimization: State of the Art Results, Research Challenges and Open Problems

“…Earlier, Azar and Regev [94] introduced Opt as an EPI in bin stretching problem and achieved UB 1.625. The latest results have been contributed in [89,91,95,96]. Important results achieved for online scheduling with known Opt is reported in Table 10.…”