We present two deterministic online algorithms for the problem of scheduling with a general machine cost function. In this problem, every machine has a cost that is given by a nonnegative cost function, and the objective function is the sum of makespan and the cost of the purchased machines. In previous work by Imreh, it was shown that no deterministic algorithm has competitive ratio below 2, and an algorithm whose competitive ratio does not exceed $$(3+\sqrt{5})/2 \approx 2.618$$ ( 3 + 5 ) / 2 ≈ 2.618 was presented. Our first algorithm imitates an optimal offline solution with respect to the number of machines used, and we show that its competitive ratio is exactly 2.5. Then, we modify our algorithm by imitating a preemptive optimal offline solution instead of a non-preemptive one. This leads to the design of a 2-competitive algorithm, which is the best possible.

中文翻译：

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具有通用机器成本函数的最优在线调度算法

我们针对具有通用机器成本函数的调度问题提出了两种确定性在线算法。在这个问题中，每台机器都有一个由非负成本函数给出的成本，目标函数是制造时间和购买机器的成本之和。Imreh 之前的工作表明，没有确定性算法的竞争率低于 2，竞争率不超过 $$(3+\sqrt{5})/2 \approx 2.618$$ ( 3 + 5 ) / 2 ≈ 2.618。我们的第一个算法针对所使用的机器数量模仿了最佳离线解决方案，我们表明其竞争比率恰好为 2.5。然后，我们通过模仿抢占式最优离线解决方案而不是非抢占式解决方案来修改我们的算法。这导致了一个 2-竞争算法的设计，