We study kill-and-restart and preemptive strategies for the fundamental scheduling problem of minimizing the sum of weighted completion times on a single machine in the non-clairvoyant setting. First, we show a lower bound of~$3$ for any deterministic non-clairvoyant kill-and-restart strategy. Then, we give for any $b > 1$ a tight analysis for the natural $b$-scaling kill-and-restart strategy as well as for a randomized variant of it. In particular, we show a competitive ratio of $(1+3\sqrt{3})\approx 6.197$ for the deterministic and of $\approx 3.032$ for the randomized strategy, by making use of the largest eigenvalue of a Toeplitz matrix. In addition, we show that the preemptive Weighted Shortest Elapsed Time First (WSETF) rule is $2$-competitive when jobs are released online, matching the lower bound for the unit weight case with trivial release dates for any non-clairvoyant algorithm. Using this result as well as the competitiveness of round-robin for multiple machines, we prove performance guarantees smaller than $10$ for adaptions of the $b$-scaling strategy to online release dates and unweighted jobs on identical parallel machines.

翻译：本文研究了非先知环境下单机最小化加权完成时间总和这一基本调度问题中的杀重启与抢占策略。首先，我们证明了任何确定性非先知杀重启策略的下界为~$3$。然后，针对任意$b > 1$，我们给出了自然$b$-缩放杀重启策略及其随机化变体的紧致分析。特别地，通过利用Toeplitz矩阵的最大特征值，我们证明了确定性的竞争比约为$(1+3\sqrt{3})\approx 6.197$，随机化策略的竞争比约为$\approx 3.032$。此外，我们证明了当作业在线释放时，抢占式加权最短已执行时间优先规则具有$2$-竞争性，这匹配了非先知算法下单位权重作业且释放时间为平凡情形时的下界。利用这一结果以及多机轮询调度的竞争性，我们为将$b$-缩放策略适配至在线释放时间和同构并行机上的未加权作业场景，证明了小于$10$的性能保证。