We contribute the first randomized algorithm that is an integration of arbitrarily many deterministic algorithms for the fully online multiprocessor scheduling with testing problem. When there are two machines, we show that with two component algorithms its expected competitive ratio is already strictly smaller than the best proven deterministic competitive ratio lower bound. Such algorithmic results are rarely seen in the literature. Multiprocessor scheduling is one of the first combinatorial optimization problems that have received numerous studies. Recently, several research groups examined its testing variant, in which each job $J_j$ arrives with an upper bound $u_j$ on the processing time and a testing operation of length $t_j$; one can choose to execute $J_j$ for $u_j$ time, or to test $J_j$ for $t_j$ time to obtain the exact processing time $p_j$ followed by immediately executing the job for $p_j$ time. Our target problem is the fully online version, in which the jobs arrive in sequence so that the testing decision needs to be made at the job arrival as well as the designated machine. We propose an expected $(\sqrt{\varphi + 3} + 1) (\approx 3.1490)$-competitive randomized algorithm as a non-uniform probability distribution over arbitrarily many deterministic algorithms, where $\varphi = \frac {\sqrt{5} + 1}2$ is the Golden ratio. When there are two machines, we show that our randomized algorithm based on two deterministic algorithms is already expected $\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)$-competitive. Besides, we use Yao's principle to prove lower bounds of $1.6682$ and $1.6522$ on the expected competitive ratio for any randomized algorithm at the presence of at least three machines and only two machines, respectively, and prove a lower bound of $2.2117$ on the competitive ratio for any deterministic algorithm when there are only two machines.

翻译：我们提出了首个能够整合任意多个确定性算法的随机化算法，用于解决带测试的完全在线多处理器调度问题。针对两台机器的情况，我们证明仅使用两个组件算法，其期望竞争比就已严格小于已知的最佳确定性算法竞争比下界。这类算法结果在文献中极为罕见。多处理器调度是最早受到广泛研究的组合优化问题之一。近年来，多个研究团队考察了其带测试的变体：每个任务$J_j$到达时附带处理时间上界$u_j$和长度为$t_j$的测试操作；可以选择直接执行$J_j$耗时$u_j$，或先测试$J_j$耗时$t_j$以获取精确处理时间$p_j$，随后立即执行该任务耗时$p_j$。我们的目标问题是完全在线版本，即任务按序到达，需在任务到达时为指定机器做出测试决策。我们提出了期望竞争比为$(\sqrt{\varphi + 3} + 1) (\approx 3.1490)$的随机化算法，该算法通过非均匀概率分布整合任意多个确定性算法，其中$\varphi = \frac {\sqrt{5} + 1}2$为黄金分割比。针对两台机器，我们证明基于两个确定性算法的随机化算法已能达到期望竞争比$\frac {3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)$。此外，我们利用Yao原理证明了：当至少有三台机器时，任意随机化算法的期望竞争比下界为$1.6682$；仅有两台机器时，下界为$1.6522$；而对于仅有两台机器的情况，任意确定性算法的竞争比下界为$2.2117$。