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 only 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 multiprocessor scheduling with testing, 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 first 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 then 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 only 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, while proving a lower bound of 2.2117 on the competitive ratio for any deterministic algorithm.

翻译：我们提出了首个集成任意多个确定性算法的随机化算法，用于求解带有测试的全在线多处理器调度问题。当仅有两台机器时，我们证明使用两个子算法即可使期望竞争比严格小于已知的最佳确定性竞争比下界。此类算法结果在文献中较为罕见。多处理器调度是最早获得广泛研究的组合优化问题之一。近期，多个研究团队探讨了其测试变种：每个作业$J_j$到达时带有处理时间上界$u_j$和测试时长$t_j$；可选择直接执行$J_j$耗时$u_j$，或先测试$J_j$耗时$t_j$获得精确处理时间$p_j$，随后立即执行该作业耗时$p_j$。我们的目标问题是带有测试的全在线多处理器调度，其中作业按序到达，需在作业到达时同时确定测试决策与分配机器。首先利用Yao原理证明：在至少三台机器和仅有两台机器的情况下，任意随机化算法的期望竞争比下界分别为1.6682和1.6522；随后提出期望竞争比为$(\sqrt{\varphi + 3} + 1) (\approx 3.1490)$的随机化算法，该算法以非均匀概率分布方式集成任意多个确定性算法，其中$\varphi = \frac{\sqrt{5} + 1}2$为黄金分割比。当仅有两台机器时，我们证明基于两个确定性算法的随机化算法即可达到期望竞争比$\frac{3 \varphi + 3 \sqrt{13 - 7\varphi}}4 (\approx 2.1839)$，同时证明任意确定性算法的竞争比下界为2.2117。