This paper considers the scheduling of stochastic jobs on parallel identical machines to minimize the expected total weighted completion time. While this is a classical problem with a significant body of research on approximation algorithms over the past two decades, constant-factor performance guarantees are currently known only under very restrictive assumptions on the input distributions, even when all job weights are identical. This algorithmic difficulty is striking given the lack of corresponding complexity results: to date, it is conceivable that the problem could be solved optimally in polynomial time. We address this gap with hardness results that demonstrate the problem's inherent intractability. For the special case of discrete two-point processing time distributions and unit weights, we prove that deciding whether there exists a scheduling policy with expected cost at most a given threshold is #P-hard. Furthermore, we show that evaluating the expected objective value of the standard (W)SEPT greedy policy is itself #P-hard. These represent the first hardness results for scheduling independent stochastic jobs and min-sum objective that do not merely rely on the intractability of the underlying deterministic counterparts.

翻译：本文研究在并行同构机器上调度随机作业以最小化期望总加权完工时间的问题。尽管这是一个经典问题，过去二十年已有大量关于近似算法的研究成果，但即使在所有作业权重相同的情况下，目前仅在对输入分布施加严格限制的假设下才能获得常数倍性能保证。考虑到缺乏相应的复杂性结果，这种算法上的困难显得尤为突出：迄今为止，该问题仍可能在多项式时间内获得最优解。我们通过证明问题的内在难解性来填补这一空白。针对离散两点处理时间分布和单位权重的特殊情况，我们证明了判断是否存在期望成本不超过给定阈值的调度策略是#P-难的。此外，我们还证明了评估标准(W)SEPT贪心策略的期望目标值本身也是#P-难的。这些结果构成了首个不依赖于底层确定性对应问题难解性的、针对独立随机作业调度与最小和目标的硬度证明。