AI 中文总结
针对伯努利作业的随机调度问题，研究团队提出了适用于任意台数机器的准多项式时间近似方案，还证明了预先固定作业顺序的策略会损失Ω(log N)因子，给出了多项式时间自适应规则等结果。
AI 中文摘要
我们研究经典问题：在m台相同机器上调度具有随机处理时间的作业，以最小化完成时间的期望总和，针对伯努利作业：作业j以概率q_j耗时p_j，否则耗时0，其结果在开始时揭晓。基准是最优自适应策略。我们针对任意数量的机器给出了准多项式时间近似方案。此前，准多项式时间已知能给出O(log N)近似，且近似方案仅在作业大小为常数时已知。该方案基于一个简单观察：只需将机器变为空闲的时间（而非作业开始的时间）四舍五入到宽度与作业大小成比例的网格，且拉伸因子接近1的最优策略已能遵守此类网格。我们还表明，预先固定作业顺序的每一种策略（即使在两台机器上）都会损失Ω(log N)的因子，因此常数因子需要根据观测结果调整顺序。进一步结果包括：比率为min{m,1+∑_j q_j}的多项式时间自适应规则、针对常数数量大小的更简单近似方案，以及计算两台机器上最优期望成本的#P-难解性。
英文摘要
We study the classical problem of scheduling jobs with random processing times on $m$ identical machines to minimize the expected sum of completion times, for Bernoulli jobs: job $j$ takes time $p_j$ with probability $q_j$ and time $0$ otherwise, and its outcome is revealed when it starts. The benchmark is an optimal adaptive policy. We give a quasi-polynomial-time approximation scheme for every number of machines. Previously, quasi-polynomial time was known to give an $O(\log N)$-approximation, and approximation schemes were known only for a constant number of distinct sizes. The scheme rests on a simple observation: it suffices to round the times at which machines become free, rather than the times at which jobs start, to a grid whose width is proportional to the job size, and an optimal policy stretched by a factor close to one already respects such grids. We also show that every policy that fixes the order of the jobs in advance loses a factor $Ω(\log N)$, already on two machines, so a constant factor requires adapting the order to the observed outcomes. Further results include a polynomial-time adaptive rule with ratio $\min\{m,1+\sum_j q_j\}$, a simpler approximation scheme for a constant number of sizes, and #P-hardness of computing the optimal expected cost on two machines.
Comments23 pages, 1 figure
