We consider the following shared-resource scheduling problem: Given a set of jobs $J$, for each $j\in J$ we must schedule a job-specific processing volume of $v_j>0$. A total resource of $1$ is available at any time. Jobs have a resource requirement $r_j\in[0,1]$, and the resources assigned to them may vary over time. However, assigning them less will cause a proportional slowdown. We consider two settings. In the first, we seek to minimize the makespan in an online setting: The resource assignment of a job must be fixed before the next job arrives. Here we give an optimal $e/(e-1)$-competitive algorithm with runtime $\mathcal{O}(n\cdot \log n)$. In the second, we aim to minimize the total completion time. We use a continuous linear programming (CLP) formulation for the fractional total completion time and combine it with a previously known dominance property from malleable job scheduling to obtain a lower bound on the total completion time. We extract structural properties by considering a geometrical representation of a CLP's primal-dual pair. We combine the CLP schedule with a greedy schedule to obtain a $(3/2+\varepsilon)$-approximation for this setting. This improves upon the so far best-known approximation factor of $2$.

翻译：摘要：我们考虑以下共享资源调度问题：给定一组作业$J$，对于每个$j\in J$，必须调度一个作业特定的处理量$v_j>0$。在任何时刻，可用资源总量为$1$。作业具有资源需求$r_j\in[0,1]$，分配给它们的资源可随时间变化。然而，资源分配不足将导致处理速度成比例降低。我们研究两种场景。第一种场景中，我们旨在在线设置下最小化完工时间：作业的资源分配必须在下一个作业到达前确定。在此，我们给出一个最优的$e/(e-1)$-竞争比算法，运行时间为$\mathcal{O}(n\cdot \log n)$。第二种场景中，我们旨在最小化总完工时间。我们采用连续线性规划（CLP）公式描述分数总完工时间，并结合先前已知的可塑作业调度中的主导性质，获得总完工时间的下界。通过考虑CLP原始-对偶对的几何表示，我们提取结构性质。将CLP调度与贪心调度相结合，得到该场景下的$(3/2+\varepsilon)$-近似比。这改进了目前已知的最优近似比$2$。