In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S with respect to j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than \(\frac{e}{e-1}\), unless \(P = NP\). On the positive side, we present polynomial-time algorithms with approximation ratios \(\frac{2e}{e-1}\) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of \(1+\varphi \) for unrelated speeds (\(\varphi \) is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the \(L_p\) norm of their speeds.

在可塑性作业调度中，作业可以在多台机器上同时执行，处理时间取决于分配的机器数量。在这种情况下，作业需要以非抢占式和一致的方式执行，因为它们在执行期间占用分配集的所有机器的相同时间间隔。在这项工作中，我们研究了受无关机器上的标准调度启发的可塑性作业调度的推广。具体来说，我们介绍了一个可塑性作业调度的通用模型，其中每台机器对每个作业都有一个（可能不同的）速度，并且作业j在一组分配的机器S上的处理时间取决于 S 的总速度相对于到j. 对于速度不相关的机器，我们表明最佳制造跨度不能在小于\(\frac{e}{e-1}\)的因子内近似，除非\(P = NP\)。积极的一面是，我们提出多项式时间算法，其近似比\(\frac{2e}{e-1}\)用于速度无关的机器，3 用于速度均匀的机器，7/3 用于相同速度的受限分配机器。我们的算法基于确定性 LP 舍入。它们会导致调度稀疏，即每台机器最多与其他机器共享一个作业。我们还证明了\(1+\varphi \)对于不相关速度 ( \(\varphi \)是黄金比例）和 2 用于统一速度和受限任务。为了表明我们方法的普遍性，我们展示了它还为一个变体产生了常数因子近似算法，其中我们根据它们速度的\(L_p\)范数来确定一组已分配机器的有效速度。