We consider scheduling monotonic moldable tasks on multiple platforms, where each platform contains a set of processors. A moldable task can be split into several pieces of equal size and processed simultaneously on multiple processors. Tasks are not allowed to be processed spanning over platforms. This scheduling model has many applications, ranging from parallel computing to the berth and quay crane allocation and the workforce assignment problem. We develop several approximation algorithms aiming at minimizing the makespan. More precisely, we provide a 2-approximation algorithm for identical platforms, a Fully Polynomial Time Approximation Scheme (FPATS) under the assumption of large processor counts and a 2-approximation algorithm for a fixed number of heterogeneous platforms. Most of the proposed algorithms combine a dual approximation scheme with a novel approach to improve the dual approximation algorithm. All results can be extended to the contiguous case, i.e., a task can only be executed by contiguously numbered processors.

我们考虑在多个平台上调度单调可塑任务，其中每个平台包含一组处理器。一个可塑的任务可以被分成几个大小相等的部分，并在多个处理器上同时处理。不允许跨平台处理任务。这种调度模型有很多应用，从并行计算到泊位和码头起重机分配以及劳动力分配问题。我们开发了几种旨在最小化完工时间的近似算法。更准确地说，我们提供了一种用于相同平台的 2-近似算法，一种在大处理器数量假设下的完全多项式时间近似方案 (FPATS) 和一种用于固定数量的异构平台的 2-近似算法。大多数提出的算法将对偶逼近方案与改进对偶逼近算法的新方法相结合。所有结果都可以扩展到连续的情况，即任务只能由连续编号的处理器执行。