Malleable Task-Graph Scheduling with a Practical Speed-Up Model

Scientific workloads are often described by Directed Acyclic task Graphs. Indeed, DAGs represent both a theoretical model and the structure employed by dynamic runtime schedulers to handle HPC applications. A natural problem is then to compute a makespan-minimizing schedule of a given graph. In this paper, we are motivated by task graphs arising from multifrontal factorizations of sparse matrices and therefore work under the following practical model. Tasks are malleable (i.e., a single task can be allotted a time-varying number of processors) and their speedup behaves perfectly up to a first threshold, then speedup increases linearly, but not perfectly, up to a second threshold where the speedup levels off and remains constant. After proving the NP-hardness of minimizing the makespan of DAGs under this model, we study several heuristics. We propose model-optimized variants for PROPSCHEDULING, widely used in linear algebra application scheduling, and FLOWFLEX. GREEDYFILLING is proposed, a novel heuristic designed for our speedup model, and we demonstrate that PROPSCHEDULING and GREEDYFILLING are 2-approximation algorithms. In the evaluation, employing synthetic data sets and task graphs arising from multifrontal factorization, the proposed optimized variants and GREEDYFILLING significantly outperform the traditional algorithms, whereby GREEDYFILLING demonstrates a particular strength for balanced graphs.