The efficient parallel execution of complex computations requires balancing the workload across processors while minimizing the communication between them. This inherent trade-off is often captured by graph partitioning or DAG scheduling problems. For the sake of model simplicity, most works on these problems assume that nodes can be assigned to only a single processor. However, in reality, replicating an operation on several processors can easily be beneficial: it may increase the computational costs only by a small amount, while significantly reducing the communication requirements. Our goal is to provide a comprehensive analysis of the impact of replication on partitioning and scheduling problems. On the theoretical side, we show that for graph partitioning, replication makes the problem significantly harder in terms of approximation complexity, whereas for scheduling, its impact on complexity seems less prominent. On the experimental side, we conduct a thorough analysis of the cost reduction obtainable by replication, on a wide range of graphs from real-world applications. For hypergraph partitioning, we use Integer Linear Programming (ILP) formulations to compare the optimal costs; our experiments show that replication can reduce the cost by 17%-65% on average, or even entirely remove the need for communication in some cases. For DAG scheduling, we similarly use ILPs on smaller graphs, and develop a sophisticated heuristic that is also applicable to much larger workloads. Our experiments here demonstrate a mean cost reduction of 11.61%-23.13% with replication, or even up to 58.17% in some cases.

翻译：复杂计算的高效并行执行需要在处理器间平衡工作负载，同时最小化处理器间的通信开销。这种固有的权衡通常通过图划分或有向无环图（DAG）调度问题来刻画。为简化模型，现有大多数研究假设节点只能分配到单个处理器。然而现实中，在多个处理器上复制操作往往是有益的：这仅会小幅增加计算成本，却能显著降低通信需求。本文旨在全面分析复制对划分与调度问题的影响。理论方面，我们证明对于图划分问题，复制在近似复杂度层面显著增加了问题难度；而对于调度问题，其对复杂度的影响则相对不明显。实验方面，我们对来自实际应用的多类图进行了系统分析，量化复制带来的成本降低程度。针对超图划分，我们使用整数线性规划（ILP）建模比较最优成本；实验表明，复制可使平均成本降低17%-65%，某些场景甚至完全消除通信需求。针对DAG调度，我们同样在较小规模图上采用ILP方法，并开发了适用于大规模工作负载的复杂启发式算法。实验结果显示，复制可使平均成本降低11.61%-23.13%，部分案例中降幅可达58.17%。