On the Volume Calculation for Conditional DAG Tasks: Hardness and Algorithms*

The hardness of analyzing conditional directed acyclic graph (DAG) tasks remains unknown so far. For example, previous researches asserted that the conditional DAG's volume can be solved in polynomial time. However, these researches all assume well-nested structures that are recursively composed by single-source-single-sink parallel and conditional components. For conditional DAGs in general that do not comply with this assumption, the hardness and algorithms of volume computation are still open. In this paper, we construct counterexamples to show that previous work cannot provide a safe upper bound of the conditional DAG's volume in general. Moreover, we prove that the volume computation problem for conditional DAGs is strongly $\mathcal{N}\mathcal{P}$-hard. Finally, we propose an exact algorithm for computing the conditional DAG's volume. Experiments show that our method can significantly improve the accuracy of the conditional DAG's volume estimation.