Optimal scheduling of arborescences using the Gangal-Ranade algorithm

The Gangal-Ranade algorithm presents the best approximation ratio for the classic scheduling problem with unit execution time and precedence constrained jobs, on a variable number of identical parallel machines, to minimize the makespan. This work presents results about the optimality of the algorithm when the acyclic directed graph (DAG) that represents the precedence constraints are arborescences (directed trees, in-tree and out-tree), reinforcing that these types of DAGs provide optimal substructures for the problem. Understanding the behavior of this algorithm for classes of arborescences can lead to optimality or better approximations for classes of larger DAGs, which is our ongoing research work. Furthermore, the search for optimal cases for algorithms such as Gangal-Ranade can provide intuition to obtain partial answers to problems that remain open, such as the famous Open 8 in the list presented in the classical Garey and Johnson book.