Open-shop scheduling for unit jobs under precedence constraints

We study open-shop scheduling for unit jobs under precedence constraints, where if one job precedes another job then it has to be finished before the other job can start to be processed. For the three-machine open-shop to minimize the makespan, we first present a simple 5/3-approximation algorithm based on a partition of the job set into agreeable layers using the natural layered representation of the precedence graph, which is directed acyclic. We then show a greedy algorithm to reduce the number of singleton job layers, resulting in an improved partition, which leads to a 4/3-approximation algorithm. Both approximation algorithms apply to the general m-machine open-shops too. (C) 2019 Elsevier B.V. All rights reserved.