An approximation algorithm for the three-machine scheduling problem with the routes given by the same partial order

The paper considers a three-machine shop scheduling problem to minimize the makespan, in which the route of a job should be feasible with respect to a machine precedence digraph with three nodes and one arc. For this NP-hard problem that is related to the classical flow shop and open shop models, we present a simple 1.5-approximation algorithm and an improved 1.4-approximation algorithm.