A tight approximation ratio of a list scheduling algorithm for a single-machine scheduling problem with a non-renewable resource

In this paper, we investigate an open problem by Gyorgyi and Kis for a single-machine scheduling problem with a non-renewable resource (NR-SSP) and total weighted completion time criterion. The problem is NP-hard, even if every job has the same processing time, and each weight is equal to its required amount of the resource. Gyorgyi and Kis prove that a simple list scheduling algorithm for this special case is a 3-approximation algorithm and conjecture that the algorithm for the case is a 2-approximation algorithm. We prove their conjecture. More strongly, we show that the approximation ratio is strictly less than 2 for any instance, and for any epsilon>0, there exists an instance for which the approximation ratio is more than 2-epsilon.