A 4-approximation for scheduling on a single machine with general cost function

We consider a single machine scheduling problem that seeks to minimize a generalized cost function: given a subset of jobs we must order them so as to minimize $\sum f_j(C_j)$, where $C_j$ is the completion time of job $j$ and $f_j$ is a job-dependent cost function. This problem has received a considerably amount of attention lately, partly because it generalizes a large number of sequencing problems while still allowing constant approximation guarantees. In a recent paper, Cheung and Shmoys provided a primal-dual algorithm for the problem and claimed that is a 2-approximation. In this paper we show that their analysis cannot yield an approximation guarantee better than $4$. We then cast their algorithm as a local ratio algorithm and show that in fact it has an approximation ratio of $4$. Additionally, we consider a more general problem where jobs has release dates and can be preempted. For this version we give a $4\kappa$-approximation algorithm where $\kappa$ is the number of distinct release dates.