Single Machine Vector Scheduling with General Penalties

In this paper, we study the single machine vector scheduling problem (SMVS) with general penalties, in which each job is characterized by a d-dimensional vector and can be accepted and processed on the machine or rejected. The objective is to minimize the sum of the maximum load over all dimensions of the total vector of all accepted jobs and the rejection penalty of the rejected jobs, which is determined by a set function. We perform the following work in this paper. First, we prove that the lower bound for SMVS with general penalties is α(n), where α(n) is any positive polynomial function of n. Then, we consider a special case in which both the diminishing-return ratio of the set function and the minimum load over all dimensions of any job are larger than zero, and we design an approximation algorithm based on the projected subgradient method. Second, we consider another special case in which the penalty set function is submodular. We propose a noncombinatorial ee−1-approximation algorithm and a combinatorial min{r,d}-approximation algorithm, where r is the maximum ratio of the maximum load to the minimum load on the d-dimensional vector.