An asymptotic PTAS for batch scheduling with nonidentical job sizes to minimize makespan

Motivated by the existence of an APTAS (Asymptotic PTAS) for bin packing problem, we consider the batch scheduling problem with nonidentical job sizes to minimize makespan. For the proportional special version, i.e., there exists a fixed number α such that pj = αsj for every 1 ≤ j ≤ n, we first present a lower bound of 3/2 for the approximation ratio and then design an APTAS for it. Our basic idea is quite simple: we first enumerate all the partial schedules of relatively large jobs; Then for every partial schedule we insert the small jobs, split them if necessary; Further then, we choose the best of all the obtained schedules; Finally, we collect the split small jobs and put them into new batches. As we can round the large jobs into only a constant number of different kinds at a reasonable expense of accuracy, the running time can be bounded. When the optimal objective value of instances in our consideration can not be arbitrarily small, infI {Pmax : Pmax is the largest processing time in I} ≠ 0 for instance, our result is perfect in the sense of worst-case performance.