Approximation algorithms for scheduling jobs with release times and arbitrary sizes on batch machines with non-identical capacities

We consider the problem of scheduling jobs with release times and arbitrary sizes on batch machines with non-identical capacities to minimize makespan. A job can only be assigned to a machine whose capacity is not less than the size of the job. A machine can process a group of jobs as a batch simultaneously, as long as the total size of the jobs in this batch does not exceed the capacity of the machine. The processing time of a batch is equal to the longest processing time of all the jobs in the batch. For the case of equal release times, we obtain a fast 5-approximation algorithm, as well as an algorithm with approximation ratio arbitrarily close to 2. For the case of unequal release times and a fixed number of different batch capacities, we obtain an algorithm with approximation ratio arbitrarily close to 2. We also study the problem of scheduling jobs with equal release times, arbitrary sizes, inclusive processing set restrictions and incompatible families on batch machines with identical capacities to minimize makespan. We obtain a 8/3-approximation algorithm, as well as an algorithm with approximation ratio arbitrarily close to 2. If only a single family is considered, a 2-approximation algorithm is presented. It is likely that these results cannot be too substantially improved upon, since all of the problems studied cannot be approximated to within a ratio smaller than 2 unless P=NP.