Scheduling Unrelated Machines of Few Different Types

A very well-known machine model in scheduling allows the machines to be unrelated, modelling jobs that might have different characteristics on each machine. Due to its generality, many optimization problems of this form are very difficult to tackle and typically APX-hard. However, in many applications the number of different types of machines, such as processor cores, GPUs, etc. is very limited. In this paper, we address this point and study the assignment of jobs to unrelated machines in the case that each machine belongs to one of a fixed number of types and the machines of each type are identical. We present polynomial time approximation schemes (PTASs) for minimizing the makespan for multidimensional jobs with a fixed number of dimensions and for minimizing the L_p-norm. In particular, our results subsume and generalize the existing PTASs for a constant number of unrelated machines and for an arbitrary number of identical machines for these problems. We employ a number of techniques which go beyond the previously known results, including a new counting argument and a method for making the concept of sparse extreme point solutions usable for a convex program.