All-norm approximation for scheduling on identical machines

We consider the problem of assigning jobs to m identical machines. The load of a machine is the sum of the weights of jobs assigned to it. The goal is to minimize the norm of the resulting load vector. It is known that for any fixed norm there is a PTAS. On the other hand, it is also known that there is no single assignment which is optimal for all norms. We show that there exists one assignment which simultaneously guarantees a 1.388-approximation of the optimal assignments for all norms. This improves the 1.5 approximation given by Chandra and Wong in 1975.