An optimal on-line algorithm for preemptive scheduling on two uniform machines in the lp norm

One of the basic and fundamental problems in scheduling is to minimize the machine completion time vector in the l(p) norm (a direct extension of the l(infinity) norm: the makespan) on uniform parallel machines. We concentrate on the on-line and preemptive version of this problem where jobs arrive one by one over a list and are allowed to be preempted. We present a best possible deterministic on-line scheduling algorithm along with a matching lower bound when there are two machines, generalizing existing results for the identical machines scheduling problem in the literature. The main difficulty in the design of the algorithm and the analysis of the resultant competitive ratio as well as the proof of the lower bound is that the competitive ratio is only known to be the root of some equation systems, which admits no analytic solution-a distinct feature from most existing literature on competitive analysis. As a consequence, we develop some new ideas to tackle this difficulty. Specifically we need to exploit the properties of the equations system that defines the competitive ratio.