On-line preemptive machine scheduling with $$\ell _p$$ ℓ p norm on two uniform machines

Abstract

One of the basic and fundamental scheduling problems is to minimize the machine completion time vector in the $$\ell _p$$ l p norm, a direct extension of the well-studied objective makespan ( $$l_{\infty }$$ l ? norm), on parallel machines. We concentrate on the on-line and preemptive version of this problem where jobs arrive one by one over a list to be allocated to two uniform machines with job preemption permitted. We present a best possible deterministic on-line scheduling algorithm for this problem along with a matching lower bound, generalizing existing results for the identical machines scheduling problem in the literature. One notable feature of this work is the highly involved technicality compared to similar analysis in the existing literature, mainly due to the intrinsic unavailability of a closed-form formula for the competitive ratio.