An Optimal Online Algorithm for Robust Flow Time Scheduling

The problem of minimizing the total flow time on a single machine is one of the few problems for which we can give an optimal online algorithm: just schedule the job with the shortest remaining processing time (SRPT). However, this requires knowledge of the true running time \(p_j\) of each job \(j\). Azar, Leonardi, and Touitou recently asked: what if we are given estimates \(\hat{p}_j\) for each job, such that the multiplicative error between \(p_j\) and \(\hat{p}_j\) (called the distortion) is at most \(\mu\)? It is easy to construct examples where no algorithm can be \(o(\mu)\) competitive; can we get \(O(\mu)\) competitiveness?