Resource augmentation for weighted flow-time explained by dual fitting

We propose a general dual-fitting technique for analyzing online scheduling algorithms in the unrelated machines setting where the objective function involves weighted flow-time, and we allow the machines of the on-line algorithm to have (1 + ε)-extra speed than the offline optimum (the so-called speed augmentation model). Typically, such algorithms are analyzed using non-trivial potential functions which yield little insight into the proof technique. We propose that one can often analyze such algorithms by looking at the dual (or Lagrangian dual) of the linear (or convex) program for the corresponding scheduling problem, and finding a feasible dual solution as the on-line algorithm proceeds. As representative cases, we get the following results : • For the problem of minimizing weighted flow-time, we show that the greedy algorithm of Chadha-GargKumar-Muralidhara is O ( 1 ε ) -competitive. This is an improvement by a factor of 1 ε on the competitive ratio of this algorithm as analyzed by them. • For the problem of minimizing weighted `k norm of flow-time, we show that a greedy algorithm gives an